# SPDEs driven by standard symmetric $\alpha$-stable cylindrical Lévy
processes: existence, Lyapunov functionals and Itô formula

Gergely Bodó Markus Riedle
Department of Mathematics
King’s College London
London WC2R 2LS
United Kingdom tomasz.kosmala<EMAIL_ADDRESS>Gergely Bodó
Department of Mathematics
King’s College
London WC2R 2LS
United Kingdom

<EMAIL_ADDRESS>
Markus Riedle1112nd affiliation: Institute of Mathematical Stochastics,
Faculty of Mathematics, TU Dresden, 01062 Dresden, Germany
Department of Mathematics
King’s College
London WC2R 2LS
United Kingdom

<EMAIL_ADDRESS>Ondřej Týbl 222The author was supported in part by
Czech Science Foundation (GA ČR) grant no./22-12790S and by the Grant schemes
at Charles University, project no. CZ.02.2.690.00.019 073/0016935.
Department of Probability and Mathematical Statistics
Charles University
Sokolovská 49/83 186 75 Prague 8
Czech Republic

<EMAIL_ADDRESS>
###### Abstract

We investigate several aspects of solutions to stochastic evolution equations
in Hilbert spaces driven by a standard symmetric $\alpha$-stable cylindrical
noise. Similarly to cylindrical Brownian motion or Gaussian white noise,
standard symmetric $\alpha$-stable noise exists only in a generalised sense in
Hilbert spaces. The main results of this work are the existence of a mild
solution, long-term regularity of the solutions via Lyapunov functional
approach, and an Itô formula for mild solutions to evolution equations under
consideration. The main tools for establishing these results are Yosida
approximations and an Itô formula for Hilbert space-valued semi-martingales
where the martingale part is represented as an integral driven by cylindrical
$\alpha$-stable noise. While these tools are standard in stochastic analysis,
due to the cylindrical nature of our noise, their application requires
completely novel arguments and techniques.

AMS 2010 subject classification: 60H15, 60G20, 60G52, 60H05

Keywords and phrases: cylindrical Lévy processes, stable processes, stochastic
partial differential equations, Itô formula, Lyapunov functional

##  1 Introduction

Standard symmetric $\alpha$-stable distributions are the natural
generalisations of Gaussian distributions for modelling random perturbations
of finite dimensional dynamical systems. They often meet various empirical
requests, such as heavy tails, self-similarity and infinite variance, but are
at the same time analytically tractable and well-understood. The importance of
these models is reflected by the available vast literature on dynamical
systems perturbed by random noises with $\alpha$-stable distributions in
various areas such as economics, biology etc.

In the infinite dimensional setting of modelling random perturbations of
partial differential equations, much fewer results are known for systems
perturbed by $\alpha$-stable distributions. In fact, only in the random field
approach, based on the seminal work by Walsh, one can find several
publications on stochastic partial differential equations (SPDEs) driven by
multiplicative $\alpha$-stable noise, e.g. Mueller [27], Mytnik [28], and more
recently Chong [7] and Chong et. al. [8]. However, in the semigroup approach,
following the spirit of Da Prato and Zabczyk, one can find several results for
equations only with additive driving noise distributed according to an
$\alpha$-stable law; see e.g. Brzeźniak and Zabczyk [6] and Riedle [33]. The
only publication in the semigroup approach for multiplicative $\alpha$-stable
perturbation is Kosmala and Riedle [23], where however the assumptions are
rather restrictive and do not correspond to the natural Lipschitz continuity
and linear growth conditions. The lack of results in the semigroup approach is
due to the fact that a random noise with a standard symmetric $\alpha$-stable
distribution does not exist as an ordinary Hilbert space-valued process but
only in the generalised sense of Gel’fand and Vilenkin [14] or Segal [37].

In this work, we investigate several aspects of solutions to equations of the
form

${\rm d}X(t)=\big{(}AX(t)+F(X(t))\big{)}\,{\rm d}t+G(X(t-))\,{\rm d}L(t),$
(1.1)

where $A$ is the generator of a $C_{0}$-semigroup in a separable Hilbert space
$H$, the coefficients $F\colon H\to H$ and $G\colon H\to\mathcal{L}_{2}(U,H)$
are mappings with $U$ being a separable Hilbert space, and $L$ is a standard
symmetric $\alpha$-stable cylindrical process in $U$ for $\alpha\in(1,2)$.

Analogously to the standard normal distribution, standard symmetric
$\alpha$-stable distributions in $\mathbb{R}^{d}$ can only be generalised to
infinite dimensional spaces as cylindrical distributions. In particular, this
means that the driving noise $L$ in (1.1) exists only in the generalised
sense; see Schwartz [36]. Since such processes do not attain values in the
underlying Hilbert space, standard results for stochastic processes in
infinite dimensional spaces are not applicable. Most notably, complications
arise from the fact that while these processes are cylindrical semi-
martingales, see Jakubowski et. al. [18], they do not enjoy a semi-martingale
decomposition in a cylindrical sense, since semi-martingale decompositions are
not invariant under linear transformations, see Jakubowski and Riedle [19, Re.
2.2]. Nevertheless, the problem of stochastic integration with respect to
cylindrical Lévy processes was solved in Jakubowski and Riedle [19] by
arguments avoiding the usual Lévy-Itô decomposition. This approach has been
further developed for standard symmetric $\alpha$-stable cylindrical process
by two of us in Bodó and Riedle [5], which enables us to integrate predictable
integrands and to derive a dominated convergence theorem for stochastic
integrals.

This work comprises of 3 main results: the existence of a mild solution to
Equation (1.1), a Lypunov functional approach for long-term regularity for
solutions to Equation (1.1), and an Itô formula for mild solutions to Equation
(1.1). The main tools for establishing these results are an Itô formula for
Hilbert space-valued semi-martingales driven by standard symmetric
$\alpha$-stable cylindrical Lévy noise and a Yosida approximation of solutions
to Equation (1.1). While these tools are standard in stochastic analysis, due
to the cylindrical nature of our noise, their application in our setting
requires completely novel arguments and techniques, which we highlight in the
following.

A classical Itô formula for semi-martingales in Hilbert spaces is well known
and easy to derive; see e.g. Metivier [26, Th. 27.2]. However, applying this
formula often requires the identification of the martingale and bounded
variation components of the process, which in the classical situation of a
semi-martingale driven by an ordinary Hilbert space-valued process can easily
be obtained via the semi-martingale decomposition of the driving process.
Since in our case, the driving cylindrical process does not enjoy a semi-
martingale decomposition, one needs to identify the martingale part of the
stochastic integral process by carrying out a deep analysis of its jump
structure.

The second major tool in our work is a Yosida approximation, which is an
often-utilised device in the classical situation with an ordinary Hilbert
space-valued process as driving noise; see e.g. Peszat and Zabczyk [30].
Convergence of the Yosida approximation is established by tightness arguments
in the space $\mathcal{C}([0,T],L^{p}(\Omega,H))$ of $p$-th mean continuous
Hilbert space-valued processes for any $p<\alpha$. It turns out that the space
$\mathcal{C}([0,T],L^{p}(\Omega,H))$ is tailor-made for analysing equations
driven by a standard symmetric $\alpha$-stable cylindrical process. The
observation that the solution is continuous in the above sense, despite having
discontinuous paths, lies at the heart of this paper. To the best of our
knowledge, we are the first to use this in the context of SPDEs driven by
cylindrical stable noise.

These two tools, the Itô formula for semi-martingales driven by a standard
symmetric $\alpha$-stable cylindrical process and convergence of the Yosida
approximation, enable us to establish the 3 main results of our work. For the
existence result, we use tightness of the Yosida approximation to establish
existence of a mild solution to Equation (1.1). In our setting, standard
methods for establishing existence of a solution, such as fix point arguments
or Grönwall’s lemma are not applicable, since the integral operator with a
standard symmetric $\alpha$-stable integrator maps to a larger space than its
domain; see Kosmala and Riedle [23] or Rosinski and Woyczynski [34].

By following the classical approach of Ichikawa in [16], we demonstrate the
power of the established tools by investigating the long-term regularity of
the mild solution to Equation (1.1) via the functional Lyapunov approach. The
functional Lyapunov approach can be used to establish various regularity
properties; in this work, we focus on exponential ultimate boundedness, but
other quantitative properties can be investigated similarly. As the mild
solution is not a semi-martingale, the derived Itô formula for semi-
martingales cannot be applied directly. However, we successfully show that the
Yosida approximations are semi-martingales, and thus the Itô formula can be
applied to these, which immediately shows their exponential ultimate
boundedness. It remains only to show that this boundedness property carries
over to the limit, for which we establish convergence of the Markov generators
in a suitable sense.

Mild solutions of SPDEs are not semi-martingales, and thus the classical Itô
formula cannot be applied. This lack of a powerful tool is often circumvented
by a specific Itô formula for mild solutions of SPDEs. One of the first
versions of such an Itô formula for mild solutions can be found in Ichikawa
[16] for the Gaussian case, and more recent versions in Da Prato et. al. [9]
for the Gaussian case and in Alberverio et. al. [1] for the case of ordinary
Lévy processes. In the last part of our work, we derive such an Itô formula
for mild solutions of equation (1.1) driven by a standard symmetric
$\alpha$-stable cylindrical process.

We outline the structure of the paper. Selected preliminaries on standard
symmetric $\alpha$-stable cylindrical processes, integration with respect to
them and underlying results on equations as well as the theory of predictable
compensators are collected in Section 2. In Sections 3 and 4, we identify the
predictable compensator and quadratic variation of the integral process. These
observations lead us directly to the Itô formula for semi-martingales driven
by a standard symmetric $\alpha$-stable cylindrical process in Section 5. In
Section 6, we prove existence of a mild solution under Lipschitz and
boundedness conditions in the space of continuous functions, where the main
result is formulated in Theorem 6.5. In Section 7, we establish conditions for
exponential ultimate boundedness in Theorem 7.1. Finally, in Section 8, an Itô
formula for mild solutions is proved.

##  2 Preliminaries

###  2.1 Standard symmetric $\alpha$-stable cylindrical Lévy processes

Let $U$ and $H$ be separable Hilbert spaces with norm
$\left\lVert\cdot\right\rVert$ and scalar product $\langle\cdot,\cdot\rangle$.
By $\overline{B}_{H}(r)$ we denote the closed ball in $H$ with radius $r>0$
and, in the special case when $r=1$, we write
$\overline{B}_{H}:=\overline{B}_{H}(1)$. The space of Hilbert-Schmidt
operators $\Phi\colon U\to H$ is denoted by $\mathcal{L}_{2}(U,H)$ and
equipped with the norm $\left\lVert\cdot\right\rVert_{\mathcal{L}_{2}(U,H)}$.

Let $S$ be a subset of $U$. For each $n\in\mathbb{N}$, elements
$u_{1},...,u_{n}\in S$ and Borel set $A\in\mathcal{B}(\mathbb{R}^{n})$, we
define

$C(u_{1},...,u_{n};A):=\\{u\in U:(\langle u,u_{1}\rangle,...,\langle
u,u_{n}\rangle)\in A\\}.$

Such sets are called cylindrical sets with respect to $S$ and the collection
of all such cylindrical sets is denoted by $\mathcal{Z}(U,S)$. It is a
$\sigma$-algebra if $S$ is finite and otherwise an algebra. We write shortly
$\mathcal{Z}(U)$ for $\mathcal{Z}(U,U)$.

A set function $\mu:\mathcal{Z}(U)\rightarrow[0,\infty]$ is called a
cylindrical measure on $\mathcal{Z}(U)$ if for each finite subset $S\subseteq
U$, the restriction of $\mu$ to the $\sigma$-algebra $\mathcal{Z}(U,S)$ is a
$\sigma$-additive measure. A cylindrical measure is said to be a cylindrical
probability measure if $\mu(U)=1$.

Let $(\Omega,\Sigma,P)$ be a complete probability space. We will denote by
$L_{P}^{0}(\Omega,U)$ the space of equivalence classes of measurable functions
$Y\colon\Omega\rightarrow U$ equipped with the topology of convergence in
probability. A cylindrical random variable $X$ in $U$ is a linear and
continuous mapping $X\colon U\rightarrow L_{P}^{0}(\Omega,\mathbb{R})$. It
defines a cylindrical probability measure $\mu_{X}$ by

$\displaystyle\mu_{X}\colon\mathcal{Z}(U)\to[0,1],\qquad\mu_{X}(Z)=P\big{(}(Xu_{1},\dots,Xu_{n})\in
A\big{)},$

for cylindrical sets $Z=C(u_{1},...,u_{n};A)$. The cylindrical probability
measure $\mu_{X}$ is called the cylindrical distribution of $X$. We define the
characteristic function of the cylindrical random variable $X$ by

$\varphi_{X}\colon
U\rightarrow\mathbb{C},\qquad\varphi_{X}(u)=E\big{[}e^{iXu}\big{]}.$

Let $T\colon U\to H$ be a linear and continuous operator. By defining

$\displaystyle TX\colon H\to
L_{P}^{0}(\Omega,\mathbb{R}),\qquad(TX)h=X(T^{\ast}h),$

we obtain a cylindrical random variable on $H$. In the special case when $T$
is a Hilbert-Schmidt operator and hence $0$-Radonifying by [39, Th. VI.5.2],
it follows from [39, Pr. VI.5.3] that the cylindrical random variable $TX$ is
induced by a genuine random variable $Y\colon\Omega\to H$, that is
$(TX)h=\langle Y,h\rangle$ for all $h\in H$.

A family $(L(t):t\geq 0)$ of cylindrical random variables $L(t)\colon
U\rightarrow L_{P}^{0}(\Omega,\mathbb{R})$ is called a cylindrical
$\left(\mathcal{F}_{t}\right)$-Lévy process if for each $n\in\mathbb{N}$ and
$u_{1},...,u_{n}\in U$, the stochastic process
$\big{(}\big{(}L(t)u_{1},...,L(t)u_{n}\big{)}:t\geq 0\big{)}$ is an
$\left(\mathcal{F}_{t}\right)$-Lévy process in $\mathbb{R}^{n}$ and the
filtration $\left(\mathcal{F}_{t}\right)_{t\geq 0}$ satisfies the usual
conditions. We denote by $\mathcal{Z}_{*}(U)$ the collection

$\big{\\{}\\{u\in U:(\langle u,u_{1}\rangle,...,\langle u,u_{n}\rangle)\in
B\\}:n\in\mathbb{N},u_{1},...,u_{n}\in
U,B\in\mathcal{B}(\mathbb{R}^{n}\setminus\\{0\\})\big{\\}}$

of cylindrical sets, which forms an algebra of subsets of $U$. For fixed
$u_{1},...,u_{n}\in U$, let $\lambda_{u_{1},\dots,u_{n}}$ be the Lévy measure
of $\big{(}(L(t)u_{1},...,L(t)u_{n}):t\geq 0\big{)}$. Define a function
$\lambda\colon\mathcal{Z}_{*}(U)\rightarrow[0,\infty]$ by

$\displaystyle\lambda(C):=\lambda_{u_{1},\dots,u_{n}}(B)\quad\text{for}\quad
C=\\{u\in U:\,(\langle u,u_{1}\rangle,...,\langle u,u_{n}\rangle)\in B\\},$

for $B\in\mathcal{B}(\mathbb{R}^{n})$. It is shown in [3] that $\lambda$ is
well defined. The set function $\lambda$ is called the cylindrical Lévy
measure of $L$.

In this paper, we restrict our attention to standard symmetric $\alpha$-stable
cylindrical Lévy processes for $\alpha\in(1,2)$, which we simply call
$\alpha$-stable cylindrical Lévy processes in the sequel. These are
cylindrical Lévy processes with characteristic function
$\varphi_{L(t)}(u)=\exp(-t\left\lVert u\right\rVert^{\alpha})$ for each $t\geq
0$ and $u\in U$. Let $(e_{k})_{k\in\mathbb{N}}$ be an orthonormal basis of
$U$.

The Lévy measure $\lambda$ of an $\alpha$-stable cylindrical Lévy process for
$\alpha\in(1,2)$ satisfies

$\displaystyle\left(\lambda\circ\Phi^{-1}\right)(\overline{B}_{H}^{c})\leq
c_{\alpha}\left\lVert\Phi\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha},\quad\Phi\in\mathcal{L}_{2}(U,H)$
(2.1)

for some $c_{\alpha}\in(0,\infty)$ depending only on $\alpha$ where
$\overline{B}_{H}^{c}$ denotes the complement of the unit ball $\\{h\in
H:\,\left\lVert h\right\rVert<1\\}$; see [23, Le. 1]. This enables us to
conclude the following technical Lemma:

######  Lemma 2.1.

Let $\lambda$ be the cylindrical Lévy measure of an $\alpha$-stable
cylindrical Lévy process for $\alpha\in(1,2)$. For every $m\in\mathbb{N}$
there exists $d_{\alpha}^{m}<\infty$, depending only on $\alpha$ and $m$, such
that

$\displaystyle\int_{\overline{B}_{H}(1/m)}{\left|\left|h\right|\right|^{2}\,(\lambda\circ\Phi^{-1})({\rm
d}h)}+\int_{\overline{B}_{H}(m)^{c}}{\left|\left|h\right|\right|\,(\lambda\circ\Phi^{-1})({\rm
d}h)}\leq
d_{\alpha}^{m}\left|\left|\Phi\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}$
(2.2)

for all $\Phi\in\mathcal{L}_{2}(U,H)$. Moreover, we have
$\lim_{m\rightarrow\infty}d_{\alpha}^{m}=0$.

###### Proof.

Let $m\in\mathbb{N}$ be fixed. We approximate the integrand of the first
integral in (2.2) by

$f_{m,n}\colon\overline{B}_{H}(\tfrac{1}{m})\rightarrow\mathbb{R},\qquad
f_{m,n}(h):=\sum_{i=0}^{m2^{n}-1}\left(\frac{i}{m2^{n}}\right)^{2}\mathbbm{1}_{(\frac{i}{m2^{n}},\frac{i+1}{m2^{n}}]}(\left\lVert
h\right\rVert).$

Since $\lambda\circ\Phi^{-1}$ is a genuine $\alpha$-stable measure in $H$, we
have for each $r>0$ that

$\displaystyle(\lambda\circ\Phi^{-1})(\overline{B}_{H}^{c}(r))=r^{-\alpha}(\lambda\circ\Phi^{-1})(\overline{B}_{H}^{c});$
(2.3)

see [24, Th. 6.2.7]. This enables us to conclude for each $n\in\mathbb{N}$
that

$\displaystyle\int_{\overline{B}_{H}(1/m)}f_{m,n}(h)$
$\displaystyle\,(\lambda\circ\Phi^{-1})({\rm d}h)$ $\displaystyle=$
$\displaystyle\frac{1}{m^{2}}(\lambda\circ\Phi^{-1})(\overline{B}_{H}^{c})\sum_{i=0}^{m2^{n}-1}\left(\frac{i}{2^{n}}\right)^{2}\left(\left(\frac{i}{m2^{n}}\right)^{-\alpha}-\left(\frac{i+1}{m2^{n}}\right)^{-\alpha}\right)$
$\displaystyle=$ $\displaystyle
m^{\alpha-2}(\lambda\circ\Phi^{-1})(\overline{B}_{H}^{c})\int_{0}^{1}\left(\sum_{i=0}^{m2^{n}-1}\left(\frac{i}{2^{n}}\right)^{2}\mathbbm{1}_{\left(\frac{i}{2^{n}},\frac{i+1}{2^{n}}\right]}(r)\right)\alpha
r^{-(\alpha+1)}\,{\rm d}r.$

The monotone convergence theorem implies

$\displaystyle\lim_{n\rightarrow\infty}\int_{\overline{B}_{H}(1/m)}f_{m,n}(h)\,(\lambda\circ\Phi^{-1})({\rm
d}h)$
$\displaystyle=m^{\alpha-2}\frac{\alpha}{2-\alpha}(\lambda\circ\Phi^{-1})(\overline{B}_{H}^{c}).$

Since another application of the monotone convergence theorem shows

$\lim_{n\rightarrow\infty}\int_{\overline{B}_{H}(1/m)}f_{m,n}(h)\,(\lambda\circ\Phi^{-1})({\rm
d}h)=\int_{\overline{B}_{H}(1/m)}\left\lVert
h\right\rVert^{2}\,(\lambda\circ\Phi^{-1})({\rm d}h),$

we obtain from (2.1) that

$\int_{\overline{B}_{H}(1/m)}\left\lVert
h\right\rVert^{2}\,(\lambda\circ\Phi^{-1})({\rm
d}h)=m^{\alpha-2}\frac{\alpha}{2-\alpha}(\lambda\circ\Phi^{-1})(\overline{B}_{H}^{c})\leq
m^{\alpha-2}\frac{\alpha}{2-\alpha}c_{\alpha}\left\lVert\Phi\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha}.$

Applying similar arguments for the integrand in the second integral in (2.2)
yields

$\int_{\overline{B}_{H}(m)^{c}}{\left|\left|h\right|\right|}\,(\nu\circ\Phi^{-1})({\rm
d}h)\leq
m^{1-\alpha}\frac{\alpha}{\alpha-1}c_{\alpha}\left|\left|\Phi\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha},$

for all $\Phi\in\mathcal{L}_{2}(U,H)$, which completes the proof as in (2.2)
we can set

$\displaystyle d_{\alpha}^{m}:=(m^{\alpha-2}+m^{1-\alpha})c_{\alpha}\to
0,\quad m\to\infty.$ (2.4)

∎

If $L$ is an $\alpha$-stable cylindrical Lévy process and $T\colon H\to U$ a
Hilbert-Schmidt operator, then the cylindrical random variable $TL(1)$ is
induced by a genuine stable random variable on $U$ with Lévy measure
$\lambda\circ T^{-1}$. This Lévy measure depends continuously on $T$ in the
following way:

######  Lemma 2.2.

Let $\lambda$ be the cylindrical Lévy measure of an $\alpha$-stable
cylindrical Lévy process for $\alpha\in(1,2)$. Then for each $r>0$, the
mapping $\Phi\mapsto\lambda\circ\Phi^{-1}\arrowvert_{\overline{B}_{H}(r)^{c}}$
is continuous from $\mathcal{L}_{2}(U,H)$ to the space of Borel measures on
$\overline{B}_{H}(r)^{c}$ equipped with the weak topology.

###### Proof.

If $\mu$ is a cylindrical probability measure and $(F_{n})_{n\in\mathbb{N}}$
is a sequence converging to $F$ in $\mathcal{L}_{2}(U,H)$ then $(\mu\circ
F_{n}^{-1})_{n\in\mathbb{N}}$ converges weakly to $\mu\circ F^{-1}$ according
to [5, Le. 2.1]. From this, the assertion follows from [29, Th. 5.5]. ∎

###  2.2 Stochastic integration

We briefly recall some facts on stochastic integration with respect to an
$\alpha$-stable cylindrical Lévy process $L$ as introduced in [5]. A process
$G\colon\Omega\times[0,T]\rightarrow\mathcal{L}_{2}(U,H)$ is called adapted
and simple if it is of the form

$\displaystyle
G=\Phi_{0}\mathbbm{1}_{\\{0\\}}+\sum_{i=1}^{N}\Phi_{i}\mathbbm{1}_{(t_{i-1},t_{i}]},$
(2.5)

where $N\in\mathbb{N}$, $0=t_{0}<t_{1}<\dots<t_{N}=T$ and $\Phi_{i}$ is an
$\mathcal{F}_{t_{i-1}}$-measurable and $\mathcal{L}_{2}(U,H)$-valued random
variable taking finitely many values. We denote by $\mathcal{S}_{\rm adp}^{\rm
HS}$ the class of all adapted, simple processes. The integral process
$\int_{0}^{\cdot}G\,{\rm d}L$ is defined as the sum of the Radonified
increments

$\displaystyle\int_{0}^{t}G{\rm
d}L:=\sum_{i=1}^{N}\Phi_{i}\big{(}L({t_{i}\land t})-L({t_{i-1}\land
t})\big{)},\quad t\in[0,T].$ (2.6)

Here, $\Phi_{i}\big{(}L({t_{i}\land t})-L({t_{i-1}\land t})\big{)}$ is defined
as the $H$-valued random variable satisfying

$\displaystyle\left\langle\Phi_{i}\big{(}L({t_{i}\land t})-L({t_{i-1}\land
t})\big{)},h\right\rangle=\big{(}L({t_{i}\land t})-L({t_{i-1}\land
t})\big{)}(\Phi_{i}^{\ast}h)\qquad\text{for all }h\in H.$

Let $\mathcal{S}_{\rm adp}^{1,\rm op}$ denote the class of adapted, simple
$\mathcal{L}(H)$-valued processes bounded in the operator norm by $1$ on
$[0,T]$. An arbitrary predictable process
$G\colon\Omega\times[0,T]\rightarrow\mathcal{L}_{2}(U,H)$ is stochastically
integrable if there exists a sequence of adapted simple processes
$(G_{n})_{n\in\mathbb{N}}\subset\mathcal{S}_{\rm adp}^{\rm HS}$ such that:

  * (i)

$(G_{n})_{n\in\mathbb{N}}$ converges to $G$ $P\otimes{\rm
Leb}|_{[0,T]}$-almost everywhere,

  * (ii)

$\displaystyle\lim_{m,n\rightarrow\infty}\sup_{\Gamma\in\mathcal{S}_{{\rm
adp}}^{1,{\rm op}}}E\Bigg{[}\left\lVert\int_{0}^{T}\Gamma(G_{m}-G_{n})\;{\rm
d}L\right\rVert\wedge 1\Bigg{]}=0$

In this case, $\int_{0}^{\cdot}G\,{\rm d}L$ is defined as the limit of
$\int_{0}^{\cdot}G_{n}\,{\rm d}L$ in the topology of uniform convergence in
probability on $[0,T]$.

It is shown in [5], that a predictable process $G$ is stochastically
integrable if and only it is an element of $L_{P}^{0}\big{(}\Omega,L_{{\rm
Leb}}^{\alpha}\big{(}[0,T],\mathcal{L}_{2}(G,H)\big{)}\big{)}$. It follows
from [23, Co. 3] that for every $0<p<\alpha$ we have

$\displaystyle\mathbb{E}\,\left[\sup_{t\in[0,T]}\left|\left|\int_{0}^{t}G\,{\rm
d}L\right|\right|^{p}\right]\leq
e_{p,\alpha}\left(\mathbb{E}\,\left[\int_{0}^{T}\left|\left|G(t)\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}\,{\rm
d}t\right]\right)^{p/\alpha},$ (2.7)

for every stochastically integrable predictable process $G$, where
$e_{p,\alpha}=\frac{\alpha}{\alpha-p}e_{2,\alpha}^{p/\alpha}$ for some
$e_{2,\alpha}\in(0,\infty)$ that depends only on $\alpha$.

######  Lemma 2.3.

If $G$ is a predictable stochastic process stochastically integrable with
respect to the $\alpha$-stable cylindrical Lévy process $L$ for some
$\alpha\in(1,2)$ then $\int_{0}^{\cdot}G\,{\rm d}L$ is a local martingale.

###### Proof.

Define the predictable stopping times
$\tau_{n}=\inf\left\\{t>0:\int_{0}^{t}\left|\left|G(s)\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}{\rm
d}s>n\right\\}$ for $n\in\mathbb{N}$. It follows from Proposition 4.22(ii) and
Lemma 1.3 in [10] that for each $n\in\mathbb{N}$ there exists a sequence of
adapted, simple processes $(G_{n,k})_{k\in\mathbb{N}}$ such that

$\displaystyle\lim_{k\to\infty}\mathbb{E}\,\left[\int_{0}^{T}\left|\left|G(s)\mathbbm{1}_{[0,\tau_{n}]}(s)-G_{n,k}(s)\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}\,{\rm
d}s\right]=0.$ (2.8)

Since inequality (2.7) guarantees for each $k,n\in\mathbb{N}$ that

$\displaystyle\mathbb{E}\,\left[\sup_{0\leq t\leq
T}\left|\left|\int_{0}^{t}G_{n,k}\,{\rm d}L\right|\right|\right]\leq
e_{1,\alpha}\left(\mathbb{E}\,\left[\int_{0}^{T}\left|\left|G_{n,k}(s)\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}{\rm
d}s\right]\right)^{1/\alpha\,}<\infty,$

the same arguments as in [32, Th. I:51] show that the processes
$\int_{0}^{\cdot}G_{n,k}\,{\rm d}L$ are martingales. Equation (2.8) shows that
$\int_{0}^{\cdot}G\mathbbm{1}_{[0,\tau_{n}]}\,{\rm d}L$ is a limit of
martingales in $L^{1}(\Omega,H)$ by (2.7), and thus a martingale. Since
standard arguments, e.g. [32, Th. I.12], establish

$\displaystyle\left(\int_{0}^{\cdot}G\,{\rm
d}L\right)^{\tau_{n}}=\int_{0}^{\cdot}G\mathbbm{1}_{[0,\tau_{n}]}\,{\rm
d}L\quad\text{a.s.},$ (2.9)

for the stopped integral process, the proof is completed. ∎

######  Theorem 2.4 (Stochastic Fubini Theorem).

Let $L$ be the standard symmetric $\alpha$-stable cylindrical Lévy process for
$\alpha\in(1,2)$. If
$G\colon\Omega\times[0,T]^{2}\rightarrow\mathcal{L}_{2}(U,H)$ is measurable,
$G(t,\cdot)$ is predictable for every $t\in[0,T],$ and
$\int_{0}^{T}\int_{0}^{T}\left\lVert
G(t,s)\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha}\,{\rm d}t\,{\rm d}s<\infty$
a.s. then it follows:

  * (a)

$G(t,\cdot)$ is stochastically integrable for every $t\in[0,T]$ and
$\int_{0}^{T}G(\cdot,s){\rm d}L(s)$ is a.s. Bochner integrable;

  * (b)

$G(\cdot,s)$ is a.s. Bochner integrable for every $s\in[0,T]$ and
$\int_{0}^{T}G(t,\cdot)\,{\rm d}t$ is stochastically integrable;

  * (c)

$\displaystyle\int_{0}^{T}\left(\int_{0}^{T}G(t,s)\,{\rm d}t\right)\,{\rm
d}L(s)=\int_{0}^{T}\left(\int_{0}^{T}G(t,s)\,{\rm d}L(s)\right)\,{\rm d}t$
a.s.

###### Proof.

The proof is similar as in finite dimensions; see [42]. ∎

###  2.3 Random measures and compensators

In this section, we briefly recall some results on random measures and their
compensators from [17, Ch. II].

######  Definition 2.5 (Random measure).

A family $\mu=\\{\mu(\omega):\omega\in\Omega\\}$ is called a random measure on
$[0,T]\times H$ if $\mu(\omega)$ is a measure on
$\mathcal{B}([0,T])\otimes\mathcal{B}(H)$ for each $\omega\in\Omega$. It is
said to be an integer-valued random measure if moreover, we have

  * (i)

$\mu(\\{t\\}\times H)\leq 1$ for all $t\in[0,T]$ $P$-a.s.;

  * (ii)

$\mu$ takes values in $\mathbb{N}\cup\\{\infty\\}$ $P$-a.s.

We denote by $\tilde{\mathcal{P}}$ (resp. $\tilde{\mathcal{O}}$) the
predictable (resp. optional) $\sigma$-algebra on $\Omega\times[0,T]\times H$
and call a function $W:\Omega\times[0,T]\times H\mapsto\mathbb{R}$ predictable
(resp. optional) if it is $\tilde{\mathcal{P}}$ (resp. $\tilde{\mathcal{O}}$)
measurable.

If $\mu$ is a random measure and $W$ is optional we define

$\displaystyle\left(\int_{0}^{t}\int_{H}W(s,h)\,\mu({\rm d}s,{\rm
d}h)\right)(\omega)$
$\displaystyle\qquad\qquad:=\begin{cases}\int_{0}^{t}\int_{H}W(\omega,s,h)\,\mu(\omega)({\rm
d}s,{\rm d}h),&\text{if}\int_{0}^{t}\int_{H}|W(\omega,s,h)|\,\mu(\omega)({\rm
d}s,{\rm d}h)<\infty,\\\ \infty,&\text{otherwise.}\end{cases}$

A random measure $\mu$ is called predictable (resp. optional) if
$(\int_{0}^{t}\int_{H}W(s,h)\,\mu({\rm d}s,{\rm d}h):\,t\in[0,T])$ is
predictable (resp. optional) for every predictable (resp. optional) function
$W$. An optional random measure $\mu$ is called $\sigma$-finite if there
exists a sequence $(A_{n})_{n\in\mathbb{N}}\subset\tilde{\mathcal{P}}$ with
$\bigcup_{n=1}^{\infty}A_{n}=\Omega\times[0,T]\times H$, such that
$\mathbb{E}\,\left[\int_{0}^{T}\int_{H}\mathbbm{1}_{A_{n}}(s,h)\,\mu({\rm
d}s,{\rm d}h)\right]<\infty$ for each $n\in\mathbb{N}$.

For each $\sigma$-finite, optional measure $\mu$ on $[0,T]\times H$ there
exists a predictable random measure $\nu$ on
$\mathcal{B}([0,T])\otimes\mathcal{B}(H)$ such that

$\displaystyle\mathbb{E}\,\left[\int_{0}^{t}\int_{H}W(s,h)\,\mu({\rm d}s,{\rm
d}h)\right]=\mathbb{E}\,\left[\int_{0}^{t}\int_{H}W(s,h)\,\nu({\rm d}s,{\rm
d}h)\right]$ (2.10)

for all $t\in[0,T]$, and any non-negative predictable function $W$. The
measure $\nu$ is determined uniquely up to a set of probability zero by (2.10)
and is called the compensator of $\mu$; see [17, th. II.1.8].

If $Y$ is an $H$-valued, adapted càdlàg process then the integer-valued random
measure $\mu^{Y}$ characterised by

$\displaystyle\mu^{Y}((0,t]\times B)=\sum_{0\leq s\leq
t}\mathbbm{1}_{B}(\Delta Y(s)),\quad t\in(0,T],B\in\mathcal{B}(H),0\notin B,$

where $\Delta Y(s):=Y(s)-\lim_{h\searrow 0+}Y(s-h)$ for $s\in[0,T]$, is an
optional and $\sigma$-finite random measure on
$\mathcal{B}([0,T])\otimes\mathcal{B}(H)$. Thus, its compensator exists which
we denote by $\nu^{Y}$.

######  Example 2.6.

Let $L$ be a genuine $H$-valued Lévy process with Lévy measure $\lambda$. Then
the compensator $\nu^{L}$ of the jump measure $\mu^{L}$ is given as the
extension of $\mu^{L}((s,t]\times B)=(t-s)\lambda(B)$, $0\leq s<t\leq T$,
$B\in\mathcal{B}(H)$ to $\mathcal{B}([0,T])\otimes\mathcal{B}(H)$.

In the sequel, we will make use of another characterisation of compensators of
jump-measures. We denote by $\mathcal{C}^{+}(H)$ the class of non-negative,
continuous functions $k:H\rightarrow\mathbb{R}$ bounded on $H$ and vanishing
inside a neighbourhood of $0$.

######  Proposition 2.7.

The compensator $\nu^{Y}$ of the jump-measure $\mu^{Y}$ of an $H$-valued
càdlàg semimartingale $Y$ is characterised by being predictable and satisfying
either of the following:

  * (i)

The process

$\displaystyle\left(\int_{0}^{t}\int_{H}k(h)\,\mu^{Y}({\rm d}s,{\rm
d}h)-\int_{0}^{t}\int_{H}k(h)\,\nu^{Y}({\rm d}s,{\rm d}h):\;t\in[0,T]\right)$

is a local martingale for every $k\in\mathcal{C}^{+}(H)$.

  * (ii)

If $W$ is predictable and the process

$\displaystyle\left(\int_{0}^{t}\int_{H}W(s,h)\,\mu^{Y}({\rm d}s,{\rm
d}h):\;t\in[0,T]\right)$ (2.11)

is locally integrable, then so is

$\displaystyle\left(\int_{0}^{t}\int_{H}W(s,h)\,\nu^{Y}({\rm d}s,{\rm
d}h):\;t\in[0,T]\right)$

and

$\displaystyle\left(\int_{0}^{t}\int_{H}W(s,h)\,\mu^{Y}({\rm d}s,{\rm
d}h)-\int_{0}^{t}\int_{H}W(s,h)\,\nu^{Y}({\rm d}s,{\rm
d}h):\;t\in[0,T]\right)$

is a local martingale.

###### Proof.

The equivalence between (i) and (ii) follows by the same argument as in the
proof of [17, Th. II.2.21.]. The fact that (ii) is an equivalent definition of
the compensator is proved in [17, Th. II.1.8.]. ∎

Proposition 2.7 justifies the following standard notation: if $W$ is
predictable and (2.11) is locally integrable, we define the following local
martingale

$\displaystyle\int_{0}^{t}\int_{H}W(s,h)\,(\mu^{Y}-\nu^{Y})({\rm d}s,{\rm
d}h):=\int_{0}^{t}\int_{H}W(s,h)\,\mu^{Y}({\rm d}s,{\rm
d}h)-\int_{0}^{t}\int_{H}W(s,h)\,\nu^{Y}({\rm d}s,{\rm d}h)$

for each $t\in[0,T$].

##  3 Predictable compensator

For an $\alpha$-stable cylindrical Lévy process $L$ for some $\alpha\in(1,2)$
and a stochastically integrable predictable process $G$, we define the
integral process $X=\int_{0}^{\cdot}G{\rm d}L$ and

$\displaystyle\nu\left((0,t]\times B\right):=\int_{0}^{t}\left(\lambda\circ
G(s)^{-1}\right)(B)\,{\rm d}s\qquad\text{for each
$t\in(0,T],B\in\mathcal{B}(H)$ with $0\notin\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu$.}$ (3.1)

The main result of this section is that $\nu$ extends to a random measure on
$\mathcal{B}([0,T])\otimes\mathcal{B}(H)$ and that the extension is the
predictable compensator of the jump measure of $X$. We will derive this result
by a couple of Lemmata.

######  Lemma 3.1.

The set function $\nu$ defined in (3.1) is well defined and extends to a
predictable random measure on $\mathcal{B}([0,T])\otimes\mathcal{B}(H)$. This
extension is unique among the class of $\sigma$-finite random measures on
$\mathcal{B}([0,T])\otimes\mathcal{B}(H)$ that assign $0$ mass to the origin.

###### Proof.

Step 1: We show that for all open sets $B\subseteq H$ with $0\notin\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu$ the process

$\displaystyle f\colon\Omega\times[0,T]\rightarrow\mathbb{R},\quad
f(\omega,t)=\left(\lambda\circ G(\omega,t)^{-1}\right)(B)$

is predictable. Since the function
$h\colon\mathcal{L}_{2}(U,H)\rightarrow\mathbb{R}$ defined by
$h(\Phi)=(\lambda\circ\Phi^{-1})(B)$ is lower semicontinuous by Lemma 2.2 and
the Portmonteau Theorem as the set $B$ assumed to be open, $h$ is measurable.
Since $G\colon\Omega\times[0,T]\rightarrow\mathcal{L}_{2}(U,H)$ is
predictable, it follows that $f=h\circ G$ is predictable.
Step 2: We show that $f$ is predictable for all
$B\in\mathcal{B}(H\setminus\\{0\\})$, which will immediately imply that (3.1)
is almost surely well defined and predictable as it is then just an integral
of a non-negative predictable process. We define

$\mathcal{D}=\left\\{B\in\mathcal{B}(H\setminus\\{0\\}):\lambda\circ
G(\cdot,\cdot)^{-1}(B)\,\text{ is predictable}\right\\},$

and claim that $\mathcal{D}$ is a $\lambda$-system. Continuity of measures
implies that $H\setminus\\{0\\}\in\mathcal{D}$ since, for all $t\in(0,T]$ and
$\omega\in\Omega$, we have

$\left(\lambda\circ
G(\omega,t)^{-1}\right)(H\setminus\\{0\\})=\lim_{n\rightarrow\infty}\left(\lambda\circ
G(\omega,t)^{-1}\right)\left(\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern
1.5mu_{H}\left(1/n\right)^{c}\right),$

where the right hand side is the limit of processes that are predictable by
Step 1. If $B\in\mathcal{D}$ then $B^{c}\in\mathcal{D}$ since

$\left(\lambda\circ G(\omega,t)^{-1}\right)(B^{c})=\left(\lambda\circ
G(\omega,t)^{-1}\right)(H\setminus\\{0\\})-\left(\lambda\circ
G(\omega,t)^{-1}\right)(B).$

The collection $\mathcal{D}$ is closed under union of increasing sequences,
which follows as above from continuity of measures and predictability of the
pointwise limit. This concludes the proof of the claim that $\mathcal{D}$ is a
$\lambda$-system.

We define the $\pi$-system.

$\mathcal{I}=\left\\{B\in\mathcal{B}(H\setminus\\{0\\}):B\,\text{ is
open}\right\\}.$

The family $\mathcal{I}$ is contained in $\mathcal{D}$, since for each
$B\in\mathcal{D}$ we have

$\left(\lambda\circ
G(\omega,t)^{-1}\right)(B)=\lim_{n\rightarrow\infty}\left(\lambda\circ
G(\omega,t)^{-1}\right)\left(B\cap\mkern
1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu_{H}(1/n)^{c}\right),$

and the right-hand side is predictable by Step 1. The Dynkin $\pi$-$\lambda$
theorem for sets, see e.g. [20, Th. 1.1] implies
$\sigma(\mathcal{I})\subseteq\mathcal{D}$, and thus
$\mathcal{D}=\mathcal{B}(H\setminus\\{0\\})$.
Step 3: Let $\omega\in\Omega$ be such that $\int_{0}^{T}\left\lVert
G(\omega,s)\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha}{\rm d}s<\infty$.
Equation (3.1) defines the set function $\nu(\omega)$ on the semi-ring

$\mathcal{S}=\left\\{(0,t]\times B:t\in[0,T]\text{ and
}B\in\mathcal{B}(H\setminus\\{0\\})\right\\}.$

The set function $\nu(\omega)$ is $\sigma$-additive by its very definition and
$\sigma$-finite, since for $n\in{\mathbb{N}}$ we have by (2.1) and (2.3) that

$\displaystyle\nu(\omega)\left((0,T]\times\overline{B}_{H}^{c}\left(1/n\right)\right)=$
$\displaystyle\int_{0}^{T}\left(\lambda\circ
G(\omega,s)^{-1}\right)\left(\overline{B}_{H}^{c}\left(1/n\right)\right)\,{\rm
d}s$ $\displaystyle=$ $\displaystyle\,n^{\alpha}\int_{0}^{T}\left(\lambda\circ
G(\omega,s)^{-1}\right)\left(\overline{B}_{H}^{c}\right)\,{\rm d}s$
$\displaystyle\leq$
$\displaystyle\,n^{\alpha}\,c_{\alpha}\int_{0}^{T}\left\lVert
G(\omega,s)\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha}{\rm d}s<\infty.$

Carathéodory’s extension theorem, see e.g. [20, Th. 2.5], implies that the set
function $\nu(\omega)$ extends uniquely to a measure on
$\mathcal{B}([0,T])\otimes\mathcal{B}(H\setminus\\{0\\})$ which we also denote
by $\nu(\omega)$.

Step 4: It remains to show that $\nu$ is predictable. Applying the monotone
class theorem as above shows that the process
$\int_{0}^{\cdot}\left(\lambda\circ G(s)^{-1}\right)(B)\,{\rm d}s$ is
predictable for each $B\in\mathcal{B}(H\setminus\\{0\\})$. Since

$\displaystyle\int_{0}^{\cdot}\mathbbm{1}_{(s,t]}(u)\mathbbm{1}_{A}\left(\lambda\circ
G(u)^{-1}\right)(B){\rm
d}u=\int_{0}^{\cdot}\left(\lambda\circ\left(\mathbbm{1}_{(s,t]}(u)\mathbbm{1}_{A}G(u)\right)^{-1}\right)(B){\rm
d}u,$

it follows that the process $\int_{0}^{\cdot}\int_{H}W(u,h)\tilde{\nu}({\rm
d}u,{\rm d}h)(\cdot)$ is predictable for all functions
$W=\mathbbm{1}_{(s,t]}\mathbbm{1}_{A}\mathbbm{1}_{B}$ with $0<s<t\leq T$,
$A\in\mathcal{F}_{s}$ and $B\in\mathcal{B}(H\setminus\\{0\\})$. An application
of the functional monotone class theorem (follows e.g. from [41, Th. 3.14])
extends this result to all predictable processes $W$ on
$\Omega\times[0,T]\times H$, which shows predictability of the random measure
$\nu$ on $\mathcal{B}([0,T])\otimes\mathcal{B}(H\setminus\\{0\\})$. Defining
$\nu((s,t]\times\\{0\\}):=0$ for any $0\leq s<t\leq T$ extends $\nu$ to a
predictable random measure on $\mathcal{B}([0,T])\otimes\mathcal{B}(H)$.

∎

To show that the random measure $\nu$ characterised by (3.1) is the
compensator of the jump-measure $\mu^{X}$ of the integral process $X$, we
first consider the case when the integrand is an adapted, simple process.

######  Lemma 3.2.

Suppose that $G$ is an adapted, simple process in $\mathcal{S}_{\rm adp}^{\rm
HS}$. Then the random measure $\nu$ obtained in Lemma 3.1 is the predictable
compensator of $\mu^{X}$.

###### Proof.

Since Lemma 3.1 guarantees that $\nu$ is predictable, it remains to show
(2.10), which by the functional monotone class theorem reduces to proving

$\displaystyle\mathbb{E}\,\left[\mathbbm{1}_{A}\sum_{s<u\leq
t}\mathbbm{1}_{B}(\Delta
X(u))\right]=\mathbb{E}\,\left[\mathbbm{1}_{A}\int_{s}^{t}\left(\lambda\circ
G(u)^{-1}\right)(B)\,{\rm d}u\right]$

for any $0<s<t\leq T$, $A\in\mathcal{F}_{s}$ and $B\in\mathcal{B}(H)$ with
$0\not\in\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu$. Let
$G$ be of the form (2.5), and assume that the points of the partition contain
$s$ and $t$; otherwise these can be added. Then $X$ takes the form (2.6), and
it follows

$\displaystyle\mathbbm{1}_{A}\sum_{s<u\leq t}\mathbbm{1}_{B}(\Delta X(u))$
$\displaystyle=\mathbbm{1}_{A}\sum_{i=1}^{N}\sum_{s\leq t_{i-1}<u\leq
t_{i}\leq t}\mathbbm{1}_{B}(\Delta G(t_{i})L(u))$
$\displaystyle=\mathbbm{1}_{A}\sum_{i=1}^{N}\sum_{s\leq t_{i-1}<u\leq
t_{i}\leq t}\mathbbm{1}_{B}(\Delta\Phi_{i}L(u)).$

For each $i\in\\{1,\dots,N\\}$, the random variable $\Phi_{i}$ is of the form
$\Phi_{i}=\sum_{j=1}^{m_{i}}\mathbbm{1}_{A_{i,j}}\varphi_{i,j}$ for some
pairwise disjoint sets $A_{i,j}\in\mathcal{F}_{t_{i-1}}$ and
$\varphi_{i,j}\in\mathcal{L}_{2}(U,H)$ for $j\in\\{1,\dots,m_{i}\\}$. Since
$0\not\in\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu$, we
have

$\displaystyle\mathbb{E}\,\left[\mathbbm{1}_{A}\sum_{t_{i-1}<u\leq
t_{i}}\mathbbm{1}_{B}(\Delta\Phi_{i}L(u))\right]$
$\displaystyle=\sum_{j=1}^{m_{i}}\mathbb{E}\,\left[\mathbbm{1}_{A\cap
A_{i,j}}\sum_{t_{i-1}<u\leq
t_{i}}\mathbbm{1}_{B}(\Delta\varphi_{i,j}L(u))\right]$
$\displaystyle=\sum_{j=1}^{m_{i}}(t_{i}-t_{i-1})\mathbb{E}\,\left[\mathbbm{1}_{A\cap
A_{i,j}}\left(\lambda\circ\varphi_{i,j}^{-1}\right)(B)\right]$
$\displaystyle=(t_{i}-t_{i-1})\mathbb{E}\,\left[\mathbbm{1}_{A}\left(\lambda\circ\Phi^{-1}_{i}\right)(B)\right],$

because $A\cap A_{i,j}\in\mathcal{F}_{t_{i-1}}$ and the compensator of the
jump measure of the Lévy process $\varphi_{i,j}L$ in $H$ is given by
$\left(\lambda\circ\varphi_{i,j}^{-1}\right)\,{\rm d}h\,{\rm d}t$ since its
Lévy measure is $\left(\lambda\circ\varphi_{i,j}^{-1}\right)$, see Example
2.6. ∎

Before we show that the result of Lemma 3.2 can be extended to general
integrands, we need to prove some technical Lemmata. Recall the class of
functions $\mathcal{C}^{+}(H)$ used in Proposition 2.7 (and defined just
before) to determine the compensator.

######  Lemma 3.3.

Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of càdlàg functions
$f_{n}\colon[0,T]\to H$ converging uniformly to $f\colon[0,T]\rightarrow H$.
Then we have for any $k\in\mathcal{C}^{+}(H)$ that

$\displaystyle\lim_{n\rightarrow\infty}\sup_{t\in[0,T]}\left|\sum_{0\leq s\leq
t}k(\Delta f_{n}(s))-\sum_{0\leq s\leq t}k(\Delta f(s))\right|=0.$ (3.2)

###### Proof.

Both sums in (3.2) are finite by the càdlàg property of $f,f_{n}$ and since
$k$ vanishes inside a neighbourhood of $0$. The assumed uniform convergence
implies

$\displaystyle\lim_{n\to\infty}\sup_{t\in[0,T]}\left\|\Delta f_{n}(t)-\Delta
f(t)\right\|=0.$ (3.3)

Denoting $\text{supp}(k):=\\{h\in H:k(h)\neq 0\\}$ and
$\delta:=\frac{1}{2}\text{dist}(0,\text{supp}(k))$, we obtain that
$\text{supp}(k)_{\delta}:=\\{h\in H,\text{dist}(h,\text{supp}(k))<\delta\\},$
is bounded away from zero, i.e. $0\notin\overline{\text{supp}(k)_{\delta}}$.
It follows that the set $D:=\\{t\in[0,T]:\Delta
f(t)\in\text{supp}(k)_{\delta}\\}$ is finite, which together with continuity
of $k$ and (3.3) implies

$\displaystyle\lim_{n\rightarrow\infty}\sup_{t\in D}{|k(\Delta
f_{n}(t))-k(\Delta f(t))|}=0.$ (3.4)

Since (3.3) guarantees that there exists $n_{0}\in\mathbb{N}$ such that we
have $\Delta f_{n}(t)\notin\text{supp}(k)$ for all $n\geq n_{0}$ and
$t\in[0,T]\setminus D$, we conclude from (3.4) for $n\geq n_{0}$ that

$\displaystyle\sup_{t\in[0,T]}\left|\sum_{0\leq s\leq t}k(\Delta
f_{n}(s))-\sum_{0\leq s\leq t}k(\Delta f(s))\right|=$
$\displaystyle\sup_{t\in[0,T]}\left|\sum_{s\in D\cap[0,t]}k(\Delta
f_{n}(s))-\sum_{s\in D\cap[0,t]}k(\Delta f(s))\right|$ $\displaystyle\leq$
$\displaystyle|D|\sup_{t\in D}{\left|k(\Delta f_{n}(t))-k(\Delta
f(t))\right|}\to 0,\quad n\to\infty.$

The proof is complete. ∎

######  Lemma 3.4.

Let $g_{n},g\in L_{\rm Leb}^{\alpha}([0,T],\mathcal{L}_{2}(U,H))$,
$n\in\mathbb{N}$, be such that $g_{n}$ converges to $g$ in $L_{\rm
Leb}^{\alpha}([0,T],\mathcal{L}_{2}(U,H))$ and pointwise for almost every
$s\in[0,T]$. Then we obtain for each $k\in\mathcal{C}^{+}(H)$ that

$\displaystyle\lim_{n\rightarrow\infty}\sup_{t\in[0,T]}\left|\int_{0}^{t}\int_{H}k(h)\,\left(\lambda\circ
g_{n}(s)^{-1}\right)\,{\rm d}h\,{\rm
d}s-\int_{0}^{t}\int_{H}k(h)\,\left(\lambda\circ g(s)^{-1}\right)\,{\rm
d}h\,{\rm d}s\right|=0.$

###### Proof.

Lemma 2.2 implies for almost each $s\in[0,T]$ and every $n\in\mathbb{N}$ that

$\displaystyle\lim_{n\rightarrow\infty}\int_{H}k(h)\,\left(\lambda\circ
g_{n}(s)^{-1}\right)\,{\rm d}h=\int_{H}k(h)\,\left(\lambda\circ
g(s)^{-1}\right)\,{\rm d}h.$

Since $k$ is bounded and vanishes in a neighbourhood of $0$, we conclude from
inequality (2.1)

$\displaystyle\int_{H}k(h)\,\left(\lambda\circ g_{n}(s)^{-1}\right)\,{\rm
d}h\leq
c_{k,\alpha}\left|\left|g_{n}(s)\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha},$

for a constant $c_{k,\alpha}$ independent of $s\in[0,T]$ and $n\in\mathbb{N}$.
Since for each $t\in[0,T]$ we have

$\displaystyle\lim_{n\rightarrow\infty}\int_{0}^{t}\left|\left|g_{n}(s)\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}\,{\rm
d}s=\int_{0}^{t}\left|\left|g(s)\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}\,{\rm
d}s,$

the generalised Lebesgue’s dominated convergence theorem, see e.g. [35, Th.
4.19], implies

$\displaystyle\lim_{n\rightarrow\infty}\int_{0}^{t}\int_{H}k(h)\,\left(\lambda\circ
g_{n}(s)^{-1}\right)\,{\rm d}h\,{\rm
d}s=\int_{0}^{t}\int_{H}k(h)\,\left(\lambda\circ g(s)^{-1}\right)\,{\rm
d}h\,{\rm d}s.$

As the functions

$\displaystyle t\mapsto\int_{0}^{t}\int_{H}k(h)\,\left(\lambda\circ
g_{n}(s)^{-1}\right)\,{\rm d}h\,{\rm d}s$

are continuous monotone and converge pointwise to a continuous limit on
$[0,T]$, the convergence is uniform by [31, p. 81/127] (or deuxième théorème
de Dini). ∎

Now we can prove the main result of this section.

######  Theorem 3.5.

Let $L$ be an $\alpha$-stable cylindrical Lévy process $L$ for some
$\alpha\in(1,2)$ and $G$ a stochastically integrable predictable process. Then
the predictable compensator $\nu^{X}$ of the jump measure $\mu^{X}$ of
$X:=\int_{0}^{\cdot}G{\rm d}L$ is characterised by (3.1).

###### Proof.

In light of Proposition 2.7, it suffices to show that the process $M^{k}$
defined by

$\displaystyle\left(M^{k}(t):=\int_{0}^{t}\int_{H}k(h)\,\mu^{X}({\rm d}s,{\rm
d}h)-\int_{0}^{t}\int_{H}k(h)\,\left(\lambda\circ G(s)^{-1}\right)\,{\rm
d}h\,{\rm d}s,\,t\in[0,T]\right),$

is a local martingale for any $k\in\mathcal{C}^{+}(H)$. Lemma 4.3 in [5]
guarantees that there exists a sequence $(G_{n})_{n\in\mathbb{N}}$ of adapted,
simple processes in $\mathcal{S}_{\rm adp}^{\rm HS}$ converging both in
$L_{\rm Leb}^{\alpha}([0,T],L_{2}(U,H))$ a.s. and $P\otimes{\rm
Leb}|_{[0,T]}-\rm a.e.$ to $G$. Letting $X_{n}:=\int_{0}^{\cdot}G_{n}{\rm d}L$
and denoting the jump-measure of $X_{n}$ by $\mu^{X_{n}}$, we define for each
$k\in\mathcal{C}^{+}(H)$ and $n\in\mathbb{N}$ a process $M_{n}^{k}$ by

$\displaystyle\left(M_{n}^{k}(t):=\int_{0}^{t}\int_{H}k(h)\,\mu^{X_{n}}({\rm
d}s,{\rm d}h)-\int_{0}^{t}\int_{H}k(h)\,\left(\lambda\circ
G_{n}(s)^{-1}\right)\,{\rm d}h\,{\rm d}s,\,t\in[0,T]\right).$

Proposition 2.7 and Lemma 3.2 imply that $M_{n}^{k}$ is a local martingale for
all $n\in\mathbb{N}$. Since for each $n\in\mathbb{N}$ and $t\in[0,T]$ we have
that $\mu^{X_{n}}(\\{t\\}\times H)\leq 1$ almost surely, it follows that

$\displaystyle\left\lVert\Delta\left(\int_{0}^{t}\int_{H}k(h)\,\mu^{X_{n}}({\rm
d}h,{\rm d}s)\right)\right\rVert\leq\left\lVert
k\right\rVert_{\infty}\quad\text{a.s.},$

which shows $\left\lVert\Delta M_{n}^{k}(t)\right\rVert\leq\left\lVert
k\right\rVert_{\infty}$ a.s. for all $n\in\mathbb{N}$.

Almost sure uniform convergence of $X_{n}$ and Lemma 3.3 guarantee that there
exists an $\Omega_{1}\subseteq\Omega$ with $P(\Omega_{1})=1$ such that, for
all $\omega\in\Omega_{1}$, we have

$\displaystyle\lim_{n\rightarrow\infty}\sup_{t\in[0,T]}\left|\int_{0}^{t}\int_{H}k(h)\,\mu^{X_{n}(\omega)}({\rm
d}h,{\rm d}s)-\int_{0}^{t}\int_{H}k(h)\,\mu^{X(\omega)}({\rm d}h,{\rm
d}s)\right|=0.$ (3.5)

In the same way, by convergence of $G_{n}$ both in $L_{\rm
Leb}^{\alpha}([0,T],L_{2}(U,H))$ a.s. and $P\otimes{\rm Leb}|_{[0,T]}-\rm
a.e.$ and Lemma 3.4 there exists an $\Omega_{2}\subseteq\Omega$ with
$P(\Omega_{2})=1$ such that, for all $\omega\in\Omega_{2}$, we have

$\displaystyle\lim_{n\rightarrow\infty}\sup_{t\in[0,T]}\left|\int_{0}^{t}\int_{H}k(h)\,\left(\lambda\circ
G_{n}(\omega,s)^{-1}\right)\,{\rm d}h\,{\rm
d}s-\int_{0}^{t}\int_{H}k(h)\,\left(\lambda\circ G(\omega,s)^{-1}\right)\,{\rm
d}h\,{\rm d}s\right|=0.$ (3.6)

Equations (3.5) and (3.6) show that $M_{n}^{k}$ converges uniformly to $M^{k}$
almost surely. As the jumps of $M^{k}_{n}$ are a.s. uniformly bounded by
$\left\lVert k\right\rVert_{\infty}$, we conclude from [17, Co. IX.1.19] that
$M^{k}$ is a local martingale and the proof is complete. ∎

##  4 Quadratic variation of the integral process

The quadratic covariation of two real-valued càdlàg semimartingales $V_{1}$
and $V_{2}$ starting from zero is the process $[V_{1},V_{2}]$ defined by

$\displaystyle\left[V_{1},V_{2}\right](t):=V_{1}(t)V_{2}(t)-\int_{0}^{t}V_{1}(s-)\,{\rm
d}V_{2}(s)-\int_{0}^{t}V_{2}(s-)\,{\rm d}V_{1}(s),\quad t\in[0,T].$

When $V:=V_{1}=V_{2}$, we call the process $\left[V\right]:=\left[V,V\right]$
the quadratic variation of $V$. The continuous part of $\left[V\right]$ is
defined by

$\displaystyle\left[V\right]^{c}(t)=\left[V\right](t)-\sum_{0\leq s\leq
t}\left(\Delta V(s)\right)^{2}\quad\text{ for each $t\in[0,T]$.}$ (4.1)

If $\left[V\right]^{c}=0$ we say that $V$ is purely discontinuous; see e.g.
[32, Se. II.6].

The concept of quadratic variation is generalised for a càdlàg semimartingale
$Z$ with values in the separable Hilbert space $H$ in [26, Se. 26]. Let
$(f_{i})_{i\in\mathbb{N}}$ denote an orthonormal basis of $H$. There exists a
unique stochastic process $\left[\left[Z\right]\right]$ with values in the
Hilbert-Schmidt tensor product of $H$ satisfying

$\displaystyle\langle\left[\left[Z\right]\right],f_{i}\otimes
f_{j}\rangle=\left[Z_{i},Z_{j}\right]\quad\text{for all }i,j\in\mathbb{N},$

where $\otimes$ denotes the tensor product and $Z_{i}(t)=\langle
Z(t),f_{i}\rangle$ for $t\in[0,T]$ are the projection processes of $Z$; see
[26, Se. 21.2] for brief introduction. The process
$\left[\left[Z\right]\right]$ does not depend on the choice of the orthonormal
basis $(f_{i})_{i\in\mathbb{N}}$. The process $\left[\left[Z\right]\right]$ is
called the tensor quadratic variation of $Z$ and its continuous part
$\left[\left[Z\right]\right]^{c}$ is defined by

$\displaystyle\langle\left[\left[Z\right]\right]^{c}(t),f_{i}\otimes
f_{j}\rangle=\langle\left[\left[Z\right]\right](t),f_{i}\otimes
f_{j}\rangle-\sum_{0\leq s\leq
t}\Delta\left(Z^{i}(s)Z^{j}(s)\right)\quad\text{for all
}t\in[0,T],\,i,j\in\mathbb{N}.$

We say that $Z$ is purely discontinuous if
$\left[\left[Z\right]\right]^{c}$=0.

######  Proposition 4.1.

Let $L$ be an $\alpha$-stable cylindrical Lévy process for some
$\alpha\in(1,2)$ and $G$ a stochastically integrable predictable process with
values in ${\mathcal{L}}_{2}(U,H)$. Then the integral process
$X:=\int_{0}^{\cdot}G\,{\rm d}L$ is purely discontinuous.

###### Proof.

We proceed in three steps.
Step 1: Assume $H=\mathbb{R}$ and $U=\mathbb{R}^{d}$ for some
$d\in\mathbb{N}$. In this case, $L$ is a $U$-valued standard symmetric
$\alpha$-stable Lévy process, and therefore purely discontinuous; see e.g.
[32, p. 71]. Pure discontinuity is preserved also for the integral process;
see e.g. [17, Se. IX.5.5a] or [32, Th. II.29].
Step 2: Assume $H=\mathbb{R}$, but without any further restrictions on $U$. In
that case, by the identification $U\simeq\mathcal{L}_{2}(U,\mathbb{R})$, the
integrand $G$ is a $U$-valued process satisfying

$\displaystyle\int_{0}^{T}\left|\left|G(t)\right|\right|^{\alpha}{\rm
d}t<\infty\quad\text{a.s.}$ (4.2)

Fix an orthonormal basis $(f_{k})_{k\in\mathbb{N}}$ in $U$ and define for each
$n\in\mathbb{N}$ the projection

$\displaystyle\pi_{n}\colon U\to U,\qquad\pi_{n}(u)=\sum_{k=1}^{n}\langle
u,f_{k}\rangle f_{k}.$

Since the projection $\pi_{n}$ is a Hilbert-Schmidt operator, there exists a
$U$-valued Lévy process $L_{n}$ with the property $\langle
L_{n},u\rangle=L(\pi_{n}^{*}u)$ for all $u\in U$. We define the approximations

$\displaystyle\quad X_{n}:=\int_{0}^{\cdot}G{\rm d}L_{n},\quad
n\in\mathbb{N}.$

Since $L_{n}$ attains values in a finite-dimensional subspace and is a
symmetric $\alpha$-stable process by [33, Le. 2.4], it follows that $X_{n}$ is
purely discontinuous by Step 1.

Let $M$ be a real-valued, continuous martingale and define for
$k\in\mathbb{N}$ the stopping times

$\displaystyle\tau_{k}=\inf\left\\{t>0:\int_{0}^{t}\left|\left|G(s)\right|\right|^{\alpha}{\rm
d}s\geq k\right\\}\land\inf\left\\{t>0:|M(t)|\geq k\right\\}\land T.$

It follows that $\tau_{k}\to T$ as $k\to\infty$ by (4.2). Since $X_{n}$ is
purely discontinuous, it follows from [17, Le. I.4.14] that
$(X_{n}M)^{\tau_{k}}$ is a local martingale for each $k$, $n\in\mathbb{N}$.
Since applying inequality (2.7) and equality (2.9) shows

$\displaystyle\mathbb{E}\,\left[\sup_{0\leq t\leq
T}{|(X_{n}M)^{\tau_{k}}(t)|}\right]$
$\displaystyle=\mathbb{E}\,\left[\sup_{0\leq t\leq
T}\left|M(t)^{\tau_{k}}\right|\left|\int_{0}^{t}\mathbbm{1}_{[0,\tau_{k}]}G\,{\rm
d}L_{n}\right|\right]$ $\displaystyle\leq
k\mathbb{E}\,\left[\int_{0}^{\tau_{k}}\left|\left|G(s)\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}\,{\rm
d}s\right]\leq k^{2}<\infty,$

we obtain that $(X_{n}M)^{\tau_{k}}$ is a martingale by [32, Th. I:51].

Noting $\int_{0}^{\cdot}G\,{\rm d}L_{n}=\int_{0}^{\cdot}G\pi_{n}\,{\rm d}L$,
inequality (2.7) and equality (2.9) establish for each $t\geq 0$ that

$\displaystyle\mathbb{E}\,\left[{|(X_{n}M-XM)^{\tau_{k}}(t)|}\right]$
$\displaystyle\leq
k\mathbb{E}\,\left[\left|\int_{0}^{t}\mathbbm{1}_{[0,\tau_{k}]}G(\pi_{n}-I)\,{\rm
d}L\right|\right]$ $\displaystyle\leq
e_{1,\alpha}k\left(\mathbb{E}\,\left[\int_{0}^{T}\left|\left|G(s)(\pi_{n}-I)\right|\right|_{\rm\mathcal{L}_{2}(U,H)}^{\alpha}{\rm
d}s\right]\right)^{1/\alpha}\to 0\quad\text{as }n\to\infty.$

It follows that the process $(XM)^{\tau_{k}}$ as a limit of martingales is
itself a martingale. Since $X$ is a local martingale according to Lemma 2.3
and $M$ is an arbitrary real-valued continuous martingale, it follows from
[17, Le. I.4.14] that $X$ is purely discontinuous.
Step 3: For the general case, we fix an orthonormal basis
$(e_{i})_{i\in\mathbb{N}}$ in $H$ and choose any $i,j\in\mathbb{N}$. Since
$\langle X(t),e_{i}\rangle=\int_{0}^{t}{G^{*}e_{i}}\,{\rm d}L$ for every
$t\geq 0$, the polarisation formula for real-valued covariation shows

$\displaystyle\langle\left[\left[X\right]\right],e_{i}\otimes e_{j}\rangle$
$\displaystyle=\left[\int_{0}^{\cdot}G^{*}e_{i}\,{\rm
d}L,\int_{0}^{\cdot}G^{*}e_{j}\,{\rm d}L\right]$
$\displaystyle=\frac{1}{2}\left(\left[\int_{0}^{\cdot}G^{*}(e_{i}+e_{j})\,{\rm
d}L\right]-\left[\int_{0}^{\cdot}G^{*}e_{i}\,{\rm
d}L\right]-\left[\int_{0}^{\cdot}G^{*}e_{j}\,{\rm d}L\right]\right).$

Linearity of the integral and binomial formula enable us to conclude

$\displaystyle\sum_{0\leq s\leq t}$ $\displaystyle{\Delta\langle
X(s),e_{i}\rangle\langle X(s),e_{j}\rangle}$ $\displaystyle=\sum_{0\leq s\leq
t}{\Delta\left(\int_{0}^{s}G^{*}e_{i}\,{\rm
d}L\right)\left(\int_{0}^{s}G^{*}e_{j}\,{\rm d}L\right)}$
$\displaystyle=\sum_{0\leq s\leq
t}\frac{1}{2}\left({\Delta\left(\int_{0}^{s}G^{*}(e_{i}+e_{j})\,{\rm
d}L\right)^{2}-\Delta\left(\int_{0}^{s}G^{*}e_{i}\,{\rm
d}L\right)^{2}-\Delta\left(\int_{0}^{s}G^{*}e_{j}\,{\rm
d}L\right)^{2}}\right).$

The very definition (4.1) of the continuous part leads us to

$\displaystyle\langle\left[\left[X\right]\right]^{c},e_{i}\otimes
e_{j}\rangle$ $\displaystyle=\langle\left[\left[X\right]\right],e_{i}\otimes
e_{j}\rangle-\sum_{0\leq s\leq t}{\Delta\langle X(s),e_{i}\rangle\langle
X(s),e_{j}\rangle}$
$\displaystyle=\frac{1}{2}\left(\left[\int_{0}^{\cdot}G^{*}(e_{i}+e_{j})\,{\rm
d}L\right]^{c}-\left[\int_{0}^{\cdot}G^{*}e_{i}\,{\rm
d}L\right]^{c}-\left[\int_{0}^{\cdot}G^{*}e_{j}\,{\rm d}L\right]^{c}\right).$

Since Step 2 guarantees that the processes
$\int_{0}^{\cdot}G^{*}(e_{i}+e_{j})\,{\rm d}L$,
$\int_{0}^{\cdot}G^{*}e_{i}\,{\rm d}L$ and $\int_{0}^{\cdot}G^{*}e_{i}\,{\rm
d}L$ are purely discontinuous, it follows that
$\langle\left[\left[X\right]\right]^{c},e_{i}\otimes e_{j}\rangle=0$ for all
$i,j\in\mathbb{N}$ which completes the proof. ∎

##  5 Strong Itô formula

In this section, we establish an Itô formula for processes that are given by a
differential driven by a standard symmetric $\alpha$-stable cylindrical Lévy
process $L$ for $\alpha\in(1,2)$ and are of the form

$\displaystyle dX(t)=F(t)\,{\rm d}t+G(t)\,{\rm d}L(t)\quad\text{for
}t\in[0,T],$ (5.1)

where $F\colon\Omega\times[0,T]\rightarrow H$,
$G\colon\Omega\times[0,T]\rightarrow\mathcal{L}_{2}(U,H)$ are predictable and
satisfy

$\displaystyle\int_{0}^{T}\left\lVert F(t)\right\rVert+\left\lVert
G(t)\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha}\,{\rm
d}t<\infty\quad\text{a.s.}$ (5.2)

We denote by $\mathcal{C}_{b}^{2}(H)$ the space of continuous functions
$f\colon H\rightarrow\mathbb{R}$ having bounded first and second Fréchet
derivatives, which are denoted by $Df$ and $D^{2}f$, respectively.

######  Theorem 5.1.

Let $X$ be a stochastic process of the form (5.1). It follows for each
$f\in\mathcal{C}_{b}^{2}(H)$ and $t\in[0,T]$ that

$\displaystyle f(X(t))$ $\displaystyle=f(X(0))+\int_{0}^{t}\langle
Df(X(s-)),F(s)\rangle\,{\rm d}s+\int_{0}^{t}\langle
G(s)^{*}Df(X(s-)),\cdot\rangle\,{\rm d}L(s)+M_{f}(t)$
$\displaystyle+\int_{0}^{t}\int_{H}\Big{(}f(X(s-)+g)-f(X(s-))-\langle
Df(X(s-)),g\rangle\Big{)}\,\left(\lambda\circ G(s)^{-1}\right)({\rm d}g)\,{\rm
d}s,$

where $M_{f}:=\big{(}M_{f}(t):\,t\in[0,T]\big{)}$ is a local martingale
defined by

$\displaystyle
M_{f}(t):=\int_{0}^{t}\int_{H}\big{(}f(X({s-})+h)-f(X(s-))-\langle
Df(X(s-)),h\rangle\big{)}\,(\mu^{X}-\nu^{X})({\rm d}s,{\rm d}h).$

######  Lemma 5.2.

Let $\lambda$ be the cylindrical Lévy measure of an $\alpha$-stable
cylindrical Lévy process for $\alpha\in(1,2)$. Then we have for each
$f\in\mathcal{C}_{b}^{2}(H)$, $h\in H$, and $\Phi\in\mathcal{L}_{2}(U,H)$ that

$\int_{H}\big{|}f(h+g)-f(h)-\langle
Df(h),g\rangle\big{|}\,\left(\lambda\circ\Phi^{-1}\right)({\rm d}g)\\\ \leq
d_{\alpha}^{1}\left(2\left|\left|Df\right|\right|_{\infty}+\frac{1}{2}\left|\left|D^{2}f\right|\right|_{\infty}\right)\left\lVert\Phi\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha},$

where $d_{\alpha}^{1}$ is a constant depending only on $\alpha$ as defined in
Inequality (2.2).

###### Proof.

Taylor’s remainder theorem in the integral form, see [2, Th. 5.8], and
Inequality (2.2) imply

$\displaystyle\int_{\overline{B}_{H}}$
$\displaystyle{\big{|}f(h+g)-f(h)-\langle
Df(h),g\rangle\big{|}}\left(\lambda\circ\Phi^{-1}\right)({\rm d}g)$
$\displaystyle=\int_{\overline{B}_{H}}\left|\int_{0}^{1}\langle
D^{2}f(h+\theta g)g,g\rangle(1-\theta)\,{\rm
d}\theta\right|\,\left(\lambda\circ\Phi^{-1}\right)({\rm d}g)$
$\displaystyle\leq\left|\left|D^{2}f\right|\right|_{\infty}\int_{\overline{B}_{H}}\left({\int_{0}^{1}\left\lVert
g\right\rVert^{2}(1-\theta)\,{\rm
d}\theta}\right)\,\left(\lambda\circ\Phi^{-1}\right)({\rm d}g)$
$\displaystyle=\frac{1}{2}\left\lVert
D^{2}f\right\rVert_{\infty}\int_{\overline{B}_{H}}{\left\lVert
g\right\rVert^{2}}\,\left(\lambda\circ\Phi^{-1}\right)({\rm d}g)$
$\displaystyle\leq d_{\alpha}^{1}\frac{1}{2}\left\lVert
D^{2}f\right\rVert_{\infty}\left\lVert\Phi\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha}.$
(5.3)

Similarly, Taylor’s remainder theorem in the integral form and Inequality 2.2
show

$\displaystyle\int_{\overline{B}_{H}^{c}}{\left|f(h+g)-f(h)\right|}\,\left(\lambda\circ\Phi^{-1}\right)({\rm
d}g)$ $\displaystyle=\int_{\overline{B}_{H}^{c}}\left|\int_{0}^{1}{\langle
Df(h+\theta
g),g\rangle}d\theta\right|\,\left(\lambda\circ\Phi^{-1}\right)({\rm d}g)$
$\displaystyle\leq\left|\left|Df\right|\right|_{\infty}\int_{\overline{B}_{H}^{c}}\left(\int_{0}^{1}\left|\left|g\right|\right|\,{\rm
d}\theta\right)\,\left(\lambda\circ\Phi^{-1}\right)({\rm d}g)$
$\displaystyle\leq
d_{\alpha}^{1}\left|\left|Df\right|\right|_{\infty}\left|\left|\Phi\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}.$
(5.4)

Another application of Inequality 2.2 shows

$\displaystyle\int_{\overline{B}_{H}^{c}}{\left|\langle
Df(h),g\rangle\right|}\,\left(\lambda\circ\Phi^{-1}\right)({\rm d}g)$
$\displaystyle\leq\left|\left|Df\right|\right|_{\infty}\int_{\overline{B}_{H}^{c}}\left|\left|g\right|\right|\,\left(\lambda\circ\Phi^{-1}\right)({\rm
d}g)$ $\displaystyle\leq
d_{\alpha}^{1}\left|\left|Df\right|\right|_{\infty}\left|\left|\Phi\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}.$
(5.5)

Combining inequalities (5) to (5) completes the proof. ∎

###### Proof of Theorem 5.1.

The stochastic process $X$ given by (5.1) is purely discontinuous as it is the
sum of a finite-variation process and a purely discontinuous process according
to Proposition 4.1. The Itô formula in [26, Th. 27.2] takes for all
$t\in[0,T]$ the form

$\displaystyle{\rm d}f(X(t))$ $\displaystyle=\langle
Df(X(t-)),\cdot\rangle\,{\rm d}X(t)$
$\displaystyle\qquad+\int_{H}\big{(}f(X({t-})+h)-f(X(t-))-\langle
Df(X(t-)),h\rangle\big{)}\,\mu^{X}({\rm d}t,{\rm d}h).$ (5.6)

One can show by approximating with simple integrands that

$\displaystyle\langle Df(X(t-)),\cdot\rangle\,{\rm d}X(t)=\langle
Df(X(t-)),F(t)\rangle\,{\rm d}t+\langle G(t)^{*}Df(X(t-)),\cdot\rangle\,{\rm
d}L(t),$

where both integrals are well defined since (5.2) guarantees

$\displaystyle\int_{0}^{T}|\langle Df(X(t-)),F(t)\rangle|+$
$\displaystyle\left\lVert\langle
G(t)^{*}Df(X(t-)),\cdot\rangle\right\rVert_{\mathcal{L}_{2}(U,\mathbb{R})}^{\alpha}\,{\rm
d}t$ $\displaystyle\leq$ $\displaystyle\left\lVert
Df\right\rVert_{\infty}\int_{0}^{T}\left\lVert F(t)\right\rVert\,{\rm
d}t+\left\lVert Df\right\rVert_{\infty}^{\alpha}\int_{0}^{T}\left\lVert
G(t)\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha}\,{\rm
d}t<\infty\quad\text{a.s.}$

The definition of the compensator $\nu^{X}$ and Lemma 5.2 imply

$\displaystyle\mathbb{E}\,\left[\int_{0}^{T}\int_{H}\big{|}f(X({s-})+h)-f(X(s-))-\langle
Df(X(s-)),h\rangle\big{|}\,\mu^{X}({\rm d}s,{\rm d}h)\right]$ (5.7)
$\displaystyle\qquad=\mathbb{E}\,\left[\int_{0}^{T}\int_{H}\big{|}f(X({s-})+h)-f(X(s-))-\langle
Df(X(s-)),h\rangle\big{|}\,\nu^{X}({\rm d}s,{\rm d}h)\right]$
$\displaystyle\qquad\leq
d_{\alpha}^{1}\left(2\left|\left|Df\right|\right|_{\infty}+\frac{1}{2}\left|\left|D^{2}f\right|\right|_{\infty}\right)\mathbb{E}\,\left[\int_{0}^{T}\left\lVert
G(s)\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha}\,{\rm d}s\right].$

The stopping times
$\tau_{n}:=\inf\left\\{t>0:\int_{0}^{t}\left|\left|G(s)\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}\,{\rm
d}s\geq n\right\\}\wedge T$ satisfy $\tau_{n}\to T$ as $n\to\infty$ by (5.2).
Since inequality (5.7) guarantees for all $n\in\mathbb{N}$ that

$\displaystyle\mathbb{E}\,\left[\int_{0}^{T\land\tau_{n}}\right.\left.\int_{H}|f(X({s-})+h)-f(X(s-))-\langle
Df(X(s-)),h\rangle|\mu^{X}({\rm d}s,{\rm d}h)\right]<\infty,$

Proposition 2.7 shows that $M_{f}$ is a local martingale. This concludes the
proof, since the claimed formula is just a different form of (5.6). ∎

##  6 Mild solutions for stochastic evolution equations

We recall that $U$ and $H$ are separable Hilbert spaces with norms
$\left\lVert\cdot\right\rVert$ and $L$ is a standard symmetric $\alpha$-stable
cylindrical $(\mathcal{F}_{t})$-Lévy process in $U$ with $\alpha\in(1,2)$. In
this section we consider the mild solution of the stochastic evolution
equation:

$\displaystyle{\rm d}X(t)$ $\displaystyle=\big{(}AX(t)+F(X(t))\big{)}\,{\rm
d}t+G(X(t-))\,{\rm d}L(t)\qquad\text{for }t\in[0,T],$ (6.1) $\displaystyle
X(0)$ $\displaystyle=x_{0},$

where $A$ is a generator of a $C_{0}$-semigroup $\left(S(t)\right)_{t\geq 0}$
in $H$, $x_{0}$ is an $\mathcal{F}_{0}$-measurable $H$-valued random variable,
$F\colon H\to H$ and $G\colon H\to\mathcal{L}_{2}(U,H)$ are measurable
mappings and $T>0$.

######  Definition 6.1.

An $H$-valued predictable process $X$ is a mild solution to (6.1) if

$\displaystyle X(t)=S(t)x_{0}+\int_{0}^{t}S(t-s)F(X(s)){\rm
d}s+\int_{0}^{t}S(t-s)G(X(s-)){\rm d}L(s)\quad\text{for every }t\in[0,T].$

We work under the following assumptions:

  * (A1)

The $C_{0}$-semigroup $\left(S(t)\right)_{t\geq 0}$ is compact, analytic and a
semigroup of contractions and $0$ is an element of the resolvent set of $A$.

  * (A2)

The mapping $F$ is Lipschitz and bounded, i.e. there exists
$K_{F}\in(0,\infty)$ such that

$\displaystyle\left\lVert F(h_{1})-F(h_{2})\right\rVert\leq K_{F}\left\lVert
h_{1}-h_{2}\right\rVert,\quad\left\lVert F(h)\right\rVert\leq K_{F}$ (6.2)

for every $h_{1},h_{2},h\in H$.

  * (A3)

The mapping $G$ is Lipschitz and bounded, i.e. there exists
$K_{G}\in(0,\infty)$ such that

$\displaystyle\left\lVert
G(h_{1})-G(h_{2})\right\rVert_{\mathcal{L}_{2}(U,H)}\leq K_{G}\left\lVert
h_{1}-h_{2}\right\rVert,\quad\left\lVert
G(h)\right\rVert_{\mathcal{L}_{2}(U,H)}\leq K_{G}$ (6.3)

for every $h_{1},h_{2},h\in H$.

  * (A4)

The initial condition $x_{0}$ has finite $p$-th moment for every $p<\alpha$.

######  Remark 6.2.

We shall use the notation $D^{\delta}:=\text{Dom}((-A)^{\delta})$ for the
domain of the fractional generator $(-A)^{\delta}$ for $\delta\in[0,1]$, and
equip $D^{\delta}$ with the norm
$\left|\left|h\right|\right|_{\delta}:=\left|\left|(-A)^{\delta}h\right|\right|$.
It follows from Assumption (A1) that the embedding of Hilbert spaces
$D^{\delta}\hookrightarrow D^{\gamma}$ is dense and compact for every
$0\leq\gamma<\delta\leq 1$, cf. [4, Cor. 3.8.2].

######  Remark 6.3.

Assumption (A1) implies, cf. [22, p. 289], that for every $\delta\geq 0$ there
exists a $c_{\delta}\in(0,\infty)$ depending only on $\delta$ such that

$\displaystyle\left\lVert S(t)\right\rVert_{\mathcal{L}(H,D^{\delta})}\leq
c_{\delta}t^{-\delta}\qquad\text{for every $t>0$.}$ (6.4)

######  Remark 6.4.

By considering the cases $\left\lVert h_{1}-h_{2}\right\rVert\leq 1$ and
$\left\lVert h_{1}-h_{2}\right\rVert>1$ separately, we conclude from
Assumptions (A2) and (A3) that there exist $K_{F},K_{G}\in(0,\infty)$ such
that for any $\beta\in(0,1)$ we have

$\displaystyle\left\lVert F(h_{1})-F(h_{2})\right\rVert\leq K_{F}\left\lVert
h_{1}-h_{2}\right\rVert^{\beta},\quad\left\lVert F(h)\right\rVert\leq K_{F},$
(6.5)

and

$\displaystyle\left\lVert
G(h_{1})-G(h_{2})\right\rVert_{\mathcal{L}_{2}(U,H)}\leq K_{G}\left\lVert
h_{1}-h_{2}\right\rVert^{\beta},\quad\left\lVert
G(h)\right\rVert_{\mathcal{L}_{2}(U,H)}\leq K_{G},$ (6.6)

for every $h_{1},h_{2},h\in H$.

The first main theorem of this article is the following existence result,
which also includes properties on the path regularity of the solution.

######  Theorem 6.5.

Under the assumptions (A1)-(A4), there exists a mild solution $X$ to (6.1).
The mild solution $X$ is an element of $\mathcal{C}([0,T],L^{p}(\Omega,H))$
for every $p<\alpha$ and has càdlàg paths in $H$.

We will obtain the solution to (6.1) by using the Yosida approximations. For
this purpose, we define $R_{n}=n\left(n{\rm I}-A\right)^{-1}$ for
$n\in\mathbb{N}$ and denote by $X_{n}$ the mild solution to

$\displaystyle{\rm d}X_{n}(t)$
$\displaystyle=\big{(}AX_{n}(t)+R_{n}F(X_{n}(t))\big{)}\,{\rm
d}t+R_{n}G(X_{n}(t-))\,{\rm d}L(t),$ (6.7) $\displaystyle X_{n}(0)$
$\displaystyle=R_{n}x_{0}.$

Existence of the mild solution $X_{n}$ to (6.7) with cádlág paths is
guaranteed by [23, Th. 12].

######  Remark 6.6.

We recall that under Assumption (A1) we have for all $\delta\in[0,1]$ that

$\displaystyle\left\lVert R_{n}\right\rVert_{\mathcal{L}(D^{\delta})}\leq
1,\quad n\in\mathbb{N}.$

This follows from the fact, that if an operator commutes with $A$ then it
commutes with $A^{\gamma}$, see e.g. [15, Pr. 3.1.1], which enables us to
conclude for every $n\in\mathbb{N}$ that

$\displaystyle\left\lVert
R_{n}\right\rVert_{\mathcal{L}(D^{\gamma})}=\sup_{\left\lVert(-A)^{\gamma}h\right\rVert\leq
1}\left\lVert n(n-A)^{-1}(-A)^{\gamma}h\right\rVert\leq\sup_{\left\lVert
h\right\rVert\leq 1}\left\lVert n(n-A)^{-1}h\right\rVert=\left\lVert
R_{n}\right\rVert_{\mathcal{L}(H)}.$

Since $(S(t))_{t\geq 0}$ is a contraction semigroup, Theorem 3.5 in [13]
guarantees $\left\lVert R_{n}\right\rVert_{\mathcal{L}(H)}\leq 1$ for all
$n\in{\mathbb{N}}$.

The solution to (6.1) will be constructed as a limit of $X_{n}$ in
$\mathcal{C}([0,T],L^{p}(\Omega,H))$ for an arbitrary but fixed $p<\alpha$. In
the first three Lemmata, we establish relative compactness of the Yosida
approximation $\\{X_{n}:n\in\mathbb{N}\\}$ in the space
$\mathcal{C}([0,T],L^{p}(\Omega,H))$.

######  Lemma 6.7.

The set $\\{X_{n}(t):n\in\mathbb{N}\\}$ is tight in $H$ for every $t\in[0,T]$.

###### Proof.

The case $t=0$ follows immediately from the strong convergence of $R_{n}$. For
the case $t\in(0,T]$ we first prove that for every $1\leq q<\alpha$, and
$0\leq\delta<1/\alpha$ we have

$\displaystyle\sup_{n\in\mathbb{N}}\mathbb{E}\,\left[\left\lVert
X_{n}(t)\right\rVert_{\delta}^{q}\right]<\infty.$ (6.8)

Applying Hölder’s inequality and inequality (2.7) shows for every
$n\in\mathbb{N}$ that

$\displaystyle\mathbb{E}\,\left[\left\lVert
X_{n}(t)\right\rVert_{\delta}^{q}\right]$ $\displaystyle\leq
3^{q-1}\left(\mathbb{E}\,\left[\left\lVert
S(t)R_{n}x_{0}\right\rVert_{\delta}^{q}\right]+t^{q-1}\mathbb{E}\,\left[\int_{0}^{t}\left\lVert
S(t-s)R_{n}F(X_{n}(s))\right\rVert_{\delta}^{q}\,{\rm d}s\right]\right.$
$\displaystyle\qquad\qquad\left.+e_{q,\alpha}\left(\mathbb{E}\,\left[\int_{0}^{t}\left\lVert
S(t-s)R_{n}G(X_{n}(s))\right\rVert_{\mathcal{L}_{2}(U,D^{\delta})}^{\alpha}\,{\rm
d}s\right]\right)^{\frac{q}{\alpha}}\right).$

Commutativity of $S$ and $R_{n}$, Remark 6.3 and Remark 6.6 verify

$\displaystyle\mathbb{E}\,\left[\left\lVert
S(t)R_{n}x_{0}\right\rVert_{\delta}^{q}\right]\leq
c_{\delta}^{q}t^{-q\delta}\sup_{n\in\mathbb{N}}\left\lVert
R_{n}\right\rVert_{\mathcal{L}(D^{\delta})}^{q}\mathbb{E}\,\left[\left\lVert
x_{0}\right\rVert^{q}\right]<\infty.$

Assumption (A2) on boundedness of $F$ together with Remark 6.3 and Remark 6.6
yield

$\displaystyle\mathbb{E}\,\left[\int_{0}^{t}\left\lVert
S(t-s)R_{n}F(X_{n}(s))\right\rVert_{\delta}^{q}\,{\rm d}s\right]\leq
c_{\delta}^{q}\frac{t^{1-q\delta}}{1-q\delta}\sup_{n\in\mathbb{N}}\left\lVert
R_{n}\right\rVert_{\mathcal{L}(D^{\delta})}^{q}K_{F}^{q}<\infty.$

Similarly, Assumption (A3) on boundedness of G implies

$\displaystyle\left(\mathbb{E}\,\left[\int_{0}^{t}\left\lVert
S(t-s)R_{n}G(X_{n}(s))\right\rVert_{\mathcal{L}_{2}(U,D^{\gamma})}^{\alpha}\,{\rm
d}s\right]\right)^{\frac{q}{\alpha}}\leq
c_{\delta}^{q}\left(\frac{t^{1-\alpha\delta}}{1-\alpha\delta}\right)^{\frac{q}{\alpha}}\sup_{n\in\mathbb{N}}\left\lVert
R_{n}\right\rVert_{\mathcal{L}(D^{\delta})}^{q}K_{G}^{q}<\infty.$

Combining the above estimates establishes (6.8), which in turn gives the
statement of the Lemma. Indeed, choose any $\delta\in(0,1/\alpha)$ and use
Markov’s inequality and (6.8) for $q=1$ to obtain for each $N>0$ that

$\sup_{n\in\mathbb{N}}P\left(\left\lVert
X_{n}(t)\right\rVert_{\delta}>N\right)\leq\frac{c}{N}$

for some constant $c\in(0,\infty)$. Since the embedding
$D^{\delta}\hookrightarrow H$ is compact according to Remark 6.2, we obtain
tightness of $\\{X_{n}(t):n\in\mathbb{N}\\}$ by Prokhorov’s theorem. ∎

######  Lemma 6.8.

The sequence $\\{X_{n}(t):n\in\mathbb{N}\\}$ is relatively compact in
$L_{P}^{0}(\Omega,H)$ for every $t\in[0,T]$.

###### Proof.

First, we impose the additional assumption that

$\displaystyle
T12^{2}e_{2,\alpha}\left(K_{F}^{\alpha}+K_{G}^{\alpha}\right)<1,$ (6.9)

where $K_{F},K_{G}$ come from (6.5), (6.6) and $e_{2,\alpha}$ is defined just
below (2.7). For $1<p<\alpha$ and $m,n\in\mathbb{N}$ we estimate the $p$-th
moment of the difference $X_{m}(t)-X_{n}(t)$ by

$\begin{split}\mathbb{E}\,\left[\left\lVert
X_{m}(t)-X_{n}(t)\right\rVert^{p}\right]\leq&6^{p-1}\Bigg{(}\mathbb{E}\,\left[\left\lVert
S(t)(R_{m}-R_{n})x_{0}\right\rVert^{p}\right]\\\
&+\mathbb{E}\,\left[\left\lVert\int_{0}^{t}S(t-s)(R_{m}-R_{n})F(X_{m}(s)){\rm
d}s\right\rVert^{p}\right]\\\
&+\mathbb{E}\,\left[\left\lVert\int_{0}^{t}S(t-s)R_{n}(F(X_{m}(s))-F(X_{n}(s))){\rm
d}s\right\rVert^{p}\right]\\\
&+\mathbb{E}\,\left[\left\lVert\int_{0}^{t}S(t-s)(R_{m}-R_{n})G(X_{m}(s-)){\rm
d}L(s)\right\rVert^{p}\right]\\\
&+\mathbb{E}\,\left[\left\lVert\int_{0}^{t}S(t-s)R_{n}(G(X_{m}(s-))-G(X_{n}(s-))){\rm
d}L(s)\right\rVert^{p}\right]\Bigg{)}.\end{split}$ (6.10)

Applying Hölder’s inequality, Jensen’s inequality, inequality (2.7), Remark
6.6, (A1) and estimates (6.5) and (6.6) we obtain

$\displaystyle\mathbb{E}\,\left[\left\lVert\int_{0}^{t}S(t-s)R_{n}(F(X_{m}(s))-F(X_{n}(s))){\rm
d}s\right\rVert^{p}\right.\left.\vphantom{\int_{0}^{t}}\right]$
$\displaystyle\qquad\leq
T^{p-\frac{p}{\alpha}}\left(\mathbb{E}\,\left[\int_{0}^{t}\left\lVert
S(t-s)R_{n}(F(X_{m}(s))-F(X_{n}(s)))\right\rVert^{\alpha}{\rm
d}s\right]\right)^{\frac{p}{\alpha}}$ $\displaystyle\qquad\leq
T^{p-\frac{p}{\alpha}}K_{F}^{p}\left(\mathbb{E}\,\left[\int_{0}^{t}\left\lVert
X_{m}(s)-X_{n}(s)\right\rVert^{p}{\rm d}s\right]\right)^{\frac{p}{\alpha}},$

and, similarly,

$\displaystyle\mathbb{E}\,\left[\left\lVert\int_{0}^{t}S(t-s)R_{n}(G(X_{m}(s-))-G(X_{n}(s-))){\rm
d}L(s)\right\rVert^{p}\right.\left.\vphantom{\int_{0}^{t}}\right]$
$\displaystyle\qquad\leq
e_{p,\alpha}K_{G}^{p}\left(\mathbb{E}\,\left[\int_{0}^{t}\left\lVert
X_{m}(s)-X_{n}(s)\right\rVert^{p}{\rm d}s\right]\right)^{\frac{p}{\alpha}}.$

Raising both sides of (6.10) to the power of $\alpha/p$ shows for each
$t\in[0,T]$ that

$\displaystyle u_{n,m,p}(t)\leq
u_{n,m,p}^{0}+w_{p}\int_{0}^{t}\left(u_{n,m,p}(s)\right)^{\frac{p}{\alpha}}{\rm
d}s,$ (6.11)

where we use

$\displaystyle u_{n,m,p}(t)$
$\displaystyle:=\left(\mathbb{E}\,\left[\left\lVert
X_{m}(t)-X_{n}(t)\right\rVert^{p}\right]\right)^{\frac{\alpha}{p}},$
$\displaystyle u_{n,m,p}^{0}$
$\displaystyle:=5^{\frac{\alpha}{p}-1}6^{\frac{\alpha}{p}(p-1)}\Bigg{(}\sup_{t\in[0,T]}\left\lVert
S(t)\right\rVert_{\mathcal{L}(H)}^{\alpha}\left(\mathbb{E}\,\left[\left\lVert(R_{m}-R_{n})x_{0}\right\rVert^{p}\right]\right)^{\frac{\alpha}{p}}$
$\displaystyle\hphantom{hhhhhhhhhhhhhh}+\left(\sup_{t\in[0,T]}\mathbb{E}\,\left[\left\lVert\int_{0}^{t}S(t-s)(R_{m}-R_{n})F(X_{m}(s)){\rm
d}s\right\rVert^{p}\right]\right)^{\frac{\alpha}{p}}$
$\displaystyle\hphantom{hhhhhhhhhhhhhh}+\left(\sup_{t\in[0,T]}\mathbb{E}\,\left[\left\lVert\int_{0}^{t}S(t-s)(R_{m}-R_{n})G(X_{m}(s-)){\rm
d}L(s)\right\rVert^{p}\right]\right)^{\frac{\alpha}{p}}\Bigg{)},$
$\displaystyle w_{p}$
$\displaystyle:=5^{\frac{\alpha}{p}-1}6^{\frac{\alpha}{p}(p-1)}\left(T^{\alpha-1}K_{F}^{\alpha}+e_{p,\alpha}^{\frac{\alpha}{p}}K_{G}^{\alpha}\right).$

It follows from (6.11) by a version of Gronwall’s inequality [40, Th. 2] that

$\displaystyle u_{n,m,p}(t)$ $\displaystyle\leq
2^{\frac{\alpha}{\alpha-p}}u_{n,m,p}^{0}+\left(2\frac{\alpha-p}{\alpha}tw_{p}\right)^{\frac{\alpha}{\alpha-p}}.$
(6.12)

If we show for each $t\in[0,T]$ that

$\displaystyle\lim_{p\to\alpha-}\left(2\frac{\alpha-p}{\alpha}tw_{p}\right)^{\frac{\alpha}{\alpha-p}}=0,$
(6.13)

and, for any $1<p<\alpha$,

$\displaystyle\lim_{m,n\to\infty}u_{n,m,p}^{0}=0,$ (6.14)

then, for each $\epsilon\in(0,1)$, we can find some $p^{*}\in(1,\alpha)$ such
that
$2^{\frac{\alpha}{\alpha-p^{*}}}\left(\frac{\alpha-p^{*}}{\alpha}tw_{p^{*}}\right)^{\frac{\alpha}{\alpha-p^{*}}}<\frac{\epsilon^{\alpha+1}}{2},$
and some $N\in\mathbb{N}$ such that for all $m,n\geq N$
$2^{\frac{\alpha}{\alpha-p^{*}}}u_{n,m,p^{*}}^{0}\leq\frac{\epsilon^{\alpha+1}}{2}.$
This completes the proof since we obtain for any $m,n\geq N$ that

$\displaystyle P\left(\left\lVert
X_{m}(t)-X_{n}(t)\right\rVert\geq\epsilon\right)\leq\frac{1}{\epsilon^{p^{*}}}\mathbb{E}\,\left[\left\lVert
X_{m}(t)-X_{n}(t)\right\rVert^{p^{*}}\right]\leq\epsilon^{\frac{p^{*}}{\alpha}}.$

It remains to establish (6.13) and (6.14), which we first do under the
additional assumption (6.9):

Argument for (6.13): Since
$e_{p,\alpha}=\frac{\alpha}{\alpha-p}e_{2,\alpha}^{p/\alpha}$ for a constant
$e_{2,\alpha}\in(0,\infty)$ independent of $\alpha$, we obtain
$T^{\alpha-1}\leq e_{p,\alpha}^{\frac{\alpha}{p}}$ for $p$ sufficiently close
to $\alpha$. In this case, we obtain

$\displaystyle\left(2\frac{\alpha-p}{p}tw_{p}\right)^{\frac{\alpha}{\alpha-p}}$
$\displaystyle\leq
5^{\frac{\alpha}{p}}\left(2\frac{\alpha-p}{\alpha}t6^{2}\left(\frac{\alpha-p}{\alpha}\right)^{-\frac{\alpha}{p}}e_{2,\alpha}\left(K_{F}^{\alpha}+K_{G}^{\alpha}\right)\right)^{\frac{\alpha}{\alpha-p}}$
$\displaystyle\leq
5^{\alpha}\left(\frac{\alpha-p}{\alpha}\right)^{-2}\left(T12^{2}e_{2,\alpha}\left(K_{F}^{\alpha}+K_{G}^{\alpha}\right)\right)^{\frac{\alpha}{\alpha-p}}.$

Since we assume (6.9), applying L’Hospital’s rule verifies (6.13).

Argument for (6.14): Let $p\in(1,\alpha)$ be fixed. By strong convergence of
$R_{n}$ and Lebesgue’s dominated convergence theorem we have

$\displaystyle\lim_{m,n\to\infty}\mathbb{E}\,\left[\left\lVert(R_{m}-R_{n})x_{0}\right\rVert^{p}\right]=0.$
(6.15)

Hölder’s inequality, strong convergence of $R_{n}$, Lemma 9.1 and Lebesgue’s
dominated convergence theorem we have

$\begin{split}\lim_{m,n\to\infty}&\left(\sup_{t\in[0,T]}\mathbb{E}\,\left[\left\lVert\int_{0}^{t}S(t-s)(R_{m}-R_{n})F(X_{m}(s)){\rm
d}s\right\rVert^{p}\right]\right)\\\ \leq&T^{p-1}\sup_{t\in[0,T]}\left\lVert
S(t)\right\rVert_{\mathcal{L}(H)}^{p}\lim_{m,n\to\infty}\left(\mathbb{E}\,\left[\int_{0}^{T}\left\lVert(R_{m}-R_{n})F(X_{m}(s))\right\rVert^{p}{\rm
d}s\right]\right)=0,\end{split}$ (6.16)

where the assumptions of Lemma 9.1 are satisfied by boundedness of $F$, see
(6.2), and tightness of $\\{F(X_{n}(s)),m\in\mathbb{N}\\}$ implied by Lemma
6.7 and continuity of $F$. Similarly, using inequality (2.7) and (6.6), we
obtain

$\begin{split}\lim_{m,n\to\infty}&\left(\sup_{t\in[0,T]}\mathbb{E}\,\left[\left\lVert\int_{0}^{t}S(t-s)(R_{m}-R_{n})G(X_{m}(s-)){\rm
d}L(s)\right\rVert^{p}\right]\right)\\\
\leq&e_{p,\alpha}\sup_{t\in[0,T]}\left\lVert
S(t)\right\rVert_{\mathcal{L}(H)}^{p}\lim_{m,n\to\infty}\left(\mathbb{E}\,\left[\int_{0}^{T}\left\lVert(R_{m}-R_{n})G(X_{m}(s))\right\rVert^{\alpha}{\rm
d}s\right]\right)^{\frac{p}{\alpha}}=0,\end{split}$ (6.17)

which establishes (6.14).

Thus, we have proved the lemma under the additional assumption (6.9). For
establishing the general case, we fix a time
$T_{0}\leq(12^{2}e_{2,\alpha}\left(K_{F}^{\alpha}+K_{G}^{\alpha}\right))^{-1}$.
If $t\in[0,T_{0}]$, relative compactness in $L_{P}^{0}(\Omega,H)$ follows from
the previous arguments. If $t\in(T_{0},2T_{0}]$ we write

$\displaystyle X_{m}(t)-X_{n}(t)$
$\displaystyle=S\left(t-T_{0}\right)\left(X_{m}\left(T_{0}\right)-X_{n}\left(T_{0}\right)\right)$
$\displaystyle\qquad+\int_{T_{0}}^{t}S(t-s)(R_{m}F(X_{m}(s)-R_{n}F(X_{n}(s)))\,{\rm
d}s$
$\displaystyle\qquad+\int_{T_{0}}^{t}S(t-s)(R_{m}G(X_{m}(s-)-R_{n}G(X_{n}(s-)))\,{\rm
d}L(s).$

Since our choice of $T_{0}$ implies that
$\\{S(t-T_{0})X_{n}(T_{0}),\,n\in\mathbb{N}\\}$ is relatively compact in
$L_{P}^{0}(\Omega,H)$, and that
$\left(t-T_{0}\right)12^{2}e_{2,\alpha}\left(K_{F}^{\alpha}+K_{G}^{\alpha}\right)<1$,
we can use the same argument as before to obtain relative compactness of
$(X_{n}(t))_{n\in\mathbbm{N}}$ in $L_{P}^{0}(\Omega,H)$ for each
$t\in(T_{0},2T_{0}]$. Using a standard induction argument, we can now cover
intervals of arbitrary length. This concludes the proof of the general case. ∎

We now step from relative compactness of $\\{X_{n}(t):n\in\mathbb{N}\\}$ in
$L_{P}^{0}(\Omega,H)$ for fixed time $t$ to relative compactness of the
processes $\\{X_{n}:n\in\mathbb{N}\\}$ using the Arzelà–Ascoli Theorem.

######  Lemma 6.9.

The collection $\\{X_{n}:n\in\mathbb{N}\\}$ is relatively compact in
$\mathcal{C}([0,T],L^{p}(\Omega,H))$ for any $0<p<\alpha$.

###### Proof.

We consider the case $1<p<\alpha$ as the case $p\leq 1$ follows from the fact
that relative compactness in $\mathcal{C}([0,T],L^{p}(\Omega,H))$ implies
relative compactness in $\mathcal{C}([0,T],L^{p^{\prime}}(\Omega,H))$ for
$p>p^{\prime}$. In light of the Arzelà–Ascoli Theorem, cf. e.g. [21, Th.
7.17]), it suffices to show that

  1. (a)

$\\{X_{n}(t):n\in\mathbb{N}\\}\subset L^{p}(\Omega,H)$ is relatively compact
for each $t\in[0,T]$;

  2. (b)

$\\{X_{n}:n\in\mathbb{N}\\}\subset\mathcal{C}([0,T],L^{p}(\Omega,H))$ is
equicontinuous.

The claim in (a) follows from [12, Cor. 3.3] by Lemmata 6.7, 6.8 and the fact
that Equation (6.8) with $\delta=0$ and any $q\in(p,\alpha)$ implies via the
Vallee-Poussin Theorem [11, Th. II.22] that the collection
$\\{X_{n}(t):n\in\mathbb{N}\\}$ is $p$-uniformly integrable and bounded in
$L^{p}(\Omega,H)$. Hence, it remains only to prove (b). To that end, we take
$t\in[0,T]$ and $h\in(0,T-t]$, and estimate

$\displaystyle\left\lVert X_{n}(t+h)-X_{n}(t)\right\rVert^{p}$
$\displaystyle\leq
5^{p-1}\Bigg{(}\left\lVert\left(S(h)-I\right)S(t)R_{n}x_{0}\right\rVert^{p}+\left\lVert\int_{t}^{t+h}S(t+h-s)R_{n}F(X_{n}(s))\,{\rm
d}s\right\rVert^{p}$
$\displaystyle\quad+\left\lVert\int_{t}^{t+h}S(t+h-s)R_{n}G(X_{n}(s-))\,{\rm
d}L(s)\right\rVert^{p}+\left\lVert\int_{0}^{t}\left(S(h)-I\right)S(t-s)R_{n}F(X_{n}(s))\,{\rm
d}s\right\rVert^{p}$
$\displaystyle\quad+\left\lVert\int_{0}^{t}\left(S(h)-I\right)S(t-s)R_{n}G(X_{n}(s-))\,{\rm
d}L(s)\right\rVert^{p}\Bigg{)}.$ (6.18)

Commutativity of $R_{n}$ and $S$ and contractivity of $S$ implies

$E\left[\left\lVert\left(S(h)-I\right)S(t)R_{n}x_{0}\right\rVert^{p}\right]\leq\sup_{n\in\mathbb{N}}\left\lVert
R_{n}\right\rVert_{\mathcal{L}(H)}^{p}\mathbb{E}\,\left[\left\lVert\left(S(h)-I\right)x_{0}\right\rVert^{p}\right].$
(6.19)

Applying Hölder’s inequality, boundedness of $F$ in Assumption (A2) and
contractivity of $S$ we get

$\mathbb{E}\,\left[\left\lVert\int_{t}^{t+h}S(t+h-s)R_{n}F(X_{n}(s))\,{\rm
d}s\right\rVert^{p}\right]\leq h^{p}\sup_{n\in\mathbb{N}}\left\lVert
R_{n}\right\rVert_{\mathcal{L}(H)}^{p}K_{F}^{p}.$ (6.20)

We conclude from Inequality (2.7) by using boundedness of $G$ in Assumption
(A3) and contractivity of $S$ that

$\displaystyle\mathbb{E}\,\left[\left\lVert\int_{t}^{t+h}S(t+h-s)R_{n}G(X_{n}(s-))\,{\rm
d}L(s)\right\rVert^{p}\right]$ (6.21) $\displaystyle\qquad\leq
e_{p,\alpha}\left(\mathbb{E}\,\left[\int_{t}^{t+h}\left\lVert
S(t+h-s)G(X_{n}(s))\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha}\,{\rm
d}s\right]\right)^{p/\alpha}\leq e_{p,\alpha}\sup_{n\in\mathbb{N}}\left\lVert
R_{n}\right\rVert_{\mathcal{L}(H)}^{p}K_{G}^{p}h^{p/\alpha}.$

It follows from Lemma 6.7 that $\\{X_{n}(s):\,n\in\mathbb{N}\\}$ is tight in
$H$ for every $s\in[0,t]$. Lemma 9.1 implies

$\displaystyle\lim_{h\searrow
0}\sup_{n\in\mathbb{N}}\mathbb{E}\,\left[\left\lVert\left(S(h)-I\right)S(t-s)R_{n}F(X_{n}(s))\right\rVert^{p}\right]=0.$

Lebesgue’s dominated convergence theorem shows

$\displaystyle\lim_{h\searrow
0}\int_{0}^{t}\sup_{n\in\mathbb{N}}\mathbb{E}\,\left[\left\lVert\left(S(h)-I\right)S(t-s)R_{n}F(X_{n}(s))\right\rVert^{p}\right]\,{\rm
d}s=0.$ (6.22)

In the same way, after applying Inequality (2.7), we obtain from Lemma 9.1

$\lim_{h\searrow
0}\sup_{n\in\mathbb{N}}\mathbb{E}\,\left[\left\lVert\int_{0}^{t}\left(S(h)-I\right)S(t-s)R_{n}G(X_{n}(s-))\,{\rm
d}L(s)\right\rVert^{p}\right]=0.$ (6.23)

Applying (6.19) - (6.23) to Inequality (6) shows uniform continuity from the
right. Similar arguments establish uniform continuity from the left, which
proves (b), and thus completes the proof.

∎

###### Proof of Theorem 6.5.

It is enough to consider the case $p\geq\alpha\beta$ where $\beta$ is the
Hölder exponent from (6.6). Lemma 6.9 guarantees that there is a subsequence
$(n_{k})_{k=1}^{\infty}$ such that

$\displaystyle\lim_{k\rightarrow\infty}\sup_{t\in[0,T]}\mathbb{E}\,\left[\left|\left|X_{n_{k}}(t)-Z(t)\right|\right|^{p}\right]=0$
(6.24)

for some $Z\in\mathcal{C}([0,T],L^{p}(\Omega,H))$. The proof will be complete
if we show that $Z$ is a mild solution to (6.1). We conclude for each
$k\in\mathbb{N}$ and $t\in[0,T]$ from Lipschitz continuity of $F$ and Hölder
continuity of $G$ in (6.2) and (6.6) and contractivity of $S$ by applying
Hölder’s inequality and Inequality (2.7) that

$\displaystyle\mathbb{E}\,$
$\displaystyle\left[\left|\left|Z(t)-S(t)x_{0}-\int_{0}^{t}S(t-s)F(Z(s))\,{\rm
d}s-\int_{0}^{t}S(t-s)G(Z(s-))\,{\rm d}L(s)\right|\right|^{p}\right]$
$\displaystyle\leq(1\wedge
3^{p-1})\Bigg{(}\mathbb{E}\,\left[\left|\left|Z(t)-X_{n_{k}}(t)\right|\right|^{p}\right]+\mathbb{E}\,\left[\left|\left|\int_{0}^{t}S(t-s)\big{(}F(Z(s))-F(X_{n_{k}}(s))\big{)}\,{\rm
d}s\right|\right|^{p}\right]$
$\displaystyle\hphantom{PHANTOMMM}+\mathbb{E}\,\left[\left|\left|\int_{0}^{t}S(t-s)\big{(}G(Z(s-))-G(X_{n_{k}}(s-))\big{)}\,{\rm
d}L(s)\right|\right|^{p}\right]\Bigg{)}$ $\displaystyle\leq(1\wedge
3^{p-1})\Bigg{(}\mathbb{E}\,\left[\left|\left|Z(t)-X_{n_{k}}(t)\right|\right|^{p}\right]+T^{p-1}\mathbb{E}\,\left[\int_{0}^{t}\left|\left|S(t-s)\big{(}F(Z(s))-F(X_{n_{k}}(s))\big{)}\right|\right|^{p}\,{\rm
d}s\right]$
$\displaystyle\hphantom{PHANTOMMM}+\left.e_{p,\alpha}\left(\mathbb{E}\,\left[\int_{0}^{t}\left|\left|S(t-s)\big{(}G(Z(s))-G(X_{n_{k}}(s))\big{)}\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}\,{\rm
d}s\right]\right)^{p/\alpha}\right)$ $\displaystyle\leq(1\wedge
3^{p-1})\Bigg{(}\mathbb{E}\,\left[\left\lVert
Z(t)-X_{n_{k}}(t)\right\rVert^{p}\right]+T^{p-1}K_{F}^{p}\sup_{t\in[0,T]}\left\lVert
S(t)\right\rVert_{\mathcal{L}(H)}^{p}\mathbb{E}\,\left[\int_{0}^{t}\left\lVert
Z(s)-X_{n_{k}}(s)\right\rVert^{p}\,{\rm d}s\right]$
$\displaystyle\hphantom{PHANTOMMM}+\left.e_{p,\alpha}K_{G}^{p}\sup_{t\in[0,T]}\left\lVert
S(t)\right\rVert_{\mathcal{L}(H)}^{p}\left(\mathbb{E}\,\left[\int_{0}^{t}\left\lVert
Z(s)-X_{n_{k}}(s)\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha\beta}\,{\rm
d}s\right]\right)^{p/\alpha}\right)$ $\displaystyle\leq(1\wedge
3^{p-1})\left(\left(1+T^{p}K_{F}^{p}\right)\sup_{t\in[0,T]}\mathbb{E}\,\left[\left\lVert
Z(t)-X_{n_{k}}(t)\right\rVert^{p}\right]\right.$
$\displaystyle\hphantom{PHANTOMMM}+\left.e_{p,\alpha}K_{G}^{p}T^{p/\alpha}\sup_{t\in[0,T]}\left(\mathbb{E}\,\left[\left\lVert
Z(t)-X_{n_{k}}(t)\right\rVert^{p}\right]\right)^{\beta}\right)$

As the last line tends to $0$ as $k\to\infty$ by (6.24), it follows that $Z$
is a mild solution to (6.1).

It remains to establish that $Z$ has càdlàg paths, but this follows
immediately from the following corollary as $X_{n}$ has càdlàg paths. ∎

At the end of this section, we present a stronger convergence result for
Yosida approximations that not only completes the proof of Theorem 6.5 but
also turns out to be useful in applications as will be seen in the following
sections.

######  Corollary 6.10.

For all $0<p<\alpha$ there exists a subsequence $(X_{n_{k}})_{k\in\mathbb{N}}$
of the Yosida approximations, which converges to a solution to (6.1) both in
$\mathcal{C}([0,T],L^{p}(\Omega,H))$ and uniformly on $[0,T]$ almost surely.

###### Proof.

Lemma 6.9 enables us to choose a subsequence $(X_{n})_{n\in\mathbb{N}}$ of the
Yosida approximations which converges in $\mathcal{C}([0,T],L^{p}(\Omega,H))$
to the mild solution $X$. To prove almost sure convergence, we fix an
arbitrary $r>0$ and estimate

$\displaystyle\begin{split}&P\left(\sup_{t\in[0,T]}\left\lVert
X(t)-X_{n}(t)\right\rVert>r\right)\leq P\left(\sup_{t\in[0,T]}\left\lVert
S(t)(I-R_{n})x_{0}\right\rVert>\frac{r}{3}\right)\\\
&\qquad\qquad+P\left(\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}S(t-s)\big{(}F(X(s))-R_{n}F(X_{n}(s))\big{)}\,{\rm
d}s\right\rVert>\frac{r}{3}\right)\\\
&\qquad\qquad+P\left(\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}S(t-s)\big{(}G(X(s-))-R_{n}G(X_{n}(s-))\big{)}\,{\rm
d}L(s)\right\rVert>\frac{r}{3}\right).\end{split}$ (6.25)

For the following arguments, we define $m:=\sup_{t\in[0,T]}\left\lVert
S(t)\right\rVert_{\mathcal{L}(H)}$. As $I-R_{n}$ converges to zero strongly as
$n\to\infty$ we obtain

$\displaystyle P\left(\sup_{t\in[0,T]}\left\lVert
S(t)(I-R_{n})x_{0}\right\rVert>\frac{r}{3}\right)\leq
P\left(m\left\lVert(I-R_{n})x_{0}\right\rVert>\frac{r}{3}\right)\to 0.$

For estimating the second term in (6.25), we apply Markov’s inequality and
Lipschitz continuity of $F$ in (A2) to obtain

$\displaystyle
P\left(\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}S(t-s)\big{(}F(X(s))-R_{n}F(X_{n}(s))\big{)}\,{\rm
d}s\right\rVert>\frac{r}{3}\right)$ $\displaystyle\leq
P\left(m\int_{0}^{T}\left\lVert F(X(s))-R_{n}F(X_{n}(s))\right\rVert{\rm
d}s>\frac{r}{3}\right)$ $\displaystyle\leq
P\left(\int_{0}^{T}\left\lVert(I-R_{n})F(X(s))\right\rVert\,{\rm
d}s>\frac{r}{6m}\right)+P\left(\int_{0}^{T}\left\lVert
R_{n}(F(X(s))-F(X_{n}(s)))\right\rVert\,{\rm d}s>\frac{r}{6m}\right)$
$\displaystyle\leq\frac{6m}{r}\mathbb{E}\,\left[\int_{0}^{T}\left\lVert(I-R_{n})F(X(s))\right\rVert\,{\rm
d}s\right]+\frac{6m}{r}\mathbb{E}\,\left[\int_{0}^{T}\left\lVert
R_{n}(F(X(s))-F(X_{n}(s)))\right\rVert\,{\rm d}s\right]$
$\displaystyle\leq\frac{6m}{r}\mathbb{E}\,\left[\int_{0}^{T}\left\lVert(I-R_{n})F(X(s))\right\rVert\,{\rm
d}s\right]+\frac{6m}{r}\left(\sup_{n\in\mathbb{N}}\left\lVert
R_{n}\right\rVert_{\mathcal{L}(H)}\right)TK_{F}\sup_{t\in[0,T]}\mathbb{E}\,\left[\left\lVert
X(t)-X_{n}(t)\right\rVert\right].$

We conclude from the last inequality by Lebesgue’s dominated convergence
theorem and convergence of $X_{n}$ to $X$ in
$\mathcal{C}([0,T],L^{1}(\Omega,H))$ that

$\displaystyle\lim_{n\to\infty}P\left(\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}S(t-s)(F(X(s))-R_{n}F(X_{n}(s)))\,{\rm
d}s\right\rVert>\frac{r}{3}\right)=0.$

To estimate the last term in (6.25), we apply the dilation theorem for
contraction semigroups, see [38, Th. I.8.1]): there exists a $C_{0}$-group
$(\hat{S}(t))_{t\in\mathbb{R}}$ of unitary operators $\hat{S}(t)$ on a larger
Hilbert space $\hat{H}$ in which $H$ is continuously embedded satisfying
$S(t)=\pi\hat{S}(t)i$ for all $t\geq 0$, where $\pi$ is the projection from
$\hat{H}$ to $H$ and $i$ is the continuous embedding of $H$ into $\hat{H}$.
Thus, if we denote
$m=\sup_{t\in[0,T]}\left\lVert\pi\hat{S}(t)\right\rVert_{\mathcal{L}(\hat{H},H)}$,
$k=\sup_{s\in[-T,0]}\left\lVert\hat{S}(s)i\right\rVert_{\mathcal{L}(H,\hat{H})}$,
we may estimate using Markov’s inequality, Inequality (2.7) and Hölder
continuity of $G$ in (6.6)

$\displaystyle
P\left(\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}S(t-s)\left(G(X(s-))-R_{n}G(X_{n}(s-))\right){\rm
d}L(s)\right\rVert>\frac{r}{3}\right)$ $\displaystyle\leq
P\left(\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}S(t-s)\left(I-R_{n}\right)G(X(s-)){\rm
d}L(s)\right\rVert>\frac{r}{6}\right)$
$\displaystyle\qquad\qquad+P\left(\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}S(t-s)R_{n}\left(G(X(s-))-G(X_{n}(s-))\right){\rm
d}L(s)\right\rVert>\frac{r}{6}\right)$
$\displaystyle=P\left(\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}\pi\hat{S}(t-s)i\left(I-R_{n}\right)G(X(s-)){\rm
d}L(s)\right\rVert>\frac{r}{6}\right)$
$\displaystyle\qquad\qquad+P\left(\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}\pi\hat{S}(t-s)iR_{n}\left(G(X(s-))-G(X_{n}(s-))\right){\rm
d}L(s)\right\rVert>\frac{r}{6}\right)$
$\displaystyle=P\left(\sup_{t\in[0,T]}\left\lVert\pi\hat{S}(t)\int_{0}^{t}\hat{S}(-s)i\left(I-R_{n}\right)G(X(s-)){\rm
d}L(s)\right\rVert>\frac{r}{6}\right)$
$\displaystyle\qquad\qquad+P\left(\sup_{t\in[0,T]}\left\lVert\pi\hat{S}(t)\int_{0}^{t}\hat{S}(-s)iR_{n}\left(G(X(s-))-G(X_{n}(s-))\right){\rm
d}L(s)\right\rVert>\frac{r}{6}\right)$ $\displaystyle\leq
P\left(\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}\hat{S}(-s)i\left(I-R_{n}\right)G(X(s-)){\rm
d}L(s)\right\rVert_{\hat{H}}>\frac{r}{6m}\right)$
$\displaystyle\qquad\qquad+P\left(\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}\hat{S}(-s)iR_{n}\left(G(X(s-))-G(X_{n}(s-))\right){\rm
d}L(s)\right\rVert_{\hat{H}}>\frac{r}{6m}\right)$
$\displaystyle\leq\frac{6m}{r}\mathbb{E}\,\left[\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}\hat{S}(-s)i\left(I-R_{n}\right)G(X(s-)){\rm
d}L(s)\right\rVert_{\hat{H}}\right]$
$\displaystyle\qquad\qquad+\frac{6m}{r}\mathbb{E}\,\left[\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}\hat{S}(-s)iR_{n}\left(G(X(s-))-G(X_{n}(s-))\right){\rm
d}L(s)\right\rVert_{\hat{H}}\right]$ $\displaystyle\leq
e_{1,\alpha}\frac{6m}{r}\left(\mathbb{E}\,\left[\int_{0}^{T}\left\lVert\hat{S}(-s)i\left(I-R_{n}\right)G(X(s-))\right\rVert_{\mathcal{L}_{2}(U,\hat{H})}^{\alpha}{\rm
d}s\right]\right)^{1/\alpha}$
$\displaystyle\qquad\qquad+e_{1,\alpha}\frac{6m}{r}\left(\mathbb{E}\,\left[\int_{0}^{T}\left\lVert\hat{S}(-s)iR_{n}\left(G(X(s-))-G(X_{n}(s-))\right)\right\rVert_{\mathcal{L}_{2}(U,\hat{H})}^{\alpha}{\rm
d}s\right]\right)^{1/\alpha}$ $\displaystyle\leq
e_{1,\alpha}\frac{6m}{r}k\left(\mathbb{E}\,\left[\int_{0}^{T}\left\lVert\left(I-R_{n}\right)G(X(s-))\right\rVert_{\mathcal{L}_{2}(U,\hat{H})}^{\alpha}{\rm
d}s\right]\right)^{1/\alpha}$
$\displaystyle\qquad\qquad+e_{1,\alpha}\frac{6m}{r}kK_{G}T^{1/\alpha}\left(\sup_{t\in[0,T]}\mathbb{E}\,\left[\left\lVert(X(t))-X_{n}(t)\right\rVert^{\alpha\beta}\right]\right)^{1/\alpha}$

We conclude from the last inequality by Lebesgue’s dominated convergence,
strong convergence of $R_{n}$ to $I$, boundedness $G$ in (6.6) and convergence
of $X_{n}$ to $X$ in $\mathcal{C}([0,T],L^{\alpha\beta}(\Omega,H))$ that

$\displaystyle\lim_{n\to\infty}P\left(\sup_{t\in[0,T]}\left\lVert\int_{0}^{t}S(t-s)(G(X(s-))-R_{n}G(X_{n}(s-)))\,{\rm
d}L(s)\right\rVert>\frac{r}{3}\right)=0,$

We have shown that all the terms on the right hand side of (6.25) converge to
zero as $n$ tends to infinity which gives uniform convergence of $X_{n}$ to
$X$ in probability on $[0,T]$. This concludes the proof, since uniform
convergence in probability implies the existence of a desired subsequence. ∎

##  7 Moment boundedness for evolution equations

In this section, we investigate stability properties of the solution for the
stochastic evolution equation (6.1) by applying the Itô’s formula derived in
Theorem 5.1. More precisely, we shall derive conditions on the coefficients
such that the mild solution $X$ is ultimately exponentially bounded in the
$p$-th moment, that is there exist constants $m_{1},m_{2},m_{3}>0$ such that

$\displaystyle\mathbb{E}\,\left[\left\lVert X(t)\right\rVert^{p}\right]\leq
m_{1}e^{-tm_{2}}\mathbb{E}\,\left[\left\lVert
x_{0}\right\rVert^{p}\right]+m_{3}\quad\text{for all }t\geq 0.$

Recall that $\mathcal{C}_{b}^{2}(H)$ denotes the space of continuous real-
valued functions defined on $H$ with bounded first and second Fréchet
derivatives. In what follows, our goal is to derive a Lyapunov-type criterion
using the following operator on $\mathcal{C}_{b}^{2}(H)$:

$\displaystyle\mathcal{L}f(h)=\langle$ $\displaystyle Df(h),Ah+F(h)\rangle$
(7.1) $\displaystyle+\int_{H}\big{(}f(h+g)-f(h)-\langle
Df(h),g\rangle\big{)}\,\left(\lambda\circ G(h)^{-1}\right)({\rm d}g),\quad
h\in D^{1}$

for $f\in\mathcal{C}_{b}^{2}(H)$. Note that the right hand side of (7.1) is
well defined by Lemma 5.2. We can now state the main result of this section,
the following general moment boundedness criterion.

######  Theorem 7.1.

Let $p\in(0,1)$ be fixed and $V$ be a function in $\mathcal{C}_{b}^{2}(H)$
satisfying for some constants $\beta_{1},\beta_{2},\beta_{3},k_{1},k_{3}>0$
the inequalities

$\displaystyle\beta_{1}\left|\left|h\right|\right|^{p}-k_{1}\leq
V(h)\leq\beta_{2}\left|\left|h\right|\right|^{p}\quad\text{for all }h\in H,$
(7.2) $\displaystyle\mathcal{L}V(h)\leq-\beta_{3}V(h)+k_{3}\quad\text{for all
}h\in D^{1}.$ (7.3)

Then the solution $X$ to (6.1) is exponentially ultimately bounded in the
$p$-th moment:

$\displaystyle\mathbb{E}\,\left[\left|\left|X(t)\right|\right|^{p}\right]\leq\frac{\beta_{2}}{\beta_{1}}e^{-\beta_{3}t}\mathbb{E}\,\left[\left\lVert
x_{0}\right\rVert^{p}\right]+\frac{1}{\beta_{1}}\left(k_{1}+\frac{k_{3}}{\beta_{3}}\right).$

Before we prove Theorem 7.1 we demonstrate its application by deriving
conditions for moment boundedness in terms of the coefficients of Equation
(6.1).

######  Corollary 7.2.

Suppose that there exists $\epsilon>0$ such that

$\displaystyle\langle
Ah+F(h),h\rangle\leq-\epsilon\left|\left|h\right|\right|^{2}\quad\text{for all
}h\in D^{1},$

then the solution to (6.1) is exponentially ultimately bounded in the $p$-th
moment for every $p\in(0,1)$.

###### Proof.

Fix $p\in(0,1)$ and let $\zeta$ be a function in $\mathcal{C}^{2}([0,\infty))$
satisfying $\zeta(x)=x^{p/2}$ for $x\geq 1$ and $\zeta(x)\leq 1$ for $x<1$. By
defining $V(h)=\zeta(\left\lVert h\right\rVert^{2})$ for all $h\in H$, we
obtain $V\in\mathcal{C}_{b}^{2}(H)$ and

$\displaystyle V(h)=\left|\left|h\right|\right|^{p}\quad\text{for all
}h\in\overline{B}_{H}^{c}\quad\text{ and }\quad 0\leq V(h)\leq 1\quad\text{for
all }h\in\overline{B}_{H}.$

It follows that (7.2) holds with $\beta_{1}=\beta_{2}=k_{1}=1$. We show that
(7.3) also holds. By the definition of $V$, it follows for each $h\in
D^{1}\cap\overline{B}_{H}^{c}$ that

$\displaystyle\langle
DV(h),Ah+F(h)\rangle=p\left|\left|h\right|\right|^{p-2}\langle
h,Ah+F(h)\rangle\leq-\epsilon p\left|\left|h\right|\right|^{p}=-\epsilon
pV(h).$

For $h\in D^{1}\cap\overline{B}_{H}$, one obtains by boundedness of $F$ in
Assumption (A2) that

$\displaystyle\langle
DV(h),Ah+F(h)\rangle\leq\left|\left|DV\right|\right|_{\infty}\left(\left|\left|A\right|\right|_{\mathcal{L}(D^{1})}+K_{F}\right).$

Since Lemma 5.2 together with boundedness of $G$ in Assumption (A3) implies
for each $h\in H$ that

$\displaystyle\int_{H}\big{(}V(h+g)-V(h)-\langle
DV(h),g\rangle\big{)}\,\left(\lambda\circ G^{-1}(h)\right)({\rm d}g)\leq
d_{\alpha}^{1}\left(2\left|\left|DV\right|\right|_{\infty}+\frac{1}{2}\left|\left|D^{2}V\right|\right|_{\infty}\right)K_{G}^{\alpha},$

we have verified Condition (7.3). ∎

In the remaining of this section, we prove Theorem 7.1 using the Yosida
approximations established in the previous sections. For this purpose, let
$X_{n}$ denote the mild solution to the approximating equations (6.7) for each
$n\in\mathbb{N}$. We may assume due to Corollary 6.10, by passing to a
subsequence if necessary, that $X_{n}$ converges to the solution $X$ of (6.1)
uniformly almost surely on $[0,T]$. In what follows, we will routinely pass on
to a subsequence without changing the indices.

######  Proposition 7.3.

The mild solution $X_{n}$ of (6.7) is a strong solution attaining values in
$D^{1}$, that is, for each $t\in[0,T]$, it satisfies

$\displaystyle
X_{n}(t)=R_{n}x_{0}+\int_{0}^{t}\big{(}AX_{n}(s)+R_{n}F(X_{n}(s))\big{)}\,{\rm
d}s+\int_{0}^{t}R_{n}G(X_{n}(s-))\,{\rm d}L(s).$

###### Proof.

Our argument will follow closely the proof of [1, Th.2]. As mild solution,
$X_{n}$ satisfies

$\displaystyle
X_{n}(t)=S(t)R_{n}x_{0}+\int_{0}^{t}S(t-s)R_{n}F(X_{n}(s))\,{\rm
d}s+\int_{0}^{t}S(t-s)R_{n}G(X_{n}(s-))\,{\rm d}L(s).$ (7.4)

The process $X_{n}$ is $\left(\mathcal{F}_{t}\right)$-measurable with càdlàg
paths and attains values in $D^{1}$. First, we obtain from (7.4) by
interchanging integrals and $A\in\mathcal{L}(D^{1})$ for $t\geq 0$ that

$\displaystyle AX_{n}(t)=AS(t)R_{n}x_{0}+\int_{0}^{t}AS(t-s)$
$\displaystyle)R_{n}F(X_{n}(s))\,{\rm d}s$ (7.5)
$\displaystyle+\int_{0}^{t}AS(t-s)R_{n}G(X_{n}(s-))\,{\rm d}L(s).$

Each term on the right hand side of (7.5) is almost surely Bochner integrable.
Indeed, integrability of the first term is immediate from the uniform
boundedness principle. For the second term, boundedness of $F$ in Condition
(A2) and commutativity of $S$ and $R_{n}$ implies

$\displaystyle\int_{0}^{t}\int_{0}^{s}$ $\displaystyle\left\lVert
AS(s-r)R_{n}F(X_{n}(r))\right\rVert_{1}\,{\rm d}r\,{\rm d}s$
$\displaystyle\leq\left\lVert A\right\rVert_{\mathcal{L}(D^{1})}\left\lVert
R_{n}\right\rVert_{\mathcal{L}(H,D^{1})}\int_{0}^{t}\int_{0}^{s}\left\lVert
S(s-r)F(X_{n}(s))\right\rVert\,{\rm d}r\,{\rm d}s<\infty\quad\text{a.s.}$

Almost sure Bochner integrability of the stochastic integral in (7.5) follows
from boundedness of $G$ in Assumption(A3), commutativity of $S$ and $R_{n}$,
and Theorem 2.4 via the estimate

$\displaystyle\mathbb{E}\,\left[\int_{0}^{t}\int_{0}^{s}\left\lVert
AS(s-r)R_{n}G(X_{n}(r))\right\rVert_{\mathcal{L}_{2}(U,D^{1})}^{\alpha}\,{\rm
d}r\,{\rm d}s\right]$ $\displaystyle\qquad\leq\left\lVert
A\right\rVert_{\mathcal{L}(D^{1})}^{\alpha}\left\lVert
R_{n}\right\rVert_{\mathcal{L}(H,D^{1})}^{\alpha}\mathbb{E}\,\left[\int_{0}^{t}\int_{0}^{s}\left\lVert
S(s-r)G(X_{n}(r))\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha}\,{\rm d}r\,{\rm
d}s\right]<\infty.$

Integrating both sides of (7.5) results in the equality

$\displaystyle\int_{0}^{t}AX_{n}(s)\,{\rm d}s$
$\displaystyle=\int_{0}^{t}AS(s)R_{n}x_{0}\,{\rm
d}s+\int_{0}^{t}\int_{0}^{s}AS(s-r)R_{n}F(X_{n}(r))\,{\rm d}r\,{\rm d}s$
$\displaystyle\qquad+\int_{0}^{t}\int_{0}^{s}AS(s-r)R_{n}G(X_{n}(r-))\,{\rm
d}L(r)\,{\rm d}s.$

Applying Fubini’s theorems, see Theorem 2.4 for the stochastic version, and
the equality $\int_{0}^{t}AS(s)R_{n}h\,{\rm d}s=S(t)R_{n}h-R_{n}h$ for all
$h\in H$ enable us to conclude

$\displaystyle\int_{0}^{t}AX_{n}(s)\,{\rm d}s$
$\displaystyle=\int_{0}^{t}AS(s)R_{n}x_{0}\,{\rm
d}s+\int_{0}^{t}\int_{r}^{t}AS(s-r)R_{n}F(X_{n}(r))\,{\rm d}s\,{\rm d}r$
$\displaystyle\qquad+\int_{0}^{t}\int_{r}^{t}AS(s-r)R_{n}G(X_{n}(r-))\,{\rm
d}s\,{\rm d}L(r)$
$\displaystyle=S(t)R_{n}x_{0}-R_{n}x_{0}+\int_{0}^{t}S(t-r)R_{n}F(X_{n}(r))\,{\rm
d}r-\int_{0}^{t}R_{n}F(X_{n}(r))\,{\rm d}r$
$\displaystyle\qquad+\int_{0}^{t}S(t-r)R_{n}G(X_{n}(r-))\,{\rm
d}L(r)-\int_{0}^{t}R_{n}G(X_{n}(r-))\,{\rm d}L(r)$
$\displaystyle=X_{n}(t)-R_{n}x_{0}-\int_{0}^{t}R_{n}F(X_{n}(r))\,{\rm
d}r-\int_{0}^{t}R_{n}G(X_{n}(r-))\,{\rm d}L(r),$

which verifies $X_{n}$ as a strong solution to (6.1). ∎

We denote by $\mathcal{L}_{n}$ the usual generator associated with the Yosida
approximations $X_{n}$, $n\in\mathbb{N}$, defined for
$f\in\mathcal{C}_{b}^{2}(H)$ and $h\in D^{1}$ by

$\displaystyle\mathcal{L}_{n}f(h)=\langle$ $\displaystyle
Df(h),Ah+R_{n}F(h)\rangle$ (7.6)
$\displaystyle+\int_{H}\big{(}f(h+R_{n}g)-f(h)-\langle
Df(h),R_{n}g\rangle\big{)}\,\left(\lambda\circ G(h)^{-1}\right)({\rm d}g).$

The right hand side of (7.6) is well defined by Lemma 5.2. Recall that the
counterpart to $\mathcal{L}_{n}$ for the mild solution $X$ denoted by
$\mathcal{L}$ was introduced in (7.1). The generators $\mathcal{L}_{n}$ and
$\mathcal{L}$ are related by the following crucial convergence result.

######  Lemma 7.4.

Let $(X_{n})_{n\in\mathbb{N}}$ be solutions of (6.7) which a.s. converges
uniformly to the solution of (6.1). It follows for each
$f\in\mathcal{C}_{b}^{2}(H)$ that

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\,\left[\int_{0}^{T}\bigg{|}\mathcal{L}_{n}f(X_{n}(s))\right.$
$\displaystyle\left.-\mathcal{L}f(X_{n}(s))\bigg{|}\,{\rm d}s\right]=0.$

###### Proof.

We obtain for each $h\in D^{1}$ that

$\displaystyle|\mathcal{L}f(h)-\mathcal{L}_{n}f(h)|\leq\left|\left|Df\right|\right|_{\infty}\left|\left|\left(I-R_{n}\right)F(h)\right|\right|$
$\displaystyle+\int_{\overline{B}_{H}}|f(h+g)-f(h+R_{n}g)-\langle
Df(h),\left(I-R_{n}\right)g\rangle|\,\left(\lambda\circ G(h)^{-1}\right)({\rm
d}g)$ (7.7)
$\displaystyle+\int_{\overline{B}_{H}^{c}}\\!|f(h+g)-f(h+R_{n}g)|\left(\lambda\circ
G(h)^{-1}\right)({\rm d}g)+\int_{\overline{B}_{H}^{c}}\\!|\langle
Df(h),\left(I-R_{n}\right)g\rangle|\left(\lambda\circ G(h)^{-1}\right)({\rm
d}g).$

Taylor’s remainder theorem in the integral form implies

$\displaystyle\int_{\overline{B}_{H}}|f(h+g)-f(h+R_{n}g)-\langle
Df(h),\left(I-R_{n}\right)g\rangle|\,\left(\lambda\circ G(h)^{-1}\right)({\rm
d}g)$ $\displaystyle\qquad\leq\int_{\overline{B}_{H}}\int_{0}^{1}|\langle
D^{2}f(h+\theta\left(I-R_{n}\right)g)\left(I-R_{n}\right)g,\left(I-R_{n}\right)g\rangle(1-\theta)\,|{\rm
d}\theta\,\left(\lambda\circ G(h)^{-1}\right)({\rm d}g)$
$\displaystyle\qquad\leq\frac{1}{2}\left|\left|D^{2}f\right|\right|_{\infty}\int_{\overline{B}_{H}}\left|\left|\left(I-R_{n}\right)g\right|\right|^{2}\,\left(\lambda\circ
G(h)^{-1}\right)({\rm d}g).$

In the same way, we obtain

$\displaystyle\int_{\overline{B}_{H}^{c}}\\!|f(h+g)-f(h+R_{n}g)|\left(\lambda\circ
G(h)^{-1}\right)({\rm
d}g)\leq\left|\left|Df\right|\right|_{\infty}\int_{\overline{B}_{H}^{c}}\\!\left|\left|\left(I-R_{n}\right)g\right|\right|\left(\lambda\circ
G(h)^{-1}\right)({\rm d}g)$

and also

$\displaystyle\int_{\overline{B}_{H}^{c}}\\!|\langle
Df(h),\left(I-R_{n}\right)g\rangle|\left(\lambda\circ G(h)^{-1}\right)({\rm
d}g)\leq\left|\left|Df\right|\right|_{\infty}\int_{\overline{B}_{H}^{c}}\\!\left|\left|\left(I-R_{n}\right)g\right|\right|\left(\lambda\circ
G(h)^{-1}\right)({\rm d}g)$

Applying the last three estimates to (7) and taking expectation on both sides,
it follows from Inequality (2.2) and for each $n\in\mathbb{N}$ that

$\displaystyle\mathbb{E}\,\left[\int_{0}^{T}\left|\mathcal{L}_{n}f(X_{n}(s))-\mathcal{L}f(X_{n}(s))\right|{\rm
d}s\right]$
$\displaystyle\quad\leq\left|\left|Df\right|\right|_{\infty}\mathbb{E}\,\left[\int_{0}^{T}\left|\left|\left(I-R_{n}\right)F(X_{n}(s))\right|\right|{\rm
d}s\right]+c\mathbb{E}\,\left[\int_{0}^{T}\left|\left|\left(I-R_{n}\right)G(X_{n}(s))\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}{\rm
d}s\right],$

where
$c:=d_{\alpha}^{1}\left(2\left|\left|Df\right|\right|_{\infty}+\frac{1}{2}\left|\left|D^{2}f\right|\right|_{\infty}\right)$.

To complete the proof, it remains to show that both

$\displaystyle\lim_{n\to\infty}\mathbb{E}\,\left[\int_{0}^{T}\left|\left|\left(I-R_{n}\right)F(X_{n}(s))\right|\right|{\rm
d}s\right]=0,$ (7.8)
$\displaystyle\lim_{n\to\infty}\mathbb{E}\,\left[\int_{0}^{T}\left|\left|\left(I-R_{n}\right)G(X_{n}(s))\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}{\rm
d}s\right]=0.$ (7.9)

Let $t\in[0,T]$ be arbitrary but fixed, and recall that we chose $X_{n}(t)$
almost surely convergent and thus
$\\{X_{m}(t)(\omega):m\in\mathbb{N}\\}\subset H$ is compact for almost all
$\omega\in\Omega$. Strong convergence of $I-R_{n}$ to zero, see [13, Le. 3.4],
continuity of $F$ and $G$ and the fact that continuous mapping converging
pointwise to a continuous mapping converge uniformly over compacts together
imply for each $t\in[0,T]$ that, almost surely, we obtain

$\displaystyle\lim_{n\to\infty}\left|\left|\left(I-R_{n}\right)F(X_{n}(t))\right|\right|\leq\lim_{n\rightarrow\infty}\sup_{m\in\mathbb{N}}\left|\left|\left(I-R_{n}\right)F(X_{m}(t))\right|\right|=0,$
$\displaystyle\lim_{n\to\infty}\left|\left|\left(I-R_{n}\right)G(X_{n}(t))\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}\leq\lim_{n\rightarrow\infty}\sup_{m\in\mathbb{N}}\left|\left|\left(I-R_{n}\right)G(X_{m}(t))\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}=0.$

Since the boundedness conditions in (A2) and (A3) guarantee

$\displaystyle\left|\left|\left(I-R_{n}\right)F(X_{n}(t))\right|\right|$
$\displaystyle\leq\left(\sup_{n\in\mathbb{N}}\left|\left|I-R_{n}\right|\right|_{\mathcal{L}(H)}\right)K_{F}\quad\text{a.s.,}$
$\displaystyle\left|\left|\left(I-R_{n}\right)G(X_{n}(t))\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}$
$\displaystyle\leq\left(\sup_{n\in\mathbb{N}}\left|\left|I-R_{n}\right|\right|_{\mathcal{L}(H)}^{\alpha}\right)K_{G}^{\alpha}\quad\text{a.s.}$

an application of Lebesgue’s dominated convergence theorem verifies (7.8) and
(7.9), which completes the proof. ∎

###### Proof of Theorem 7.1.

Let $(X_{n})_{n\in\mathbb{N}}$ be the solutions of (6.7). Because of Corollary
6.10, we can assume that $(X_{n})_{n\in\mathbb{N}}$ converges uniformly to the
solution of (6.1) a.s. Proposition 7.3 enables us to apply the Itô formula in
Theorem 5.1 to $X_{n}$, which results in

$\displaystyle
V(X_{n}(t))=V(X_{n}(0)+\int_{0}^{t}\mathcal{L}_{n}V(X_{n}(s)){\rm
d}s+\int_{0}^{t}\langle
G(X_{n}(s-))^{*}R_{n}^{*}DV(X_{n}(s-)),\cdot\rangle\,{\rm d}L(s)$
$\displaystyle\quad+\int_{0}^{t}\int_{H}V(X_{n}(s-)+h)-V(X_{n}(s-))-\langle
DV(X_{n}(s-)),h\rangle(\mu^{X_{n}}-\nu^{X_{n}})\,({\rm d}s,{\rm d}h)$ (7.10)

almost surely for all $t\geq 0$. Applying the product formula to the real-
valued semi-martingale $V(X_{n}(\cdot))$ and the function $t\mapsto
e^{\beta_{3}t}$ and taking expectations on both sides of (7.10) shows

$\displaystyle e^{\beta_{3}t}\mathbb{E}\,\left[V(X_{n}(t))\right]=$
$\displaystyle\mathbb{E}\,\left[V(X_{n}(0))\right]+\mathbb{E}\,\left[\int_{0}^{t}e^{\beta_{3}s}\left(\beta_{3}V(X_{n}(s))+\mathcal{L}_{n}V(X_{n}(s))\right){\rm
d}s\right].$ (7.11)

Here, we used the fact that the last two integrals in (7.10) define
martingales, and thus have expectation zero. This follows from the observation
that they are local martingales according to Lemma 2.3 and Theorem 5.1 and are
uniformly bounded. The latter is guaranteed by the boundedness of $G$ in (A3),
since

$\displaystyle\mathbb{E}\,\left[\int_{0}^{t}\left|\left|\langle
G(X_{n}(s))^{*}R_{n}^{*}DV(X_{n}(s)),\cdot\rangle\right|\right|_{\mathcal{L}_{2}(U,\mathbb{R})}^{\alpha}\,{\rm
d}s\right]$
$\displaystyle=\mathbb{E}\,\left[\int_{0}^{t}\left|\left|G(X_{n}(s))^{*}R_{n}^{*}DV(X_{n}(s))\right|\right|^{\alpha}\,{\rm
d}s\right]$
$\displaystyle\leq\left|\left|R_{n}\right|\right|_{\mathcal{L}(H)}^{\alpha}\left|\left|DV\right|\right|_{\infty}^{\alpha}TK_{G}^{\alpha}<\infty,$

and similarly, by using Lemma 5.2,

$\displaystyle\mathbb{E}\,\left[\int_{0}^{t}\right.$
$\displaystyle\left.\int_{H}|V(X_{n}(s-)+h)-V(X_{n}(s-))-\langle
DV(X_{n}(s-),h\rangle|\,\nu^{X_{n}}({\rm d}s,{\rm d}h)\right]$
$\displaystyle\leq
d_{\alpha}^{1}\left(2\left|\left|DV\right|\right|_{\infty}+\frac{1}{2}\left|\left|D^{2}V\right|\right|_{\infty}\right)\left|\left|R_{n}\right|\right|_{\mathcal{L}(H)}^{\alpha}\mathbb{E}\,\left[\int_{0}^{t}\left|\left|G(X_{n}(s))\right|\right|_{\mathcal{L}_{2}(U,H)}^{\alpha}\,{\rm
d}s\right]<\infty.$

The first term on the right hand side in (7.11) is finite since

$\displaystyle\mathbb{E}\,\left[V(X_{n}(0))\right]\leq\beta_{2}\left\lVert
R_{n}\right\rVert_{\mathcal{L}(H)}^{p}\mathbb{E}\,\left[\left\lVert
x_{0}\right\rVert^{p}\right]<\infty.$

The same holds for the second term, which can be shown using the same
arguments as in the proof of Lemma 7.4. By applying Inequality (7.3) to
(7.11), we conclude

$\displaystyle e^{\beta_{3}t}\mathbb{E}\,\left[[V(X_{n}(t))\right]$
$\displaystyle\leq\mathbb{E}\,\left[V(X_{n}(0))\right]+\mathbb{E}\,\Bigg{[}\int_{0}^{t}e^{\beta_{3}s}\bigg{(}-\mathcal{L}V(X_{n}(s))+\mathcal{L}_{n}V(X_{n}(s))+k_{3}\bigg{)}{\rm
d}s\Bigg{]}$
$\displaystyle\leq\mathbb{E}\,\left[V(X_{n}(0))\right]+e^{\beta_{3}T}\mathbb{E}\,\Bigg{[}\int_{0}^{t}\bigg{|}\mathcal{L}_{n}V(X_{n}(s))-\mathcal{L}V(X_{n}(s))\bigg{|}{\rm
d}s\Bigg{]}+\frac{k_{3}}{\beta_{3}}\left(e^{\beta_{3}t}-1\right).$

Lemma 7.4 together with Fatou’s lemma implies

$\displaystyle\mathbb{E}\,\left[V(X(t))\right]\leq\liminf_{n\to\infty}\mathbb{E}\,\left[V(X_{n}(t))\right]\leq
e^{-\beta_{3}t}\mathbb{E}\,\left[V(x_{0})\right]+\frac{k_{3}}{\beta_{3}}.$

Applying Assumption (7.2) completes the proof. ∎

##  8 Mild Itô formula

In this section, we prove an Itô formula for mild solutions of Equation (6.1)
and mappings $f\in\mathcal{C}_{b}^{2}(H)$ such that the second derivative
$D^{2}f$ is not only continuous but satisfies

$\displaystyle\lim_{n\rightarrow\infty}\left\lVert
g_{n}-g\right\rVert=0\implies\lim_{n\rightarrow\infty}\sup_{h\in\overline{B}_{H}}\left\lVert
D^{2}f(g_{n}+h)-D^{2}f(g+h)\right\rVert_{\mathcal{L}(H)}=0.$ (8.1)

The subspace of all these functions is denoted by $\mathcal{C}_{b,u}^{2}(H)$.

######  Theorem 8.1 (Itô formula for mild solutions).

A mild solution $X$ of (6.1) satisfies for each $f\in\mathcal{C}_{b,u}^{2}(H)$
and $t\geq 0$ that

$\displaystyle f(X(t))$ $\displaystyle=f(x_{0})+\int_{0}^{t}\langle
G(X(s-)^{*}Df(X(s-))),\cdot\rangle\,{\rm d}L(s)$ (8.2)
$\displaystyle\qquad+\int_{0}^{t}\int_{H}\big{(}f(X(s-)+h)-f(X(s-))-\langle
Df(X(s-)),h\rangle\big{)}\,\left(\mu^{X}-\nu^{X}\right)({\rm d}s,{\rm d}h)$
$\displaystyle\qquad+\lim_{n\to\infty}\left(\int_{0}^{t}\langle
Df(X_{n}(s)),AX_{n}(s)\rangle\,{\rm d}s\right)+\int_{0}^{t}\langle
Df(X(s)),F(X(s))\rangle\,{\rm d}s$
$\displaystyle\qquad+\int_{0}^{t}\int_{H}\big{(}f(X(s)+h)-f(X(s))-\langle
Df(X(s)),h\rangle\big{)}\,\left(\lambda\circ G(X(s))^{-1}\right)({\rm
d}h)\,{\rm d}s,$

where the limit is taken in $L_{P}^{0}(\Omega,\mathbb{R})$.

######  Remark 8.2.

Note that while $X$ may not be a semimartingale, the compensated measure
$\mu^{X}-\nu^{X}$ in (8.2) still exists as $X$ is both adapted and càdlàg; see
[17, Chap. II].

######  Remark 8.3.

Unlike in similar situation with the driving noise being Gaussian e.g. in [25]
we do not identify the limit in (8.2) as then the imposed assumptions on $f$
are very restrictive. In many applications (see e.g. [1]), it is enough to
identify some bound on

$\displaystyle\lim_{n\to\infty}\left(\int_{0}^{t}\langle
Df(X_{n}(s)),AX_{n}(s)\rangle\,{\rm d}s\right)$

which leads to natural assumptions on the generator $A$.

We divide the proof of the above theorem in some technical lemmas. To simplify
the notation, we introduce the function $T_{f}\colon H\times
H\rightarrow\mathbb{R}$ for $f\in\mathcal{C}_{b,u}^{2}(H)$ defined by

$\displaystyle T_{f}(g,h)=f(g+h)-f(g)-\langle Df(g),h\rangle,\quad g,h\in H.$

######  Lemma 8.4.

Let $\lambda$ be the cylindrial Lévy measure of $L$. It follows for every
$f\in\mathcal{C}_{b}^{2}(H)$, $\varphi\in\mathcal{L}_{2}(U,H)$ and $g,h\in H$
that

$\displaystyle\int_{H}$
$\displaystyle\left|T_{f}(g,b)-T_{f}(h,b)\right|\left(\lambda\circ\varphi^{-1}\right)({\rm
d}b)$ $\displaystyle\leq
2d_{\alpha}^{1}\left\lVert\varphi\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha}\left(\sup_{b\in\overline{B}_{H}}\left\lVert
D^{2}f(g+b)-D^{2}f(h+b)\right\rVert_{\mathcal{L}(H)}+\left\lVert
D^{2}f\right\rVert_{\infty}\left\lVert g-h\right\rVert\right).$

###### Proof.

Taylor’s remainder theorem in the integral form and Inequality (2.2) imply

$\displaystyle\int_{\overline{B}_{H}}$
$\displaystyle\left|T_{f}(g,b)-T_{f}(h,b)\right|\,\left(\lambda\circ\varphi^{-1}\right)({\rm
d}b)$
$\displaystyle=\int_{\overline{B}_{H}}\left|\int_{0}^{1}\langle(D^{2}f(g+\theta
b)-D^{2}f(h+\theta b))b,b\rangle(1-\theta)\,{\rm
d}\theta\right|\,\left(\lambda\circ\varphi^{-1}\right)({\rm d}b)$
$\displaystyle\leq\frac{1}{2}\sup_{b\in\overline{B}_{H}}\left\lVert
D^{2}f(g+b)-D^{2}f(h+b)\right\rVert_{\mathcal{L}(H)}\int_{\overline{B}_{H}}\left\lVert
b\right\rVert^{2}\,\left(\lambda\circ\varphi^{-1}\right)({\rm d}b)$
$\displaystyle\leq\frac{1}{2}d_{\alpha}^{1}\left(\sup_{b\in\overline{B}_{H}}\left\lVert
D^{2}f(g+b)-D^{2}f(h+b)\right\rVert_{\mathcal{L}(H)}\right)\left\lVert\varphi\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha}.$
(8.3)

A similar argument yields

$\displaystyle\int_{\overline{B}_{H}^{c}}$
$\displaystyle\left|T_{f}(g,b)-T_{f}(h,b)\right|\,\left(\lambda\circ\varphi^{-1}\right)({\rm
d}b)$ (8.4)
$\displaystyle\leq\int_{\overline{B}_{H}^{c}}\left|\int_{0}^{1}\langle
Df(g+\theta b)-Df(h+\theta b),b\rangle\,{\rm
d}\theta\right|\,\left(\lambda\circ\varphi^{-1}\right)({\rm d}b)$
$\displaystyle\qquad\qquad+\int_{\overline{B}_{H}^{c}}\left|\langle
Df(g)-Df(h),b\rangle\right|\,\left(\lambda\circ\varphi^{-1}\right)({\rm d}b)$
$\displaystyle\leq\left(\sup_{b\in H}\left\lVert
Df(g+b)-Df(h+b)\right\rVert+\left\lVert
Df(g)-Df(h)\right\rVert\right)\int_{\overline{B}_{H}^{c}}\left\lVert
b\right\rVert\,\left(\lambda\circ\varphi^{-1}\right)({\rm d}b)$
$\displaystyle\leq 2d_{\alpha}^{1}\left\lVert
D^{2}f\right\rVert_{\infty}\left\lVert
g-h\right\rVert\left\lVert\varphi\right\rVert_{\mathcal{L}_{2}(U,H)}^{\alpha}.$

Combining Inequalities (8.3) and (8.4) completes the proof. ∎

######  Lemma 8.5.

Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of càdlàg processes in $H$ which
converges to a process $X$ both in $\mathcal{C}([0,T],L^{p}(\Omega,H))$ and
uniformly on $[0,T]$ almost surely. Then it follows for any
$f\in\mathcal{C}_{b,u}^{2}(H)$ and $t\in[0,T]$ that

$\lim_{n\to\infty}\int_{0}^{t}\int_{H}T_{f}(X_{n}(s-),h)\,\mu^{X_{n}}({\rm
d}s,{\rm d}h)=\int_{0}^{t}\int_{H}T_{f}(X(s-),h)\,\mu^{X}({\rm d}s,{\rm
d}h)\qquad\text{in }L_{P}^{0}(\Omega,\mathbb{R}).$

###### Proof.

Theorem 3.5 guarantees for each $n\in\mathbb{N}$ that

$\displaystyle\mathbb{E}\,\left[\left|\int_{0}^{t}\int_{H}\big{(}T_{f}(X_{n}(s-),h)-T_{f}(X(s-),h)\big{)}\,\mu^{X_{n}}({\rm
d}s,{\rm d}h)\right|\right]$
$\displaystyle\qquad\leq\mathbb{E}\,\left[\int_{0}^{t}\int_{H}\big{|}T_{f}(X_{n}(s-),h)-T_{f}(X(s-),h)\big{|}\,\mu^{X_{n}}({\rm
d}s,{\rm d}h)\right]$
$\displaystyle\qquad=\mathbb{E}\,\left[\int_{0}^{t}\int_{H}\big{|}T_{f}(X_{n}(s),h)-T_{f}(X(s),h)\big{|}\,\nu^{X_{n}}({\rm
d}s,{\rm d}h)\right]$
$\displaystyle\qquad=\mathbb{E}\,\left[\int_{0}^{t}\int_{H}\big{|}T_{f}(X_{n}(s),h)-T_{f}(X(s),h)\big{|}\,\left(\lambda\circ(R_{n}G(X_{n}(s-)))^{-1}\right)({\rm
d}h)\,{\rm d}s\right].$ (8.5)

Since Remark 6.6 guarantees
$c:=2d_{\alpha}^{1}K_{G}^{\alpha}\sup_{n\in\mathbb{N}}\left\lVert
R_{n}\right\rVert_{\mathcal{L}(H)}^{\alpha}<\infty$, we obtain from Lemma 8.4
for $P\otimes\rm{Leb}$-a.a. $(\omega,s)\in\Omega\times[0,T]$ that

$\displaystyle\int_{H}\left|T_{f}(X_{n}(s)(\omega),h)-T_{f}(X(s)(\omega),h)\right|\,\left(\lambda\circ(R_{n}G(X_{n}(s-)(\omega)))^{-1}\right)({\rm
d}h)$ $\displaystyle\leq c\Bigg{(}\sup_{b\in\overline{B}_{H}}\left\lVert
D^{2}f(X_{n}(s)(\omega)+b)-D^{2}f(X(s)(\omega)+b)\right\rVert_{\mathcal{L}(H)}$
$\displaystyle\qquad\qquad\qquad\qquad+\left\lVert
D^{2}f\right\rVert_{\infty}\left\lVert
f(X_{n}(s)(\omega))-f(X(s)(\omega))\right\rVert\Bigg{)}.$

Since $f$ satisfies (8.1), Lebesgue’s dominated convergence theorem implies

$\displaystyle\lim_{n\to\infty}E\left[\left|\int_{0}^{t}\int_{H}\big{(}T_{f}(X_{n}(s-),h)-T_{f}(X(s-),h)\big{)}\,\mu^{X_{n}}({\rm
d}s,{\rm d}h)\right|\right]=0.$ (8.6)

For the next step, fix $\epsilon,\epsilon^{\prime}>0$, and use for any
$m,n\in\mathbb{N}$ the decomposition

$\displaystyle\left(\left|\int_{0}^{t}\int_{H}T_{f}(X(s-),h)\left(\mu^{X_{n}}({\rm
d}s,{\rm d}h)-\mu^{X}({\rm d}s,{\rm d}h)\right)\right|>\epsilon\right)$
$\displaystyle\qquad\leq
P\left(\left|\int_{0}^{t}\int_{\overline{B}_{H}(1/m)}T_{f}(X(s-),h)\,\mu^{X_{n}}({\rm
d}s,{\rm d}h)\right|>\frac{\epsilon}{3}\right)$
$\displaystyle\qquad\qquad\qquad+P\left(\left|\int_{0}^{t}\int_{\overline{B}_{H}(1/m)}T_{f}(X(s-),h)\,\mu^{X}({\rm
d}s,{\rm d}h)\right|>\frac{\epsilon}{3}\right)$
$\displaystyle\qquad\qquad\qquad+P\left(\left|\int_{0}^{t}\int_{\overline{B}_{H}(1/m)^{c}}T_{f}(X(s-),h)\,\left(\mu^{X_{n}}({\rm
d}s,{\rm d}h)-\mu^{X}({\rm d}s,{\rm
d}h)\right)\right|>\frac{\epsilon}{3}\right).$ (8.7)

Since Taylor’s remainder theorem in the integral form implies
$|T_{f}(X(s-),h)|\leq\frac{1}{2}\left\lVert
D^{2}f\right\rVert_{\infty}\left\lVert h\right\rVert^{2}$ for all $h\in H$, we
obtain by applying Theorem 3.5 and Inequality (2.2) that

$\displaystyle\mathbb{E}\,\left[\left|\int_{0}^{t}\int_{\overline{B}_{H}(1/m)}T_{f}(X(s-),h)\,\mu^{X_{n}}({\rm
d}s,{\rm d}h)\right|\right]$ $\displaystyle\qquad\leq\frac{1}{2}\left\lVert
D^{2}f\right\rVert_{\infty}\mathbb{E}\,\left[\int_{0}^{t}\int_{\overline{B}_{H}(1/m)}\left\lVert
h\right\rVert^{2}\,\left(\lambda\circ\left(R_{n}G(X_{n}(s))\right)^{-1}\right)({\rm
d}h){\rm d}s\right]\leq d_{\alpha}^{m}K_{G}^{\alpha}\frac{1}{2}\left\lVert
D^{2}f\right\rVert_{\infty}T.$

Since the last line is independent of $n\in\mathbb{N}$ and $d_{\alpha}^{m}\to
0$ as $m\to\infty$ according to Inequality (2.2), Markov’s inequality implies
that there exists $m_{1}\in\mathbb{N}$ such that for all $m\geq m_{1}$ and all
$n\in\mathbb{N}$

$\displaystyle
P\left(\left|\int_{0}^{t}\int_{\overline{B}_{H}(1/m)}T_{f}(X(s-),h)\,\mu^{X_{n}}({\rm
d}s,{\rm d}h)\right|>\frac{\epsilon}{3}\right)\leq\epsilon^{\prime}.$ (8.8)

Exactly the same arguments establish that for all $m\geq m_{1}$

$\displaystyle
P\left(\left|\int_{0}^{t}\int_{\overline{B}_{H}(1/m)}T_{f}(X(s-),h)\,\mu^{X}({\rm
d}s,{\rm d}h)\right|>\frac{\epsilon}{3}\right)\leq\epsilon^{\prime}.$ (8.9)

For the last term in (8.7) we calculate for each $m,n\in\mathbb{N}$ that

$\displaystyle\int_{0}^{t}\int_{\overline{B}_{H}(1/m)^{c}}T_{f}(X(s-),h)\,\left(\mu^{X_{n}}({\rm
d}s,{\rm d}h)-\mu^{X}({\rm d}s,{\rm d}h)\right)$ $\displaystyle=\sum_{0\leq
s\leq t}T_{f}(X(s-),\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}({\Delta X_{n}(s)})$
$\displaystyle\qquad-\sum_{0\leq s\leq t}T_{f}(X(s-),\Delta
X(s))\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}({\Delta X(s)})$
$\displaystyle=\sum_{0\leq s\leq t}T_{f}(X(s-),\Delta
X_{n}(s))\left(\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}({\Delta
X_{n}(s)})-\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}({\Delta X(s)})\right)$
$\displaystyle\qquad+\sum_{0\leq s\leq t}\big{(}T_{f}(X(s-),\Delta
X_{n}(s))-T_{f}(X(s-),\Delta
X(s))\big{)}\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}({\Delta X(s)}).$ (8.10)

For estimating the first term in the last line, we use the equality
$\mathbbm{1}_{A}(x)-\mathbbm{1}_{A}(y)=\mathbbm{1}_{A}(x)\mathbbm{1}_{A^{c}}(y)-\mathbbm{1}_{A^{c}}(x)\mathbbm{1}_{A}(y)$.
For the first term, resulting from the application of this identity, we
conclude from Taylor’s remainder theorem in the integral form that

$\displaystyle\left|\sum_{0\leq s\leq t}T_{f}\left(X(s-\right),\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}(\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)}(\Delta X(s))\right|$
$\displaystyle\leq\sum_{0\leq s\leq t}\big{|}T_{f}(X(s-),\Delta
X_{n}(s))\big{|}\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}(\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)}(\Delta X(s))$
$\displaystyle\leq\frac{1}{2}\left\lVert
D^{2}f\right\rVert_{\infty}\sum_{0\leq s\leq t}\left\lVert\Delta
X_{n}(s)\right\rVert^{2}\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}(\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)}(\Delta X(s))$ $\displaystyle\leq
c_{f}\\!\\!\sum_{0\leq s\leq t}\\!\\!\left(\left\lVert\Delta
X(s)\right\rVert^{2}+\left\lVert\Delta X_{n}(s)-\Delta
X(s)\right\rVert^{2}\right)\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}(\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)}(\Delta X(s)),$ (8.11)

where we used the notation $c_{f}:=\left\lVert D^{2}f\right\rVert_{\infty}$.
Applying Theorem 3.5 and using the boundedness assumption on $G$ in (A3)
result in

$\displaystyle\mathbb{E}\,\left[\sum_{0\leq s\leq t}\left\lVert\Delta
X(s)\right\rVert^{2}\mathbbm{1}_{\overline{B}_{H}(1/m)}(\Delta
X(s))\right]=\mathbb{E}\,\left[\int_{0}^{t}\int_{\overline{B}_{H}(1/m)}\left\lVert
h\right\rVert^{2}\,\mu^{X}({\rm d}s,{\rm d}h)\right]$
$\displaystyle\qquad=\mathbb{E}\,\left[\int_{0}^{t}\int_{\overline{B}_{H}(1/m)}\left\lVert
h\right\rVert^{2}\,\big{(}\lambda\circ G(X(s-))^{-1}\big{)}({\rm d}h,{\rm
d}s)\right]\leq d_{\alpha}^{m}TK_{G}^{\alpha}.$

Since $d_{\alpha}^{m}\to 0$ as $m\to\infty$, Markov’s inequality implies that
there exists $m_{2}\in\mathbb{N}$ with $m_{2}\geq m_{1}$ such that for all
$m\geq m_{2}$ and all $n\in\mathbb{N}$

$\displaystyle P\left(\sum_{0\leq s\leq t}\left\lVert\Delta
X(s)\right\rVert^{2}\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}(\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)}({\Delta
X(s)})\geq\frac{\epsilon}{24c_{f}}\right)\leq\frac{\epsilon^{\prime}}{8}.$
(8.12)

In the remaining part of the proof fix $m\geq m_{2}\geq m_{1}$ such that
(8.12) is satisfied. There exists $n_{1}\in\mathbb{N}$ such that the set
$A_{n}:=\\{\sup_{s\in[0,t]}\left\lVert\Delta X_{n}(s)-\Delta
X(s)\right\rVert\leq\tfrac{1}{2m}\\}$ satisfies
$P(A_{n}^{c})\leq\tfrac{\epsilon^{\prime}}{16}$ for all $n\geq n_{1}$. If
$\sup_{s\in[0,t]}\left\lVert\Delta X_{n}(s)\right\rVert\geq\tfrac{1}{m}$ then
we obtain on $A_{n}$ for $n\geq n_{1}$ that

$\displaystyle\sup_{s\in[0,t]}\left\lVert\Delta
X(s)\right\rVert\geq-\sup_{s\in[0,t]}\left\lVert\Delta X(s)-\Delta
X_{n}(s)\right\rVert+\sup_{s\in[0,t]}\left\lVert\Delta
X_{n}(s)\right\rVert\geq\frac{1}{2m}.$

Consequently, we obtain for all $n\geq n_{1}$ that

$\displaystyle P\left(\sum_{0\leq s\leq t}\left\lVert\Delta X_{n}(s)-\Delta
X(s)\right\rVert^{2}\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}(\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)}(\Delta
X(s))\geq\frac{\epsilon}{24c_{f}}\right)$ $\displaystyle\qquad\leq
P\left(\sum_{0\leq s\leq t}\left\lVert\Delta X_{n}(s)-\Delta
X(s)\right\rVert^{2}\mathbbm{1}_{\overline{B}_{H}(1/2m)^{c}}(\Delta
X(s))\geq\frac{\epsilon}{24c_{f}}\right)+\frac{\epsilon^{\prime}}{16}.$

Since $X$ has only finitely many jumps in $\overline{B}_{H}(1/2m)^{c}$ on
$[0,t]$ and $\Delta X_{n}(s)$ converges to $\Delta X(s)$ for all $s\in[0,t]$,
there exists $n_{2}$ such that for all $n\geq n_{2}$

$\displaystyle P\left(\sum_{0\leq s\leq t}\left\lVert\Delta X_{n}(s)-\Delta
X(s)\right\rVert^{2}\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}(\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)}(\Delta
X(s))\geq\frac{\epsilon}{24c_{f}}\right)\leq\frac{\epsilon^{\prime}}{8}.$

Applying this together with (8.12) to Inequality (8) proves that for $m\geq
m_{2}$ there exists $n_{2}$ such that for all $n\geq n_{2}$

$\displaystyle P\left(\sum_{0\leq s\leq t}T_{f}\left(X(s-\right),\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}(\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)}(\Delta
X(s))\geq\frac{\epsilon}{12}\right)\leq\frac{\epsilon^{\prime}}{4}.\quad$
(8.13)

As $\Delta X_{n}$ converges to $\Delta X$ uniformly on $[0,T]$ almost surely
we obtain that for almost all $\omega\in\Omega$ we have

$\displaystyle\mathbbm{1}_{\overline{B}_{H}(1/m)}(\Delta
X_{n}(s)(\omega))\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}(\Delta
X(s)(\omega))=0$

if $n$ is large enough. Therefore

$\displaystyle\lim_{n\to\infty}\left(\sum_{0\leq s\leq t}T_{f}(X(s-),\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)}(\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}(\Delta
X(s))\right)=0\quad\text{a.s.}$

and thus we obtain that there exists $n_{3}$ such that for all $n\geq n_{3}$
we have

$\displaystyle P\left(\sum_{0\leq s\leq t}T_{f}(X(s-),\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)}(\Delta
X_{n}(s))\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}(\Delta
X(s))\geq\frac{\epsilon}{12}\right)\leq\frac{\epsilon^{\prime}}{4}.$

Combining this with (8.13) shows that for $m\geq m_{2}$ and
$n\geq\max\\{n_{2},n_{3}\\}$ we have

$\displaystyle P\left(\sum_{0\leq s\leq t}T_{f}(X(s-),\Delta
X_{n}(s))\left(\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}({\Delta
X_{n}(s)})-\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}({\Delta
X(s)})\right)\geq\frac{\epsilon}{6}\right)\leq\frac{\epsilon^{\prime}}{2}.$
(8.14)

Since $X$ has only finitely many jumps in $\overline{B}_{H}(1/m)^{c}$ on
$[0,t]$ and $\Delta X_{n}(s)$ converges to $\Delta X(s)$ for all $s\in[0,t]$,
there exits $n_{4}$ such that for all $n\geq n_{4}$

$\displaystyle P\left(\sum_{0\leq s\leq t}\big{(}T_{f}(X(s-),\Delta
X_{n}(s))-T_{f}(X(s-),\Delta
X(s))\big{)}\mathbbm{1}_{\overline{B}_{H}(1/m)^{c}}({\Delta
X(s)})\geq\frac{\epsilon}{6}\right)\leq\frac{\epsilon^{\prime}}{2}.$

Applying this together with (8.14) to (8) shows

$\displaystyle
P\left(\int_{0}^{t}\int_{\overline{B}_{H}(1/m)^{c}}T_{f}(X(s-),h)\,\left(\mu^{X_{n}}({\rm
d}s,{\rm d}h)-\mu^{X}({\rm d}s,{\rm
d}h)\right|\geq\frac{\epsilon}{3}\right)\leq\epsilon^{\prime}.$ (8.15)

By applying Equations (8.8),(8.9) and (8.15) to (8.7), the proof is now
complete. ∎

###### Proof of Theorem 8.1.

Let $(X_{n})_{n\in\mathbb{N}}$ be the solutions to (6.7). According to
Corollary 6.10, we can assume that $(X_{n})_{n\in\mathbb{N}}$ converges both
in $\mathcal{C}([0,T],L^{p}(\Omega,H))$ and uniformly on $[0,T]$ almost surely
to the mild solution $X$, which has càdlàg paths. Since $X_{n}$ is a strong
solution to (6.7) according to Proposition 7.3, the Itô formula in Theorem 5.1
implies for all $t\geq 0$ and $n\in\mathbb{N}$ that

$\displaystyle f(X_{n}(t))$ $\displaystyle=f(R_{n}x_{0})+\int_{0}^{t}\langle
G(X_{n}(s-))^{*}R_{n}^{*}Df(X_{n}(s-)),\cdot\rangle\,{\rm d}L(s)$
$\displaystyle\qquad+\int_{0}^{t}\langle Df(X_{n}(s)),AX_{n}(s)\rangle\,{\rm
d}s+\int_{0}^{t}\langle Df(X_{n}(s)),R_{n}F(X_{n}(s))\rangle\,{\rm d}s$ (8.16)
$\displaystyle\qquad+\int_{0}^{t}\int_{H}\bigg{(}f(X_{n}(s-)+h)-f(X_{n}(s-))-\langle
Df(X_{n}(s-)),h\rangle\bigg{)}\,\mu^{X_{n}}({\rm d}s,{\rm d}h).$

Continuity of $f$ shows that $f(X_{n}(t))\to f(X(t))$ and $f(R_{n}x_{0})\to
f(x_{0})$ a.s. Inequality (2.7) implies for the first integral in (8) that

$\displaystyle\mathbb{E}\,$ $\displaystyle\left[\left\lVert\int_{0}^{t}\langle
G(X_{n}(s-))^{*}R_{n}^{*}Df(X_{n}(s-)),\cdot\rangle\,{\rm
d}L(s)-\int_{0}^{t}\langle G(X(s-))^{*}Df(X(s-)),\cdot\rangle\,{\rm
d}L(s)\right\rVert\right]$ $\displaystyle\leq
e_{1,\alpha}\left(\mathbb{E}\,\left[\int_{0}^{t}\left\lVert
G(X_{n}(s))^{*}R_{n}^{*}Df(X_{n}(s))-G(X(s))^{*}Df(X(s))\right\rVert_{\mathcal{L}_{2}(U,\mathbb{R})}^{\alpha}\,{\rm
d}s\right]\right)^{1/\alpha},$

which tends to zero by a similar argument as in the proof of Lemma 7.4. It
follows in $L_{P}^{1}(\Omega,\mathbb{R})$ that

$\displaystyle\lim_{n\to\infty}\int_{0}^{t}\langle
G(X_{n}(s-))^{*}R_{n}^{*}Df(X_{n}(s-)),\cdot\rangle\,{\rm
d}L(s)=\int_{0}^{t}\langle G(X(s-))^{*}Df(X(s-)),\cdot\rangle\,{\rm d}L(s).$

Lemma 8.5 implies in $L_{P}^{0}(\Omega,\mathbb{R})$ that

$\displaystyle\lim_{n\to\infty}\int_{0}^{t}\int_{H}\big{(}f(X_{n}(s-)+h)-f(X_{n}(s-))-\langle
Df(X_{n}(s-)),h\rangle\big{)}\,\mu^{X_{n}}({\rm d}s,{\rm d}h)$
$\displaystyle\qquad=\int_{0}^{t}\int_{H}\big{(}f(X(s-)+h)-f(X(s-))-\langle
Df(X(s-)),h\rangle\big{)}\,\mu^{X}({\rm d}s,{\rm d}h).$

Convergence of $(X_{n})_{n\in\mathbb{N}}$ and Lebesgue’s dominated convergence
theorem yields

$\displaystyle\lim_{n\to\infty}\int_{0}^{t}\langle
Df(X_{n}(s)),R_{n}F(X_{n}(s))\rangle\,{\rm d}s=\int_{0}^{t}\langle
Df(X(s)),F(X(s))\rangle\,{\rm d}s\quad\text{a.s.}$

As all terms in (8) converge in $L_{P}^{0}(\Omega,\mathbb{R})$, it follows
that the remaining term

$\displaystyle\int_{0}^{t}\langle Df(X_{n}(s)),AX_{n}(s)\,\rangle{\rm d}s$

also converges in $L_{P}^{0}(\Omega,\mathbb{R})$, which completes the proof. ∎

##  9 Appendix

######  Lemma 9.1.

Let $V$ be a separable Hilbert space with the norm
$\left\lVert\cdot\right\rVert_{V}$ and let $A_{m}\in\mathcal{L}(V)$ be a
sequence of operators converging strongly to $0$. If
$(B_{n})_{n\in\mathbb{N}}$ is a tight sequence of uniformly bounded $V$-valued
random variables, then it follows for all $p>0$ that

$\lim_{m\rightarrow\infty}\sup_{n\in\mathbb{N}}\mathbb{E}\,\left[\left\lVert
A_{m}B_{n}\right\rVert_{V}^{p}\right]=0.$

###### Proof.

Let $\epsilon>0$ be fixed. Our assumptions guarantee that there exists a