# New Stochastic Fubini Theorems

Tahir Choulli
Mathematical and Statistical Sciences Dept.
University of Alberta  Edmonton  Canada
<EMAIL_ADDRESS>Martin Schweizer
ETH Zürich
Departement Mathematik
ETH-Zentrum  HG G 51.2
Rämistrasse 101
CH – 8092 Zürich
<EMAIL_ADDRESS>
###### Abstract

The classic stochastic Fubini theorem says that if one stochastically
integrates with respect to a semimartingale $S$ an $\eta(dz)$-mixture of
$z$-parametrized integrands $\psi^{z}$, the result is just the
$\eta(dz)$-mixture of the individual $z$-parametrized stochastic integrals
$\int\psi^{z}{d}S.$ But if one wants to use such a result for the study of
Volterra semimartingales of the form $X_{t}=\int_{0}^{t}\Psi_{t,s}dS_{s},t\geq
0,$ the classic assumption that one has a fixed measure $\eta$ is too
restrictive; the mixture over the integrands needs to be taken instead with
respect to a stochastic kernel on the parameter space. To handle that
situation and prove a corresponding new stochastic Fubini theorem, we
introduce a new notion of measure-valued stochastic integration with respect
to a general multidimensional semimartingale. As an application, we show how
this allows to handle a class of quite general stochastic Volterra
semimartingales.

Keywords: stochastic Fubini theorem, measure-valued stochastic integrals,
weak-Fubini property Volterra semimartingales.

MSC 2000 Classification Numbers: 60H05, 28B05, 60G48

##  1 Introduction

The standard setup for a stochastic Fubini theorem is that one has a
semimartingale $S$, a parametric family $\psi^{z}$ of processes indexed by $z$
from some space $\cal{Z}$, and a (fixed) measure $\eta$ on $\cal{Z}$. For each
$z$, the process $\psi^{z}$ is assumed to be $S$-integrable, and so is the
mixture $\bar{\psi}^{\eta}:=\int\psi^{z}\eta(dz)$. The typical classic
stochastic Fubini theorem then has the form

$\int_{(0,\cdot]}\left(\int_{\cal{Z}}\psi^{z}\eta(dz)\right)dS_{t}=\int_{(0,\cdot]}\bar{\psi}^{\eta}_{t}{d}S_{t}=\int_{\cal{Z}}\left(\int_{(0,\cdot]}\psi^{z}_{t}{d}S_{t}\right)\eta(dz);$
(0.1)

see for instance Théorème 5.44 of Jacod (1979), Lebedev (1995) [which corrects
an error in Jacod (1979)], Theorem IV.65 of Protter (2005), or Bichteler/Lin
(1995) [which extends the result in (the first, 1990 edition of) Protter
(2005) from an $L^{2}$\- to an $L^{1}$-setting]. A good overview with very
weak assumptions has recently been given by Veraar (2012).

The Fubini property (0.1) says that if one stochastically integrates a mixture
(over $z$, with respect to $\eta$) of integrands, the result is given by the
analogous mixture of the individual stochastic integrals. This looks very
general, but it actually imposes one very restrictive assumption — one has a
single fixed measure ? which mixes over the parameter $z$, and which does not
depend on the randomness $\omega$ (nor on time $t$). Some very natural
applications like the study of Volterra semimartingales of the form
$X_{t}=\int_{0}^{t}\Psi_{s,t}dS_{s}$, $t\geq 0$, lead almost immediately
outside the scope of the above standard setup. In this paper, we therefore
study what can be done if one replaces $\eta(dz)$ by a kernel
$\rho(\omega,t,dz)$ from $\Omega\times(0,\infty)\times{\cal{P}})$ to
$\cal{Z}$. One can then still easily make sense of the left-hand side of (0.1)
by just mixing the integrands with $\rho$ instead of $\eta$, so that one
considers $\bar{\psi}^{\rho}:=\int\psi^{z}_{t}\rho_{t}(dz)$; but it is a
priori not clear how the result can be expressed on the right-hand side sonce
it makes no sense (and is not even well defined) to take there a mixture of
$\int\psi^{z}_{t}{d}S_{t}$ with $\rho_{t}(dz)$.

To answer this question, we have to take a new viewpoint and develop a new
theory. Our solution will be to view
$\varphi_{t}(dz):=\psi^{z}_{t}\rho_{t}(dz)$ as a measure-valued stochastic
integral of $\varphi$ with respect to $S$, and to obtain a weak* stochastic
Fubini theorem by plugging in test functions $f$ or indicators $I_{D}$ of sets
$D\subseteq{\cal{Z}}$. This will allow us to make sense of a generalized form
of (0.1). We explain below in more detail why this needs in particular a new
stochastic integral.

Our approach combines and generalises two ideas from the existing literature
in a nontrivial way. The idea of viewing stochastic Fubini theorems via more
abstract integrands can be traced back to van Neerven/Veraar (2006) who proved
a stochastic Fubini theorem for an infinite-dimensional integrator $S$ (a
cylindrical Brownian motion). But they still work with a fixed reference
measure $\eta$ on $\cal{Z}$, and so their integrands have values in
$L^{1}({\cal{Z}},\eta)$, since one can identify a finite measure $m\ll\eta$
with its Radon–Nikodým derivative ${{dm}\over{d\eta}}$. Moreover, their whole
approach does not extend substantially beyond the case of Brownian motion as
integrator. For more general measure-valued integrands, Björk/Di
Masi/Kabanov/Runggaldier (1997) develop a stochastic integration theory with
respect to a function-valued integrator $S$, with the resulting integral being
real-valued. The construction and setup for our measure-valued stochastic
integral is strongly inspired by that work. However, Björk/Di
Masi/Kabanov/Runggaldier (1997) did not look at the issue of more general
stochastic Fubini theorems at all, and their integral processes are real-
valued which makes their definition and construction reasonably
straightforward.

One important application as well as motivation is the study of Volterra
semimartingales of the form $X_{t}=\int_{0}^{t}\Psi_{s,t}dS_{s}$ with a two-
parameter process $\Psi=(\Psi_{t,s})$. If $\Psi$ is, with respect to $s$,
predictable and $S$-integrable for each $t$, including on the diagonal $s=t$,
and, with respect to $t$, of finite and $S$-integrable variation, we show that
$X$ is again a semimartingale, and that we can say more about its
decomposition. This substantially generalizes a result by Protter (1985) which
was obtained under the restrictive condition that each $\Psi_{t,s}$ is, with
respect to $t$, absolutely continuous with respect to Lebesgue measure. That
assumption allows to reduce the question to a classic stochastic Fubini
theorem, which is no longer possible if absolute continuity (with respect to a
fixed reference measure) is generalized to finite variation.

Our results rely on a stochastic integral of measure-valued processes with
respect to a general $\mathbb{R}^{d}$-valued semimartingale, with the integral
also taking values in a space of measures. This is an unusual setup and to the
best of our knowledge not covered by existing literature. One often finds
stochastic integrals where either integrand or integrator are measure-valued
and integrator respectively integrand take values in a corresponding dual
space; see for instance Mikulevicius/Rozovskii (1998). We also mention the
paper by Krylov (2011) who proves an Itô-Wentzell formula for distribution-
valued processes by plugging in test functions and using or extending classic
results. The crucial point here is that the integral processes in all these
cases are still real-valued. If one looks instead at integrals with values in
a Banach space, the construction of a stochastic integral (even with respect
to Brownian motion) typically becomes nontrivial and depends on the geometry
of the Banach space; see for example Dettweiler (1985). Some recent papers
have managed to overcome this difficulty by reformulating the Itô isometry
property via an operator-theoretic approach. But this has been done only for
the case where the integrator is a (possibly infinite-dimensional) Brownian
motion, and the approach also crucially exploits the underlying Gaussian
structure; see for instance van Neerven/Veraar/Weis (2007). A recent overview
of this theory with many references can be found in the survey paper by van
Neerven/Veraar/Weis (2013). For semimartingale integrators, corresponding
results do not seem to exist yet.

This paper is organized as follows. We present the basic setup and some
preliminary results in Section 2. Section 3 develops the stochastic integral
of measure-valued processes $\varphi$ with respect to the semimartingale $S$,
and this integral is vital for our new stochastic Fubini theorems. Section 4
contains new stochastic Fubini theorems. We start with the case when $S$ is a
locally square integrable local martingale, in two variants for different
applications. Then, we extend the results to the semimartingale case
afterwards. Section 5 shows how these results can be used to study the above
Volterra semimartingale $X$, and we also explain how they help to obtain a
decomposition for $X$. Section 6 discusses the connection between our new
stochastic Fubini theorem and the classical stochastic Fubini theorem, while
Section 7 illustrates an example.

##  2 Preliminaries

In this section, we collect some basic notations and concepts.

###  2.1 Notations

Let $(\Omega,\mathcal{F},P)$ be a probability space with a filtration
$\mathbb{F}=\left(\mathcal{F}_{t}\right)_{t\geq 0}$ satisfying the usual
conditions of right-continuity and completeness. Throughout the paper, we set
$\mathcal{F}_{\infty}:=\bigvee_{t\geq
0}\mathcal{F}_{t}=\sigma\left(\bigcup_{t\geq 0}\mathcal{F}_{t}\right)$ and
denote by $\mathcal{P}$ the predictable $\sigma$-field on
$\bar{\Omega}:=\Omega\times(0,\infty)$. Standard terminology and results from
stochastic calculus can be found in Dellacherie/Meyer [9, Chapters VI-VIII]
and Jacod/Shiryaev [16, Chapter I]. Identifying as usual processes that are
indistinguishable, we define

$\mathcal{R}^{0}:=\left\\{\mbox{all real-valued adapted RCLL processes}\
Z=\left(Z_{t}\right)_{t\geq 0}\right\\}.$

For any $Z\in\mathcal{R}^{0}$, we set $Z_{t}^{*}:=\sup_{0\leq s\leq
t}\left|Z_{s}\right|$ and $Z_{t-}^{*}:=\sup_{0\leq s<t}\left|Z_{s}\right|$ for
$t\geq 0$, with $Z_{\infty}^{*}=Z_{\infty-}^{*}:=\sup_{t\geq
0}\left|Z_{t}\right|$. Note that $Z^{*}=\left(Z_{t}^{*}\right)_{t\geq 0}$ is
in $\mathcal{R}^{0}$, but $Z_{\infty}^{*}$ may take the value $+\infty$. We
equip $\mathcal{R}^{0}$ with the topology of ucp-convergence (i.e.,
$Z^{n}\rightarrow Z$ if $\left(Z^{n}-Z\right)_{t}^{*}\rightarrow 0$ in
probability for every $t\geq 0$ ). We also need the Banach space

$\mathcal{R}^{2}:=\left\\{Z\in\mathcal{R}^{0}:\|Z\|_{\mathcal{R}^{2}}:=\left\|Z_{\infty}^{*}\right\|_{L^{2}}<\infty\right\\}.$

For any nonnegative increasing process $A$ in $\mathcal{R}^{0}$ and any
(possibly multidimensional) product-measurable $H=\left(H_{t}\right)_{t>0}$ on
$\bar{\Omega}$, we define the process $\mathcal{D}(H;A)$ by

$\mathcal{D}_{t}(H;A):=\int_{0}^{t}\left|H_{r}\right|^{2}\mathrm{~{}d}A_{r}:=\int_{(0,t]}\left|H_{r}\right|^{2}\mathrm{~{}d}A_{r}\in[0,\infty]\quad\text{
for all }t\geq 0$

Note that $\mathcal{D}(H;A)$ is always increasing, null at 0 and may take the
value $+\infty$; it is in addition adapted if $H$ is progressively measurable
(and in particular if $H$ is predictable). For any stopping time $\tau$, we
define the measure $\mu_{\tau,A}$ on the product $\sigma$-field of
$\bar{\Omega}$ by

$\mu_{\tau,A}(C):=E\left[A_{\tau-}\mathcal{D}_{\tau-}\left(I_{C};A\right)\right]=E\left[A_{\tau-}(\omega)\int_{0}^{\tau-(\omega)}I_{C}(\omega,r)\mathrm{d}A_{r}(\omega)\right]$

so that

$\|H\|_{L^{2}\left(\mu_{\tau,A}\right)}^{2}=E\left[A_{\tau-}\int_{0}^{\tau-}\left|H_{r}\right|^{2}\mathrm{~{}d}A_{r}\right]=E\left[A_{\tau-}\mathcal{D}_{\tau-}(H;A)\right]$

Note that both $A$ and $\mathcal{D}(1;A)=A-A_{0}$ are in $\mathcal{R}^{0}$.
Therefore the product $A\mathcal{D}(1;A)$ is prelocally bounded, and we can
find a localizing sequence $\left(\tau_{M}\right)_{M\in\mathbb{N}}$ of
stopping times such that $\mu_{\tau_{M},A}$ is a finite measure for each
$M\in\mathbb{N}$. Note that if $H$ is bounded by a constant $C$, say, then
$\mathcal{D}(H;A)\leq C^{2}A$ so that any bounded process is in
$L^{2}\left(\mu_{\tau,A}\right)$ as soon as $A_{\tau-}\in L^{2}$.

###  2.2 Semimartingales and stochastic integrals

For semimartingales and stochastic integration, we use the approach due to
Métivier and Pellaumail as presented in the textbook by Métivier [21]. First,
an $\mathbb{R}^{d}$-valued simple predictable process is of the form

$H=\sum_{\ell=0}^{L-1}h_{\ell}I_{A_{\ell}\times\left(t_{\ell},t_{\ell+1}\right]}$
(2.1)

where $L\in\mathbb{N},0\leq
t_{0}<t_{1}<\cdots<t_{L}<\infty,A_{\ell}\in\mathcal{F}_{t_{\ell}}$ and
$h_{\ell}$ is $\mathbb{R}^{d}$-valued, bounded and
$\mathcal{F}_{t_{\ell}}$-measurable. If the $h_{\ell}$ are nonrandom, we call
$H$ very simple and denote by $\mathcal{H}$ the family of all very simple
predictable processes. For any $\mathbb{R}^{d}$-valued adapted RCLL process
$S=\left(S_{t}\right)_{t\geq 0}$ and $H$ as in (2.1), we define the stochastic
integral process $H\cdot S=\int H\mathrm{~{}d}S$ by

$H\cdot S_{t}:=(H\cdot
S)_{t}:=\sum_{\ell=0}^{L-1}I_{A_{\ell}}h_{\ell}^{\top}\left(S_{t_{\ell+1}\wedge
t}-S_{t_{\ell}\wedge
t}\right)=\sum_{\ell=0}^{L-1}I_{A_{\ell}}h_{\ell}^{\top}\left(S_{t_{\ell+1}}^{t}-S_{t_{\ell}}^{t}\right),\quad
t\geq 0$

It is clear that $H\cdot S$ is in $\mathcal{R}^{0}$ and null at 0 . A control
process is a nonnegative increasing process $V=\left(V_{t}\right)_{t\geq 0}$
in $\mathcal{R}^{0}$ such that for any stopping time $\tau$,

$E\left[\sup_{0\leq
t<\tau}\left|\int_{0}^{t}H_{r}\mathrm{~{}d}S_{r}\right|^{2}\right]\leq
E\left[V_{\tau-}\int_{0}^{\tau-}\left|H_{r}\right|^{2}\mathrm{~{}d}V_{r}\right]\quad\text{
for any }H\in\mathcal{H}\text{. }$ (2.2)

We denote by $\mathcal{V}(S)$ the family of all control processes for $S$. By
Métivier [21, Theorem 23.14], $S$ is a semimartingale if and only if it admits
a control process, i.e., $\mathcal{V}(S)\neq\emptyset$. Indeed, with
$\mathbb{H}=\mathbb{R}^{d}$, the "if" part follows from part
$\left(2^{\circ}\right)$ of that theorem, and the "only if" part follows from
part $\left(1^{\circ}\right)$ with $\mathbb{G}=\mathbb{R}^{d}$, noting that
one can easily extend (2.2) from very simple to simple predictable processes.

Now suppose $S$ is an $\mathbb{R}^{d}$-valued semimartingale. Using the
notations from Section 2.1 and choosing $V\in\mathcal{V}(S)\neq\emptyset$, we
can rewrite (2.2) compactly, for any stopping time $\tau$, as

$\left\|H\cdot S^{\tau-}\right\|_{\mathcal{R}^{2}}=\left\|(H\cdot
S)^{\tau-}\right\|_{\mathcal{R}^{2}}\leq\|H\|_{L^{2}\left(\mu_{\tau,V}\right)}\quad\text{
for any }H\in\mathcal{H}$ (2.3)

If $\tau$ is such that $V_{\tau-}\in L^{2}$ (or, equivalently, $\mu_{\tau,V}$
is a finite measure), then by [21, Sections 24.1 and 26.1], any
$\mathbb{R}^{d}$-valued predictable process $H$ with
$\|H\|_{L^{2}\left(\mu_{\tau,V}\right)}<\infty$ is integrable with respect to
$S$ on $\llbracket 0,\tau\llbracket$ so that $(H\cdot S)^{\tau-}$ is well
defined and a real-valued semimartingale. As $S=S^{\tau-}$ on $\llbracket
0,\tau\llbracket$, we can also write $(H\cdot S)^{\tau-}=H\cdot S^{\tau-}$ and
view this equivalently either as the stochastic integral of $H$ with respect
to $S$ on $\llbracket 0,\tau\llbracket$, or as the stochastic integral of $H$
with respect to $S^{\tau-}$ on $\llbracket
0,\infty\rrbracket=[0,\infty)\times\Omega$. From the construction in [21], we
also have, extending (2.3), that (again for $V_{\tau-}\in L^{2}$)

$\left\|H\cdot
S^{\tau-}\right\|_{\mathcal{R}^{2}}\leq\|H\|_{L^{2}\left(\mu_{\tau,V}\right)}\
\text{ for any }\mathbb{R}^{d}\text{-valued predictable }H\in
L^{2}\left(\mu_{\tau,V}\right)$ (2.4)

To get rid of the stopping time $\tau$, fix a control process
$V\in\mathcal{V}(S)$. Take any $\mathbb{R}^{d}$-valued predictable process $H$
such that $\mathcal{D}(H;V)=\int\left|H_{r}\right|^{2}\mathrm{~{}d}V_{r}$ is
finite-valued (i.e., in $\mathcal{R}^{0}$ ) and note that both $V$ and
$\mathcal{D}(H;V)$ are in $\mathcal{R}^{0}$ and hence prelocally bounded.
Choose a localising sequence $\left(\tau_{M}\right)_{M\in\mathbb{N}}$ with
$V_{\tau_{M}-}\in L^{2}$ and $H\in L^{2}\left(\mu_{\tau_{M},V}\right)$, for
each $M\in\mathbb{N}$. This allows us to define the stochastic integral
$H\cdot S=\int H\mathrm{~{}d}S$ on every stochastic interval $\llbracket
0,\tau_{M}\llbracket$ and hence on $\llbracket
0,\infty\rrbracket=[0,\infty)\times\Omega$ by pasting together. It is shown in
$[21$, Section 24.1$]$ that this stochastic integral does not depend on the
choice of localising sequence, nor on the choice of the control process
$V\in\mathcal{V}(S)$.

###  2.3 Measure-valued processes

Fix a compact metric space $\mathcal{K}$ equipped with its Borel
$\sigma$-field $\mathcal{B}(\mathcal{K})$, denote by
$\mathbb{M}:=\mathbb{M}(\mathcal{K})$ the space of signed and finite measures
$m$ on $\mathcal{B}(\mathcal{K})$ and recall that each $m\in\mathbb{M}$ can be
written as $m=m^{+}-m^{-}$for two unique finite measures $m^{\pm}$on
$\mathcal{B}(\mathcal{K})$ with disjoint supports. The total variation measure
of $m$ is $|m|=m^{+}+m^{-}$. Write $C(\mathcal{K}):=C(\mathcal{K};\mathbb{R})$
for the space of continuous functions $f:\mathcal{K}\rightarrow\mathbb{R}$
with the sup-norm $\|f\|_{\infty}$ and denote the closed unit ball in
$C(\mathcal{K})$ by $U_{1}:=\left\\{f\in C(\mathcal{K}):\|f\|_{\infty}\leq
1\right\\}$. The integral of $f$ with respect to $m$ is

$m(f):=\int
f\mathrm{~{}d}m:=\int_{\mathcal{K}}f(z)m(\mathrm{~{}d}z)\quad\text{ for }f\in
C(\mathcal{K})\text{ and }m\in\mathbb{M}$

The variation norm of $m\in\mathbb{M}$ (not to be confused with $|m|$) is then
given by

$\|m\|_{\mathrm{var}}:=\sup\left\\{m(f):f\in U_{1}\right\\}=|m|(\mathcal{K}).$

We take on $\mathbb{M}$ the $\sigma$-field $\mathcal{M}$ generated by the
weak* topology, i.e. by all the mappings $m\mapsto m(f)$ with $f\in
C(\mathcal{K})$. For $d\in\mathbb{N}$ and
$m=\left(m^{i}\right)_{i=1,\ldots,d}\in\mathbb{M}^{d}$, we set

$\|m\|_{\mathrm{var}}:=\left(\left\|m^{i}\right\|_{\mathrm{var}}\right)_{i=1,\ldots,d},\quad
m(f):=\left(m^{i}(f)\right)_{i=1,\ldots,d}$

Finally, we denote by $bB_{1}(\mathcal{K})$ the space of all bounded
$g:\mathcal{K}\rightarrow\mathbb{R}$ of Baire class 1 . Recall that $g\in
bB_{1}(\mathcal{K})$ if and only if $g$ is a pointwise limit of continuous
functions; see e.g. Kuratowski [19, Theorems II.VIII. 1 and II.VIII.7].

For ease of reference, we list here some properties used later and give the
corresponding references to the textbook by Aliprantis/Border [2]. Because
$\mathcal{K}$ is compact and metric, it is Hausdorff and Polish by [2, Theorem
3.28 and Lemma 3.26], and $C(\mathcal{K})$ is separable for the sup-norm by
$[2$, Lemma 3.99]. In turn, separability of $C(\mathcal{K})$ implies that the
closed unit ball of its dual space $\mathbb{M}(\mathcal{K})=\mathbb{M}$ is
metrisable for the weak* topology by [2, Theorem 6.30], and the same also
holds for $\mathbb{M}^{d}=\left((C(\mathcal{K}))^{d}\right)^{*}$. Because
$\mathcal{K}$ is Polish, every finite (nonnegative) measure $\mu$ on
$\mathcal{B}(\mathcal{K})$ is regular by $\left[2\right.$, Theorem 12.7], and
therefore $C(\mathcal{K})$ is dense in $L^{p}(\mu)$ for every
$p\in[1,\infty)$, but not for $p=\infty$; see $[2$, Theorem 13.9].

We want to use (signed) measure-valued processes as integrands for a
stochastic integral. A process $\varphi=\left(\varphi_{t}\right)_{t\geq 0}$ on
$(\Omega,\mathcal{F},P)$ with values in $\mathbb{M}^{d}$ is weak $k^{*}$
predictable if the $\mathbb{R}^{d}$-valued process
$\varphi(f)=\left(\varphi_{t}(f)\right)_{t\geq 0}$ is predictable for each
$f\in C(\mathcal{K})$. In view of the definition of the $\sigma$-field
$\mathcal{M}$, this is equivalent to saying that
$\varphi:\bar{\Omega}\rightarrow\mathbb{M}^{d}$ is $\mathcal{P}$ \-
$\mathcal{M}^{d}$-measurable. We remark for later use that the
$\mathbb{R}^{d}$-valued process $\|\varphi\|_{\text{var
}}=\left(\left\|\varphi_{t}\right\|_{\text{var }}\right)_{t\geq 0}$ is then
predictable. Indeed, by separability of $C(\mathcal{K})$, we can take a
countable dense subset $\underline{u}=\left(u_{k}\right)_{k\in\mathbb{N}}$ of
$U_{1}$ and obtain $\sup_{f\in
U_{1}}G(f)=\sup_{k\in\mathbb{N}}G\left(u_{k}\right)$ for any continuous
function $G:C(\mathcal{K})\rightarrow\mathbb{R}$. Taking
$G(f)=\varphi_{t}^{i}(f)(\omega)$ for $i=1,\ldots,d$ thus yields for each
$(\omega,t)\in\bar{\Omega}$ that

$\left\|\varphi^{i}\right\|_{\operatorname{var}}(\omega,t)=\left\|\varphi_{t}^{i}\right\|_{\operatorname{var}}(\omega)=\sup_{f\in
U_{1}}\left(\varphi_{t}^{i}(f)\right)(\omega)=\sup_{k\in\mathbb{N}}\left(\varphi^{i}\left(u_{k}\right)\right)(\omega,t)$

As each $\varphi\left(u_{k}\right)$ is predictable, so is then
$\|\varphi\|_{\text{var }}$ as a countable supremum.

##  3 Stochastic integrals

In this section, we introduce the stochastic integral of measure-valued
processes $\varphi$ with respect to the semimartingale $S$. This will be
achieved in three subsections. The first subsection defines this integral for
elementary measure-valued processes $\varphi$. The second subsection presents
some approximation results for general measure-valued processes $\varphi$,
while the third (last) subsection uses these approximation to deal with the
integral of general measure-valued integrands $\varphi$.

###  3.1 Simple integrands and integrals

To start our integration theory, let $\mathcal{E}$ be the family of
$\mathbb{M}^{d}$-valued processes $\varphi$ of the form

$\varphi=\sum_{\ell=0}^{L-1}m_{\ell}I_{A_{\ell}\times\left(t_{\ell},t_{\ell+1}\right]}=:\sum_{\ell=0}^{L-1}m_{\ell}I_{D_{\ell}}$
(3.1)

with $L\in\mathbb{N},0\leq t_{0}<t_{1}<\cdots<t_{L}<\infty$, each
$m_{\ell}\in\mathbb{M}^{d}$ and each $A_{\ell}\in\mathcal{F}_{t_{\ell}}$ so
that $D_{\ell}\in\mathcal{P}$. Note that the coefficients $m_{\ell}$ do not
depend on $\omega$. For every $f\in C(\mathcal{K})$, the process

$\varphi(f)=\sum_{\ell=0}^{L-1}m_{\ell}(f)I_{D_{\ell}}=\sum_{\ell=0}^{L-1}m_{\ell}(f)I_{A_{\ell}\times\left(t_{\ell},t_{\ell+1}\right]}$

is then an $\mathbb{R}^{d}$-valued very simple predictable process (i.e., in
$\mathcal{H}$ ), which justifies calling $\varphi\in\mathcal{E}$ a very simple
$\mathbb{M}^{d}$-valued weak* predictable process. Similarly, we call a
process $N=\left(N_{t}\right)_{t\geq 0}$ a weak* semimartingale, or a weak
process in $\mathcal{R}^{2}$, if the real-valued process
$N(f)=\left(N_{t}(f)\right)_{t\geq 0}$ is a semimartingale, respectively in
$\mathcal{R}^{2}$, for each $f\in C(\mathcal{K})$. Here, each $N_{t}(\omega)$
is assumed to be a linear (but not necessarily continuous) functional on
$C(\mathcal{K})$, hence can be viewed as a (signed and finite) finitely
additive measure on $\mathcal{B}(\mathcal{K})$, and then
$N_{t}(f):=\int_{\mathcal{K}}f(z)N_{t}(\mathrm{~{}d}z)$. Such functionals are
sometimes also called finite charges, and we sometimes call such a process $N$
a charge-valued process.

In order to define a stochastic integral $\varphi\bullet S$ for a fixed
$\mathbb{R}^{d}$-valued semimartingale and suitably general
$\mathbb{M}^{d}$-valued $\varphi$, we start with $\varphi\in\mathcal{E}$ and
then extend to a larger class of integrands. Note that while both $\varphi$
and $S$ are $d$-dimensional, the resulting integral $\varphi\bullet S$ is one-
dimensional, in analogy to the classic vector stochastic integral.

Because the description (2.2) of the semimartingale property is in prelocal
form (i.e., on a right-open stochastic interval $\llbracket 0,\tau\llbracket$
), we also construct the stochastic integral $\varphi\bullet S$ on $\llbracket
0,\tau\llbracket$ and only piece things together globally at the end.

######  Lemma 3.1.

Fix an $\mathbb{R}^{d}$-valued semimartingale $S$, a control process $V$ for
$S$ and a stopping time $\tau$ with $V_{\tau-}\in L^{2}$. For each
$\varphi\in\mathcal{E}$, there exists a process $\varphi\bullet
S^{\tau-}=\left(\varphi\bullet S_{t}^{\tau-}\right)_{t\geq 0}$ with values in
$\mathbb{M}$ which is a weak* semimartingale in $\mathcal{R}^{2}$. This
process has the regular weak* Fubini property

$\left(\varphi\bullet S_{t}^{\tau-}\right)(f)=\varphi(f)\cdot
S_{t}^{\tau-}\quad\text{ for }t\geq 0\text{ and }f\in C(\mathcal{K})$ (3.2)

More precisely, (3.2) states that for each $f\in C(\mathcal{K})$, the two
processes $\left(\varphi\bullet S^{\tau-}\right)(f)$ and $\varphi(f)\cdot
S^{\tau-}$ are indistinguishable. Written out in terms of integrals, (3.2)
takes the form

$\int_{\mathcal{K}}f(z)\left(\int_{0}^{t}\varphi_{r}\mathrm{~{}d}S_{r}^{\tau-}\right)(\mathrm{d}z)=\int_{0}^{t}\left(\int_{\mathcal{K}}f(z)\varphi_{r}(\mathrm{~{}d}z)\right)\mathrm{d}S_{r}^{\tau-}\quad\text{
for }t\geq 0\text{ and }f\in C(\mathcal{K})$

which explains the terminology. In particular, we have for
$\varphi\in\mathcal{E}$ the inequality

$\left\|\left(\varphi\bullet
S^{\tau-}\right)(f)\right\|_{\mathcal{R}^{2}}\leq\|\varphi(f)\|_{L^{2}\left(\mu_{\tau,V}\right)}\quad\text{
for }f\in C(\mathcal{K})$ (3.3)

Moreover, $\varphi\bullet S^{\tau-}$ also has the general weak* Fubini
property

$\left(\varphi\bullet S_{t}^{\tau-}\right)(g)=\varphi(g)\cdot
S_{t}^{\tau-}\quad\text{ for }t\geq 0\text{ and }g\text{ bounded and
}\mathcal{B}(\mathcal{K})\text{-measurable. }$ (3.4)

Finally, (3.3) also holds with $f\in C(\mathcal{K})$ replaced by $g$ bounded
and $\mathcal{B}(\mathcal{K})$-measurable.

###### Proof.

For
$\varphi=\sum_{\ell=0}^{L-1}m_{\ell}I_{D_{\ell}}=\sum_{\ell=0}^{L-1}m_{\ell}I_{A_{\ell}\times\left(t_{\ell},t_{\ell+1}\right]}$,
it is clear that we set

$\varphi\bullet
S_{t}^{\tau-}:=\sum_{\ell=0}^{L-1}I_{A_{\ell}}\left(S_{t_{\ell+1}\wedge
t}^{\tau-}-S_{t_{\ell}\wedge t}^{\tau-}\right)^{\top}m_{\ell}\quad\text{ for
}t\geq 0$

which clearly takes values in $\mathbb{M}$. Then we get for each $f\in
C(\mathcal{K})$ that

$\left(\varphi\bullet
S_{t}^{\tau-}\right)(f)=\sum_{\ell=0}^{L-1}I_{A_{\ell}}m_{\ell}(f)^{\top}\left(S_{t_{\ell+1}\wedge
t}^{\tau-}-S_{t_{\ell}\wedge
t}^{\tau-}\right)=\int_{0}^{t}H_{r}\mathrm{~{}d}S_{r}^{\tau-}$ (3.5)

with the process

$\displaystyle
H_{r}(\omega):=\sum_{\ell=0}^{L-1}m_{\ell}(f)I_{D_{\ell}}(\omega,r)$
$\displaystyle=\sum_{\ell=0}^{L-1}\int_{\mathcal{K}}I_{A_{\ell}\times\left(t_{\ell},t_{\ell+1}\right]}(\omega,r)f(z)m_{\ell}(\mathrm{d}z)$
(3.6)
$\displaystyle=\left(\int_{\mathcal{K}}f(z)\varphi_{r}(\mathrm{~{}d}z)\right)(\omega)=\varphi_{r}(f)(\omega).$

Combining (3.5) with (3.6) gives (3.2). Moreover, the predictable process
$H=\varphi(f)$ is bounded uniformly in $(\omega,r)$ by

$C:=\max\left\\{\left\|m_{\ell}^{i}\right\|_{\text{var
}}\|f\|_{\infty}:i=1,\ldots,d,\ell=0,\ldots,L-1\right\\}$

and hence in $L^{2}\left(\mu_{\tau,V}\right)$ as $V_{\tau-}\in L^{2}$.
Therefore $\left(\varphi\bullet S^{\tau-}\right)(f)$ is well defined as a
semimartingale and in $\mathcal{R}^{2}$ by (2.4), which also directly gives
(3.3). Finally, all operations in the above argument only involve finite sums,
and so we can replace $f\in C(\mathcal{K})$ by any bounded
$\mathcal{B}(\mathcal{K})$-measurable $g$ and obtain the same results. This
completes the proof.∎

######  Remark 3.2.

1) For elementary integrands $\varphi\in\mathcal{E}$, the regular weak* Fubini
property (3.2) and the general one in (3.4) are equally easy to obtain. But
for an extension to a larger class of integrands, it is simpler to start with
(3.2) because the class $C(\mathcal{K})$ of test functions is separable,
whereas the class of bounded $\mathcal{B}(\mathcal{K})$-measurable functions
is not.
2) The above construction and setup as well as the subsequent extension to
more general $\varphi$ are strongly inspired by the work of Björk et al. [8].
However, there is an important difference. In [8], the goal is to develop
stochastic integration with respect to an integrator ( $\bar{S}$, say) taking
values in $C([0,T])$; the natural integrands ( $\bar{\varphi}$, say) then take
values in the dual space $C([0,T])^{*}={\mathbb{M}}([0,T])$, and the resulting
integral $\int\bar{\varphi}\mathrm{d}\bar{S}$ has values in $\mathbb{R}$. (In
the application in [8], $\bar{S}_{t}$ is the bond price curve at time $t$ of a
term structure model, $\bar{\varphi}_{t}$ is a portfolio of bonds held at time
$t$, and $\int\bar{\varphi}\mathrm{d}\bar{S}$ describes the cumulative
gains/losses from trading over time via the dynamic strategy
$\bar{\varphi}=\left(\bar{\varphi}_{t}\right)$.) In contrast, our integrator
$S$ is $\mathbb{R}^{d}$-valued, and so using the analogous elementary
$\mathbb{M}^{d}$-valued integrands $\varphi$ leads to an integral process
$\varphi\bullet S$ which (at least for elementary $\varphi$ ) is again
measure-valued (and charge-valued in general). As a consequence, the resulting
space of integral processes has a more complicated structure, and in
particular completeness, for a natural seminorm, is no longer clear.

###  3.2 Approximation of general integrands

To extend both $\varphi\mapsto\varphi\bullet S$ and the regular weak* Fubini
property (3.2) from $\varphi\in\mathcal{E}$ to more general integrands, we use
approximations and hence need appropriate seminorms on $\varphi$ and on real-
valued weak* semimartingales $N$ in $\mathcal{R}^{2}$ like $\varphi\bullet
S^{\tau-}$. We start on the side of the integrands. Throughout this section,
we fix an $\mathbb{R}^{d}$-valued semimartingale $S$, a control process $V$
for $S$ and a stopping time $\tau$ with $V_{\tau-}\in L^{2}$, and denote by
$\underline{u}=\left(u_{k}\right)_{k\in\mathbb{N}}$
a countable dense subset of the unit ball $U_{1}$ of $C(\mathcal{K})$ and by
$\underline{\gamma}=\left(\gamma_{k}\right)_{k\in\mathbb{N}}$ a sequence in
$(0,\infty)$ with $\sum_{k=1}^{\infty}\gamma_{k}=1$.

######  Definition 3.3.

For any $\mathbb{M}^{d}$-valued weak* predictable process
$\varphi=\left(\varphi_{t}\right)_{t\geq 0}$, we define

$q_{\underline{\gamma},\underline{u};\tau,V}(\varphi):=\left(\sum_{k=1}^{\infty}\gamma_{k}\left\|\varphi\left(u_{k}\right)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}^{2}\right)^{1/2}$

and call $L^{2}(\underline{\gamma},\underline{u};\tau,V)$ the space of all
$\mathbb{M}^{d}$-valued weak* predictable $\varphi$ with
$q_{\underline{\gamma},\underline{u};\tau,V}(\varphi)<\infty$. We denote by
$\Phi(\tau,V)$ the class of all $\mathbb{M}^{d}$-valued weak* predictable
$\varphi$ satisfying

$C_{\varphi}:=\sup_{f\in
U_{1}\backslash\\{0\\}}\frac{\|\varphi(f)\|_{L^{2}\left(\mu_{\tau,V}\right)}}{\|f\|_{\infty}}<\infty$
(3.7)

For any $c>0$ and with $\mathbf{1}=(1,\ldots,1)\in\mathbb{R}^{d}$, we also
define the set $\mathcal{U}_{c}^{d}\subseteq\mathbb{M}^{d}$ by

$\mathcal{U}_{c}^{d}:=\left\\{m\in\mathbb{M}^{d}:\|m\|_{\operatorname{var}}\leq
c\mathbf{1}\right\\}$

######  Remark 3.4.

1) Any $\varphi\in\mathcal{E}$ has values in $\mathcal{U}_{c}^{d}$ with
$c\leq\max_{i=1,\ldots,d,\ell=0,1,\ldots,L-1}\left\|m_{\ell}^{i}\right\|_{\text{var
}}$. Next, if a weak* predictable $\varphi$ has values in
$\bigcup_{c>0}\mathcal{U}_{c}^{d}$, the process $H:=\varphi(f)$ is predictable
and $\mathbb{R}^{d}$-valued for any $f\in C(\mathcal{K})$ and satisfies
$\left|H_{t}^{i}\right|=\left|\varphi_{t}^{i}(f)\right|\leq\left\|\varphi_{t}^{i}\right\|_{\text{var
}}\|f\|_{\infty}\leq c\|f\|_{\infty}$, $i=1,\ldots,d$, for some $c>0$. This
yields

$\|\varphi(f)\|_{L^{2}\left(\mu_{\tau,V}\right)}=\|H\|_{L^{2}\left(\mu_{\tau,V}\right)}\leq
c\|f\|_{\infty}\mu_{\tau,V}(\bar{\Omega})=c\left\|V_{\tau-}\right\|_{L^{2}}\|f\|_{\infty}$
(3.8)

so that $\varphi\in\Phi(\tau,V)$, with $C_{\varphi}\leq
c\left\|V_{\tau-}\right\|_{L^{2}}$. Finally, any $\varphi\in\Phi(\tau,V)$ is
clearly in $L^{2}(\underline{\gamma},\underline{u};\tau,V)$ with
$q_{\underline{\gamma},\underline{u};\tau,V}(\varphi)\leq C_{\varphi}$. So we
have

$\mathcal{E}\subseteq\left\\{\left(\bigcup_{c>0}\mathcal{U}_{c}^{d}\right)\text{-valued
weak* predictable }\varphi\right\\}\subseteq\Phi(\tau,V)\subseteq
L^{2}(\underline{\gamma},\underline{u};\tau,V)$ (3.9)

2) From (3.7), we see that an $\mathbb{M}^{d}$-valued weak ∗ predictable
process $\varphi$ is in $\Phi(\tau,V)$ if and only if $\varphi$ as a (clearly
linear) mapping from $C(\mathcal{K})$ to the space of $\mathbb{R}^{d}$-valued
predictable processes is continuous for
$\|\cdot\|_{L^{2}\left(\mu_{\tau,V}\right)}$, meaning that we have
$\|\varphi(f)\|_{L^{2}\left(\mu_{\tau,V}\right)}\leq c\|f\|_{\infty}$ for all
$f\in C(\mathcal{K})$, for some $c>0$ which can depend on $\varphi$. Clearly,
we can take $c=C_{\varphi}$.
3) The space $L^{2}(\underline{\gamma},\underline{u};\tau,V)$ as well as
$q_{\underline{\gamma},\underline{u};\tau,V}$ obviously depend on
$\underline{\gamma}$ and $\underline{u}$ as well as on $V$ and $\tau$. But the
above relations in 1) and 2) hold for any fixed choice of
$\underline{\gamma},\underline{u}$ and $V,\tau$.

######  Lemma 3.5.

Under our standing assumptions in this subsection, we have:
1) $q_{\underline{\gamma},\underline{u};\tau,V}$ is a seminorm on
$L^{2}(\underline{\gamma},\underline{u};\tau,V)$.
2) For each $c>0$, the functional
$q_{\underline{\gamma},\underline{u};\tau,V}$ is sequentially continuous with
respect to the weak* topology on the space of all weak predictable $\varphi$
with values in $\mathcal{U}_{c}^{d}$ : If a sequence
$\left(\varphi^{n}\right)_{n\in\mathbb{N}}$ of such processes converges
$\mu_{\tau,V}$-a.e. for the weak $k^{*}$ topology to some
$\mathbb{M}^{d}$-valued weak* predictable $\varphi$, then also $\varphi$ takes
values in $\mathcal{U}_{c}^{d}$ and
$\lim_{n\rightarrow\infty}q_{\underline{\gamma},\underline{u};\tau,V}\left(\varphi^{n}-\varphi\right)=0$.

###### Proof.

1) Clearly $q_{\underline{\gamma},\underline{u};\tau,V}$ is positively
homogeneous. For the triangle inequality, we write

$q_{\underline{\gamma},\underline{u};\tau,V}(\varphi)=\left(\sum_{k=1}^{\infty}\gamma_{k}\left\|\varphi\left(u_{k}\right)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}^{2}\right)^{1/2}=\left(\sum_{k=1}^{\infty}\gamma_{k}x_{k}^{2}\right)^{1/2}=\|\underline{x}\|_{\ell^{2}(\gamma)}$

with
$x_{k}:=\left\|\varphi\left(u_{k}\right)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}$,
and analogously for $\varphi^{\prime}$ with $\underline{x}^{\prime}$. So
$q_{\underline{\gamma},\underline{u};\tau,V}(\varphi)$ is just the
$\ell^{2}$-norm of the sequence
$\underline{x}=\left(x_{k}\right)_{k\in\mathbb{N}}$ for the probability
measure on $\mathbb{N}$ having weights $\gamma_{k}$ for $k\in\mathbb{N}$. But
$y_{k}:=\left\|\varphi\left(u_{k}\right)+\varphi^{\prime}\left(u_{k}\right)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}\leq
x_{k}+x_{k}^{\prime}$ by the triangle inequality in
$L^{2}\left(\mu_{\tau,V}\right)$, and so

$\displaystyle
q_{\underline{\gamma},\underline{u};\tau,V}\left(\varphi+\varphi^{\prime}\right)$
$\displaystyle=\|\underline{y}\|_{\ell^{2}(\gamma)}\leq\left(\sum_{k=1}^{\infty}\gamma_{k}\left(x_{k}+x_{k}^{\prime}\right)^{2}\right)^{1/2}=\left\|\underline{x}+\underline{x}^{\prime}\right\|_{\ell^{2}(\gamma)}$
$\displaystyle\leq\|\underline{x}\|_{\ell^{2}(\gamma)}+\left\|\underline{x}^{\prime}\right\|_{\ell^{2}(\gamma)}=q_{\underline{\gamma},\underline{u};\tau,V}(\varphi)+q_{\underline{\gamma},\underline{u};\tau,V}\left(\varphi^{\prime}\right).$

This proves the triangle inequality for
$q_{\underline{\gamma},\underline{\underline{u}};\tau,V}$ on
$L^{2}(\underline{\gamma},\underline{u};\tau,V)$.
2) Take a sequence $\left(\varphi^{n}\right)_{n\in\mathbb{N}}$ of
$\mathbb{M}^{d}$-valued weak∗ predictable processes with
$\left\|\varphi^{n}\right\|_{\text{var }}\leq c\mathbf{1}$ for all $n$ and
suppose that $\left(\varphi^{n}\right)_{n\in\mathbb{N}}$ converges
$\mu_{\tau,V}$-a.e. to some $\varphi$ for the weak* topology. Then
$\varphi^{n}(f)\rightarrow\varphi(f)\mu_{\tau,V}$-a.e. for every $f\in
C(\mathcal{K})$, and thus for $i=1,\ldots,d$,

$\left|\varphi_{t}^{i}(f)\right|=\lim_{n\rightarrow\infty}\left|\varphi_{t}^{n;i}(f)\right|\leq\limsup_{n\rightarrow\infty}\left\|\varphi_{t}^{n;i}\right\|_{\text{var
}}\|f\|_{\infty}\leq c\|f\|_{\infty}\quad\mu_{\tau,V}\text{-a.e. }$ (3.10)

Hence $\varphi$ also has values in $\mathcal{U}_{c}^{d}$ and is thus in
$L^{2}(\underline{\gamma},\underline{u};\tau,V)$ by (3.9). Replacing
$\varphi^{n}$ by $\varphi^{n}-\varphi$ and $c$ by $2c$, we can and do assume
without loss of generality that $\varphi\equiv 0$. Then
$\lim_{n\rightarrow\infty}\varphi^{n}\left(u_{k}\right)=0\mu_{\tau,V}$-a.e.
for all $k\in\mathbb{N}$, and using $\left\|\varphi^{n}\right\|_{\text{var
}}\leq 2c\mathbf{1}$ for all $n\in\mathbb{N}$ yields as in (3.10) that
$\sup_{n\in\mathbb{N}}\left|\varphi^{n}\left(u_{k}\right)\right|\leq\sup_{n\in\mathbb{N}}\left|\left\|\varphi^{n}\right\|_{\operatorname{var}}\right|\left\|u_{k}\right\|_{\infty}\leq
2c|\mathbf{1}|\in L^{2}\left(\mu_{\tau,V}\right)$ due to $V_{\tau-}\in L^{2}$.
So dominated convergence gives
$\lim_{n\rightarrow\infty}\varphi^{n}\left(u_{k}\right)=0$ in
$L^{2}\left(\mu_{\tau,V}\right)$, for each $k\in\mathbb{N}$. Finally, (3.8)
for $\varphi^{n}$ and $u_{k}\in U_{1}$ with $c$ replaced by $2c$ yields
$\left\|\varphi^{n}\left(u_{k}\right)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}^{2}\leq
4c^{2}E\left[V_{\tau-}^{2}\right]$. Now
first take $K_{0}$ large enough to obtain
$4c^{2}E\left[V_{\tau-}^{2}\right]\sum_{k=K_{0}+1}^{\infty}\gamma_{k}\leq\varepsilon^{2}$
and then use $\varphi^{n}\left(u_{k}\right)\rightarrow 0$ in
$L^{2}\left(\mu_{\tau,V}\right)$ as $n\rightarrow\infty$, for each $k$, to
take $n$ large enough to get
$\gamma_{k}\left\|\varphi^{n}\left(u_{k}\right)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}^{2}\leq\varepsilon^{2}/K_{0}$
for $k=1,\ldots,K_{0}$. Then the definition of
$q_{\underline{\gamma},\underline{u};\tau,V}$ and the above estimates imply
that

$\left(q_{\underline{\gamma},\underline{u};\tau,V}\left(\varphi^{n}\right)\right)^{2}=\sum_{k=1}^{\infty}\gamma_{k}\left\|\varphi^{n}\left(u_{k}\right)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}^{2}\leq
2\varepsilon^{2}$

This proves the assertion.∎

The next approximation result is crucial for our construction of stochastic
integrals of measure-valued processes.

######  Lemma 3.6.

Under our standing assumptions in this subsection, the set $\mathcal{E}$ is
dense in $L^{2}(\underline{\gamma},\underline{u};\tau,V)$ with respect to the
seminorm $q_{\underline{\gamma},\underline{u};\tau,V}$.

###### Proof.

This is similar to the very concisely written result in Björk et al. [8, Lemma
2.3], but for completeness and readability, we give the proof in detail. First
of all, $\mathcal{E}\subseteq L^{2}(\underline{\gamma},\underline{u};\tau,V)$
as seen in (3.9). If we consider very simple $\mathcal{P}$-measurable
$\varphi$ as in (3.1) where we only have sets $D_{\ell}\in\mathcal{P}$ instead
of predictable rectangles
$D_{\ell}=A_{\ell}\times\left(t_{\ell},t_{\ell+1}\right]$, these $\varphi$ are
still in $L^{2}(\underline{\gamma},\underline{u};\tau,V)$ if they are valued
in $\mathcal{U}_{c}^{d}$ for some $c>0$, again by (3.9).
To prove denseness, we start with $\varphi\in
L^{2}(\underline{\gamma},\underline{u};\tau,V)$. Then
$\underline{\gamma}\subseteq(0,\infty)$ and

$q_{\underline{\gamma},\underline{u};\tau,V}(\varphi)=\left(\sum_{k=1}^{\infty}\gamma_{k}\left\|\varphi\left(u_{k}\right)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}^{2}\right)^{1/2}<\infty$

implies that $\varphi\left(u_{k}\right)\in L^{2}\left(\mu_{\tau,V}\right)$ for
all $k\in\mathbb{N}$. For an arbitrary $c>0$, the process

$\varphi^{(c)}:=\varphi I_{\left\\{\|\varphi\|_{\mathrm{var}}\leq
c\mathbf{1}\right\\}}\quad\text{ (defined coordinatewise) }$

is weak∗ predictable like $\varphi$ because $\|\varphi\|_{\text{var }}$ is
predictable, and $\varphi-\varphi^{(c)}=\varphi
I_{\left\\{\|\varphi\|_{\text{var }}>c1\right\\}}$ (again coordinatewise).
This implies that
$\lim_{c\rightarrow\infty}\left(\varphi-\varphi^{(c)}\right)(f)=0\mu_{\tau,V}$-a.e.
for every $f\in C(\mathcal{K})$ because $\varphi$ takes values in
$\mathbb{M}^{d}$, and
$\left|\left(\varphi-\varphi^{(c)}\right)\left(u_{k}\right)\right|\leq\left|\varphi\left(u_{k}\right)\right|\in
L^{2}\left(\mu_{\tau,V}\right)$ for each $k$. Dominated convergence hence
gives $\left(\varphi-\varphi^{(c)}\right)\left(u_{k}\right)\rightarrow 0$ in
$L^{2}\left(\mu_{\tau,V}\right)$ as $c\rightarrow\infty$, for each $k$. Next,
$q_{\underline{\gamma},\underline{u};\tau,V}(\varphi)<\infty$ allows us to
again use dominated convergence, for the measure on $\mathbb{N}$ with weights
$\left(\gamma_{k}\right)_{k\in\mathbb{N}}$, and obtain
$q_{\underline{\gamma},\underline{u};\tau,V}\left(\varphi-\varphi^{(c)}\right)\rightarrow
0$ as $c\rightarrow\infty$. Each $\varphi^{(c)}$
takes values in
$\mathcal{U}_{c}^{d}=\left\\{m\in\mathbb{M}^{d}:\|m\|_{\text{var }}\leq
c\mathbf{1}\right\\}$, and so it is enough to show that $\mathcal{E}$ is dense
with respect to $q_{\underline{\gamma},\underline{\underline{u}};\tau,V}$ in
the set of all weak* predictable processes $\varphi$ valued in
$\mathcal{U}_{c}^{d}$.

Now by the Banach-Alaoglu theorem, each ball $\mathcal{U}_{c}^{d}$ is compact
in the weak* topology on
$\mathbb{M}^{d}=\left((C(\mathcal{K}))^{d}\right)^{*}$, and metrisable because
$C(\mathcal{K})$ is separable; see Section 2.3. Hence $\mathcal{U}_{c}^{d}$ is
separable for the weak* topology and we can choose a countable weak* dense
subset $\left(m_{j}\right)_{j\in\mathbb{N}}$ in $\mathcal{U}_{c}^{d}$. Define
$\mathcal{M}_{n}:=\left\\{m_{1},m_{2},\ldots,m_{n}\right\\}$ for all
$n\in\mathbb{N}$ and note that because
$\underline{u}=\left(u_{k}\right)_{k\in\mathbb{N}}$ is dense in $U_{1}$, the
metric
$\delta\left(m,m^{\prime}\right):=\sum_{k=1}^{\infty}2^{-k}\left|m\left(u_{k}\right)-m^{\prime}\left(u_{k}\right)\right|$
on $\mathcal{U}_{c}^{d}$ induces the weak* topology on any
$\mathcal{U}_{c}^{d}$. If $\varphi$ is a weak* predictable process valued in
$\mathcal{U}_{c}^{d},\delta(\varphi,m)$ is predictable for every fixed
$m\in\mathcal{U}_{c}^{d}$, and so is then the process
$\delta\left(\varphi,\mathcal{M}_{n}\right)=\min_{i=1,\ldots,n}\delta\left(\varphi,m_{i}\right)$.
As $\left(m_{j}\right)_{j\in\mathbb{N}}$ is weak ∗ dense in
$\mathcal{U}_{c}^{d}$, we know that
$\delta\left(\varphi,\mathcal{M}_{n}\right)\searrow 0\mu_{\tau,V}$-a.e. as
$n\rightarrow\infty$, and the pairwise disjoint sets

$C_{j}^{n}:=\left\\{\delta\left(\varphi,\mathcal{M}_{n}\right)=\delta\left(\varphi,m_{j}\right)\text{
and }\delta\left(\varphi,m_{i}\right)>\delta\left(\varphi,m_{j}\right)\text{
for }i<j\right\\}\subseteq\bar{\Omega}$

where the distance of $\varphi$ to $\mathcal{M}_{n}$ is realized for the first
time in $m_{j}\in\mathcal{M}_{n}$ are in $\mathcal{P}$. The process
$\Psi^{n}:=\sum_{j=1}^{n}m_{j}I_{C_{j}^{n}}$ is thus a very simple
$\mathcal{P}$-measurable $\mathcal{U}_{c}^{d}$-valued process, and

$\delta\left(\varphi,\Psi^{n}\right)=\sum_{j=1}^{n}I_{C_{j}^{n}}\delta\left(\varphi,m_{j}\right)=\delta\left(\varphi,\mathcal{M}_{n}\right)\longrightarrow
0\quad\mu_{\tau,V}\text{-a.e. }$

shows that $\Psi^{n}\rightarrow\varphi$ in the weak* topology which is
metrised by $\delta$. This proves that because the weak* predictable process
$\varphi$ takes values in $\mathcal{U}_{c}^{d}$, it is actually strongly
$\mathcal{P}$-measurable in the sense (usual for Banach-valued random
processes) that it is an a.e. pointwise limit of elementary
$\mathcal{P}$-measurable processes. As a consequence, Lemma 3.5, 2) implies
that $\left(\Psi^{n}\right)_{n\in\mathbb{N}}$ converges to $\varphi$ with
respect to $q_{\underline{\gamma},\underline{u};\tau,V}$.

The above sequence $\left(\Psi^{n}\right)$ is not yet in $\mathcal{E}$ because
the sets $C_{j}^{n}$ are in $\mathcal{P}$, but not necessarily predictable
rectangles of the form $A_{s}\times(s,t]$ with $s\leq t$ and
$A_{s}\in\mathcal{F}_{s}$. But each $\Psi^{n}$ has the form
$\Psi^{n}=\sum_{j=1}^{J(n)}m_{j}I_{C_{j}^{n}}$ with
$m_{j}\in\mathcal{U}_{c}^{d}$ and $C_{j}^{n}\in\mathcal{P}$ pairwise disjoint.
This is a finite linear combination, and so it is enough to consider
$\Psi=mI_{C}$ with $m\in\mathcal{U}_{c}^{d},C\in\mathcal{P}$ and show that
this can be approximated with respect to
$q_{\underline{\gamma},\underline{u};\tau,V}$ by a sequence
$\left(\varphi^{i}\right)_{i\in\mathbb{N}}$ in $\mathcal{E}$. For this, note
that the predictable $\sigma$-field $\mathcal{P}$ is generated by the sets of
the form $D_{\ell}=A_{\ell}\times\left(t_{\ell},t_{\ell+1}\right]$ with
$A_{\ell}\in\mathcal{F}_{t_{\ell}}$, so that the real-valued predictable
processes of the form
$H=\sum_{\ell=0}^{L-1}c_{\ell}I_{A_{\ell}\times\left(t_{\ell},t_{\ell+1}\right]}$
with $L\in\mathbb{N}$ and $c_{\ell}\in\mathbb{R}$ are dense in
$L^{2}\left(\mu_{\tau,V}\right)$. (This uses that the measure $\mu_{\tau,V}$
is finite as $V_{\tau-}\in L^{2}$.) So we can choose a sequence
$\left(H^{i}\right)_{i\in\mathbb{N}}$ of such processes with $H^{i}\rightarrow
I_{C}$ in $L^{2}\left(\mu_{\tau,V}\right)$, and by passing to a subsequence,
still called $\left(H^{i}\right)$, we can achieve that $H^{i}\rightarrow
I_{C}$ $\mu_{\tau,V^{-}}$-a.e. Moreover, as the limit $I_{C}$ is bounded by 1
, we can even modify the $H^{i}$ so that they are bounded by $1+\varepsilon$,
uniformly in $\omega,t,i$. If we now set $\varphi^{i}:=mH^{i}$, each
$\varphi^{i}$ takes values in $\mathcal{U}_{c(1+\varepsilon)}^{d}$ because
$m\in\mathcal{U}_{c}^{d}$, and recalling that $\Psi=mI_{C}$, we get
$\mu_{\tau,V^{-}}$a.e. that

$\left|\varphi^{i}\left(u_{k}\right)-\Psi\left(u_{k}\right)\right|=\left|m\left(u_{k}\right)\right|\left|H^{i}-I_{C}\right|\leq\left|\|m\|_{\operatorname{var}}\right|\left\|u_{k}\right\|_{\infty}\left|H^{i}-I_{C}\right|\longrightarrow
0$

simultaneously for all and uniformly in $k\in\mathbb{N}$; note that
$\left\|u_{k}\right\|_{\infty}\leq 1$. Therefore

$\delta\left(\varphi^{i},mI_{C}\right)=\sum_{k=1}^{\infty}2^{-k}\left|\varphi^{i}\left(u_{k}\right)-\Psi\left(u_{k}\right)\right|\leq\left|\|m\|_{\mathrm{var}}\right|\left|H^{i}-I_{C}\right|\longrightarrow
0\quad\mu_{\tau,V^{-}}\text{a.e., }$

and we have already seen above that $\delta$ metrises the weak* topology on
$\mathcal{U}_{c(1+\varepsilon)}^{d}$. So we get $\varphi^{i}\rightarrow
mI_{C}\mu_{\tau,V^{-}}$-a.e. for the weak* topology as $i\rightarrow\infty$,
and Lemma 3.5,2) then yields that
$q_{\underline{\gamma},\underline{u};\tau,V}\left(\varphi^{i}-mI_{C}\right)\rightarrow
0$ as $i\rightarrow\infty$.
Combining all the above results implies the assertion and completes the
proof.∎

By Remark 3.4,2), a weak* predictable process $\varphi$ is in
$\Phi(\tau,V)\subseteq L^{2}(\underline{\gamma},\underline{u};\tau,V)$ if and
only if it is continuous as a mapping from $C(\mathcal{K})$ to
$L^{2}\left(\mu_{\tau,V}\right)$. The next result sharpens the approximation
obtained in Lemma 3.6 - it shows that for $\varphi\in\Phi(\tau,V)$, one can
find an approximating sequence $\left(\varphi^{n}\right)_{n\in\mathbb{N}}$
with the same type of continuity as $\varphi$, uniformly in $n$.

######  Corollary 3.7.

Under our standing assumptions in this subsection, for every
$\varphi\in\Phi(\tau,V)$, there exists a sequence
$\left(\varphi^{n}\right)_{n\in\mathbb{N}}\subseteq\mathcal{E}$ with
$\lim_{n\rightarrow\infty}q_{\underline{\gamma},\underline{u};\tau,V}\left(\varphi^{n}-\varphi\right)=0$
and

$\sup_{n\in\mathbb{N}}\left\|\varphi^{n}(f)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}\leq
c\|f\|_{\infty}\quad\text{ for all }f\in C(\mathcal{K})$ (3.11)

with some constant $c\in(0,\infty)$.

###### Proof.

We go back to the proof of Lemma 3.6 and examine in more detail how the
approximating sequence is constructed there. Start with
$\varphi\in\Phi(\tau,V)$, fix $b>1$, take $c:=b-1$ and set
$\varphi^{(c)}:=\varphi I_{\left\\{\|\varphi\|_{\mathrm{var}}\leq
c1\right\\}}$. Approximate, with respect to
$q_{\underline{\gamma},\underline{u};\tau,V}$, first $\varphi^{(c)}$ by a
sequence
$\left(\Psi^{n}\right)_{n\in\mathbb{N}}$ of the form
$\Psi^{n}=\sum_{j=1}^{J(n)}m_{j}I_{C_{j}^{n}}$, and then each $\Psi^{n}$ by
some $\varphi^{n}\in\mathcal{E}$. Note (from the proof of Lemma 3.6) that
$\Psi^{n}$ takes values in $\mathcal{U}_{c}^{d}$ and $\varphi^{n}$ hence in
$\mathcal{U}_{c(1+\varepsilon)}^{d}\subseteq\mathcal{U}_{b}^{d}$ for
$\varepsilon\leq\frac{1}{b-1}$ because then $c(1+\varepsilon)\leq b$. This
gives
$\lim_{n\rightarrow\infty}q_{\underline{\gamma},\underline{u};\tau,V}\left(\varphi^{n}-\varphi\right)=0$,
and for any $f\in C(\mathcal{K})$,

$\left|\varphi^{n}(f)\right|\leq\left|\varphi^{n}(f)-\Psi^{n}(f)\right|+\left|\Psi^{n}(f)-\varphi^{(c)}(f)\right|+\left|\varphi^{(c)}(f)-\varphi(f)\right|+|\varphi(f)|.$

Because $\varphi^{n}$ has values in $\mathcal{U}_{b}^{d}$ and $\Psi^{n}$ in
$\mathcal{U}_{c}^{d}\subseteq\mathcal{U}_{b}^{d}$, we get
$\left|\varphi^{n}(f)-\Psi^{n}(f)\right|\leq|b\mathbf{1}|\|f\|_{\infty}$, and
in the same way,
$\left|\Psi^{n}(f)-\varphi^{(c)}(f)\right|\leq|c\mathbf{1}|\|f\|_{\infty}\leq|b\mathbf{1}|\|f\|_{\infty}$.
Moreover, by the definition of $\varphi^{(c)}$, we have
$\left|\varphi^{(c)}(f)-\varphi(f)\right|\leq|\varphi(f)|$, and so we obtain
from (3.8) and $\varphi\in\Phi(\tau,V)$ that

$\left\|\varphi^{n}(f)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}\leq
2b\left\|V_{\tau-}\right\|_{L^{2}}\|f\|_{\infty}+2\|\varphi(f)\|_{L^{2}\left(\mu_{\tau,V}\right)}\leq\left(2b\left\|V_{\tau-}\right\|_{L^{2}}+2C_{\varphi}\right)\|f\|_{\infty}.$

As the constant on the right-hand side does not depend on $n$, this completes
the proof.∎

###  3.3 Limits of measure-valued semimartingales

In the last section, we have constructed a stochastic integral process
$\varphi\bullet S^{\tau-}$ for measure-valued integrands
$\varphi\in\mathcal{E}$. This yields a measure-valued weak* semimartingale in
$\mathcal{R}^{2}$, and we have also seen that any $\varphi\in
L^{2}(\underline{\gamma},\underline{u};\tau,V)$ can be approximated by a
sequence $\left(\varphi^{n}\right)_{n\in\mathbb{N}}\subseteq\mathcal{E}$. It
seems natural to try and define $\varphi\bullet S^{\tau-}$ for more general
$\varphi$ than in $\mathcal{E}$ as a limit, in a suitable sense, of
$\left(\varphi^{n}\bullet S^{\tau-}\right)_{n\in\mathbb{N}}$, and so we first
need a concept of convergence.

As in Section 3.2, throughout this subsection, we fix an
$\mathbb{R}^{d}$-valued semimartingale $S$, a control process $V$ for $S$ and
a stopping time $\tau$ with $V_{\tau-}\in L^{2}$, and we denote by
$\underline{u}=\left(u_{k}\right)_{k\in\mathbb{N}}$ a countable dense subset
of the unit ball $U_{1}$ of $C(\mathcal{K})$ and by
$\underline{\gamma}=\left(\gamma_{k}\right)_{k\in\mathbb{N}}$ a sequence in
$(0,\infty)$ with $\sum_{k=1}^{\infty}\gamma_{k}=1$.

######  Definition 3.8.

Definition 3.8. For a weak* process $N=\left(N_{t}\right)_{t\geq 0}$ in
$\mathcal{R}^{2}$, we define

$r_{\underline{\gamma},\underline{u}}(N):=\left(\sum_{k=1}^{\infty}\gamma_{k}\left\|N\left(u_{k}\right)\right\|_{\mathcal{R}^{2}}^{2}\right)^{1/2}$

and call $\mathcal{N}^{2}(\underline{\gamma},\underline{u})$ the space of all
weak* processes $N$ in $\mathcal{R}^{2}$ with
$r_{\underline{\gamma},\underline{u}}(N)<\infty$.

######  Lemma 3.9.

Under our standing assumptions in this subsection, we have:
1) $r_{\underline{\gamma},\underline{u}}$ is a seminorm on
$\mathcal{N}^{2}(\underline{\gamma},\underline{u})$.
2) For any $\varphi\in\mathcal{E}$, we have the inequality

$r_{\underline{\gamma},\underline{u}}\left(\varphi\bullet S^{\tau-}\right)\leq
q_{\underline{\gamma},\underline{u};\tau,V}(\varphi).$ (3.12)

###### Proof.

1) This is completely analogous to the proof of 1) in Lemma 3.5, just
replacing
$x_{k}=\left\|\varphi\left(u_{k}\right)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}$
by $x_{k}:=\left\|N\left(u_{k}\right)\right\|_{\mathcal{R}^{2}}$.
2) The inequality (3.3) in Lemma 3.1 gives $\left\|\left(\varphi\bullet
S^{\tau-}\right)\left(u_{k}\right)\right\|_{\mathcal{R}^{2}}\leq\left\|\varphi\left(u_{k}\right)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}$
for each $\varphi\in\mathcal{E}$ and all $k\in\mathbb{N}$. So (3.12) follows
from the definitions of $r_{\underline{\gamma},\underline{u}}$ and
$q_{\underline{\gamma},\underline{u};\tau,V}$. ∎

The next result is crucial for taking limits in
$\mathcal{N}^{2}(\underline{\gamma},\underline{u})$.

######  Lemma 3.10.

Under our standing assumptions in this subsection, suppose that the sequence
$\left(N^{n}\right)_{n\in\mathbb{N}}\subseteq\mathcal{N}^{2}(\underline{\gamma},\underline{u})$
is a Cauchy sequence for $r_{\underline{\gamma},\underline{u}}$. If it also
satisfies

$\sup_{n\in\mathbb{N}}\left\|N^{n}(f)\right\|_{\mathcal{R}^{2}}\leq
c\|f\|_{\infty}\quad\text{ for all }f\in C(\mathcal{K})$ (3.13)

with a constant $c\in(0,\infty)$ (that can depend on the sequence), then
$\left(N^{n}\right)_{n\in\mathbb{N}}$ converges with respect to
$r_{\underline{\gamma},\underline{u}}$ to some
$N^{\infty}\in\mathcal{N}^{2}(\underline{\gamma},\underline{u})$ which also
satisfies

$\left\|N^{\infty}(f)\right\|_{\mathcal{R}^{2}}\leq c\|f\|_{\infty}\quad\text{
for all }f\in C(\mathcal{K})$ (3.14)

###### Proof.

From $\underline{\gamma}\subseteq(0,\infty)$ and the definition of
$r_{\underline{\gamma},\underline{u}}$, we see that the Cauchy property of
$\left(N^{n}\right)_{n\in\mathbb{N}}$ for
$r_{\underline{\gamma},\underline{u}}$ implies for each $k\in\mathbb{N}$ that
$\left(N^{n}\left(u_{k}\right)\right)_{n\in\mathbb{N}}$ is a Cauchy sequence
in $\mathcal{R}^{2}$ and therefore has a limit $N^{\infty,k}$ in
$\mathcal{R}^{2}$. We define $N^{\infty}$ on $\underline{u}\subseteq
C(\mathcal{K})$ by

$N_{t}^{\infty}\left(u_{k}\right):=N_{t}^{\infty,k}\quad\text{ for all }t\geq
0\text{ and }k\in\mathbb{N}$

and get that $N^{\infty}\left(u_{k}\right)=N^{\infty,k}$ is in
$\mathcal{R}^{2}$ for each $k\in\mathbb{N}$. We now claim that the definition
of $N^{\infty}$ can be extended from $\underline{u}$ to all of
$C(\mathcal{K})$ in a linear way. To see this, take any $f\in U_{1}$ (which is
enough by using linearity) and set

$N^{\infty}(f):=\lim_{\ell\rightarrow\infty}N^{\infty}\left(u_{\ell}^{\prime}\right)\quad\text{
in }\mathcal{R}^{2}$ (3.15)

for any sequence
$\left(u_{\ell}^{\prime}\right)_{\ell\in\mathbb{N}}\subseteq\underline{u}$
which converges to $f$ in $C(\mathcal{K})$. Such a sequence exists because
$\underline{u}$ is dense in $U_{1}$, and so it remains to show that the limit
in (3.15) exists and does not depend on the approximating sequence
$\left(u_{\ell}^{\prime}\right)_{\ell\in\mathbb{N}}$.
Because $u_{\ell}^{\prime}\rightarrow f$ in
$C(\mathcal{K}),\left(u_{\ell}^{\prime}\right)_{\ell\in\mathbb{N}}$ is Cauchy
in $C(\mathcal{K})$. Now write

$N^{\infty}\left(u_{\ell}^{\prime}\right)-N^{\infty}\left(u_{j}^{\prime}\right)=\left(N^{\infty}\left(u_{\ell}^{\prime}\right)-N^{n}\left(u_{\ell}^{\prime}\right)\right)+\left(N^{n}\left(u_{\ell}^{\prime}\right)-N^{n}\left(u_{j}^{\prime}\right)\right)+\left(N^{n}\left(u_{j}^{\prime}\right)-N^{\infty}\left(u_{j}^{\prime}\right)\right)$
(3.16)

and note that the first and third differences converge to 0 in
$\mathcal{R}^{2}$ as $n\rightarrow\infty$ by the construction of $N^{\infty}$.
The second difference can be estimated in $\mathcal{R}^{2}$ by

$\sup_{n\in\mathbb{N}}\left\|N^{n}\left(u_{\ell}^{\prime}\right)-N^{n}\left(u_{j}^{\prime}\right)\right\|_{\mathcal{R}^{2}}=\sup_{n\in\mathbb{N}}\left\|N^{n}\left(u_{\ell}^{\prime}-u_{j}^{\prime}\right)\right\|_{\mathcal{R}^{2}}\leq
c\left\|u_{\ell}^{\prime}-u_{j}^{\prime}\right\|_{\infty}$

thanks to the linearity of each $N^{n}$ and (3.13), and so
$\left(N^{\infty}\left(u_{\ell}^{\prime}\right)\right)_{\ell\in\mathbb{N}}$ is
a Cauchy sequence in $\mathcal{R}^{2}$ and hence convergent in
$\mathcal{R}^{2}$. Thus the limit in (3.15) exists, and we claim that it does
not depend on the chosen approximating sequence
$\left(u_{\ell}^{\prime}\right)_{\ell\in\mathbb{N}}$ for $f$. Indeed, if
$\left(\tilde{u}_{\ell}\right)_{\ell\in\mathbb{N}}$ is another approximating
sequence for $f$ with limit
$\tilde{N}^{\infty}(f):=\lim_{\ell\rightarrow\infty}N^{\infty}\left(\tilde{u}_{\ell}\right)$,
we can write

$\tilde{N}^{\infty}(f)-N^{\infty}(f)=\left(\tilde{N}^{\infty}(f)-N^{\infty}\left(\tilde{u}_{\ell}\right)\right)+\left(N^{\infty}\left(\tilde{u}_{\ell}\right)-N^{\infty}\left(u_{\ell}^{\prime}\right)\right)+\left(N^{\infty}\left(u_{\ell}^{\prime}\right)-N^{\infty}(f)\right).$

Then the first and third differences converge to 0 in $\mathcal{R}^{2}$ as
$\ell\rightarrow\infty$ by the definitions of $\tilde{N}^{\infty}$ and
$N^{\infty}$, respectively, and the second difference converges to 0 in
$\mathcal{R}^{2}$ as $\ell\rightarrow\infty$ by the same argument as for the
Cauchy property in (3.16); note that
$\left\|\tilde{u}_{\ell}-u_{\ell}^{\prime}\right\|_{\infty}\rightarrow 0$ as
$\ell\rightarrow\infty$ because both
$\left(\tilde{u}_{\ell}\right)_{\ell\in\mathbb{N}}$ and
$\left(u_{\ell}^{\prime}\right)_{\ell\in\mathbb{N}}$ converge to $f$ in
$C(\mathcal{K})$. So we obtain $\tilde{N}^{\infty}(f)=N^{\infty}(f)$.

By the preceding arguments, $N^{\infty}$ is well defined on all of
$C(\mathcal{K})$, and linear and a weak* process in $\mathcal{R}^{2}$, both by
construction. To show that the sequence $\left(N^{n}\right)_{n\in\mathbb{N}}$
converges to $N^{\infty}$ for $r_{\underline{\gamma},\underline{u}}$, we first
note that $N^{n}\left(u_{k}\right)\rightarrow N^{\infty}\left(u_{k}\right)$ in
$\mathcal{R}^{2}$ as $n\rightarrow\infty$, for each $k\in\mathbb{N}$. For
fixed $\varepsilon>0$ and each finite $K$, we thus obtain for large $n$ that

$\displaystyle\sum_{k=1}^{K}\gamma_{k}\left\|N^{n}\left(u_{k}\right)-N^{\infty}\left(u_{k}\right)\right\|_{\mathcal{R}^{2}}^{2}$
$\displaystyle=\lim_{m\rightarrow\infty}\sum_{k=1}^{K}\gamma_{k}\left\|N^{n}\left(u_{k}\right)-N^{m}\left(u_{k}\right)\right\|_{\mathcal{R}^{2}}^{2}$
$\displaystyle\leq\limsup_{m\rightarrow\infty}\sum_{k=1}^{\infty}\gamma_{k}\left\|N^{n}\left(u_{k}\right)-N^{m}\left(u_{k}\right)\right\|_{\mathcal{R}^{2}}^{2}$
$\displaystyle=\limsup_{m\rightarrow\infty}\left(r_{\underline{\gamma},\underline{u}}\left(N^{n}-N^{m}\right)\right)^{2}\leq\varepsilon^{2}$

independently of $K$, by the Cauchy property of
$\left(N^{n}\right)_{n\in\mathbb{N}}$ for
$r_{\underline{\gamma},\underline{u}}$. This shows by letting
$K\rightarrow\infty$ that
$r_{\underline{\gamma},\underline{u}}\left(N^{n}-N^{\infty}\right)\leq\varepsilon$
for $n$ large enough. So $\left(N^{n}\right)_{n\in\mathbb{N}}$ converges to
$N^{\infty}$ with respect to $r_{\underline{\gamma},\underline{u}}$ and
$N^{\infty}$ lies in $\mathcal{N}^{2}(\underline{\gamma},\underline{u})$.

Finally, to prove (3.14), we fix $f\in C(\mathcal{K})$ and take a sequence
$\left(u_{\ell}^{\prime}\right)_{\ell\in\mathbb{N}}\subseteq\underline{u}$
converging to $f$ in $C(\mathcal{K})$. Then

$\displaystyle\left\|N^{\infty}(f)\right\|_{\mathcal{R}^{2}}\leq$
$\displaystyle\left\|N^{\infty}(f)-N^{\infty}\left(u_{\ell}^{\prime}\right)\right\|_{\mathcal{R}^{2}}+\left\|N^{\infty}\left(u_{\ell}^{\prime}\right)-N^{n}\left(u_{\ell}^{\prime}\right)\right\|_{\mathcal{R}^{2}}$
$\displaystyle+\sup_{n\in\mathbb{N}}\left\|N^{n}\left(u_{\ell}^{\prime}\right)-N^{n}(f)\right\|_{\mathcal{R}^{2}}+\sup_{n\in\mathbb{N}}\left\|N^{n}(f)\right\|_{\mathcal{R}^{2}}$

and the fourth summand is at most $c\|f\|_{\infty}$ by (3.13). The other
summands all converge to 0 as $\ell\rightarrow\infty$ and $n\rightarrow\infty$
\- the first by the definition (3.15) of $N^{\infty}(f)$, the second as in
(3.16) by the construction of $N^{\infty}$, and the third by (3.13) because
$u_{\ell}^{\prime}\rightarrow f$ in $C(\mathcal{K})$. This shows (3.14) and
completes the proof.∎

######  Remark 3.11.

1) By the construction in the proof of Lemma 3.10,
$N^{\infty}:C(\mathcal{K})\rightarrow\mathcal{R}^{2}$ is continuous, meaning
that $f^{n}\rightarrow f$ in $C(\mathcal{K})$ implies that
$N^{\infty}\left(f^{n}\right)\rightarrow N^{\infty}(f)$ in $\mathcal{R}^{2}$.
However, this is different from continuity of
$N_{t}^{\infty}(\omega):C(\mathcal{K})\rightarrow\mathbb{R}$. In other words,
the proof of Lemma 3.10 does not show (and it is not true in general) that the
limit process $N^{\infty}$ takes values in $\mathbb{M}$. This is the reason
why the stochastic integral $\varphi\bullet S$ we construct below is not
always a measure-valued process, but only charge-valued in general. We give a
counterexample later in Section 7.
2) Because we construct it pointwise on $C(\mathcal{K})$ as a limit in
$\mathcal{R}^{2}$, the process $N^{\infty}$ need not be a weak* semimartingale
in general even if the sequence $\left(N^{n}\right)_{n\in\mathbb{N}}$ consists
of weak* semimartingales. This is because the subspace of semimartingales is
not closed in $\mathcal{R}^{2}$. But see ${}^{***}[$ check what can be said
here] below for some positive results.
3) As in Remark 3.4, 3), the space
$\mathcal{N}^{2}(\underline{\gamma},\underline{u})$ as well as
$r_{\underline{\gamma},\underline{u}}$ depend on $\underline{\gamma}$ and
$\underline{u}$. But the above results again hold for any fixed choice of
$\underline{\gamma}$ and $\underline{u}$.

##  4 Two new stochastic Fubini theorems

In this section, we combine the preceding results to construct a stochastic
integral process $\varphi\bullet S$ for fairly general measure-valued
integrands $\varphi$, and we use this to prove new stochastic Fubini theorems.
The idea to get $\varphi\bullet S$ is simple. We first prelocalise,
approximate $\varphi$ on a stochastic interval $\llbracket 0,\tau\llbracket$
by a sequence $\left(\varphi^{n}\right)_{n\in\mathbb{N}}$ of elementary
integrands in $\mathcal{E}$ and define $\varphi\bullet S^{\tau-}$ to be the
limit of the sequence $\left(\varphi^{n}\bullet
S^{\tau-}\right)_{n\in\mathbb{N}}$. While the approximation in $\mathcal{E}$
can be done for any $\varphi\in
L^{2}(\underline{\gamma},\underline{u};\tau,V)$, the existence of the limit of
$\left(\varphi^{n}\bullet S^{\tau-}\right)_{n\in\mathbb{N}}$ needs the
stronger assumption that $\varphi\in\Phi(\tau,V)$. In the end, we paste things
together.
Let $S=\left(S_{t}\right)_{t\geq 0}$ be a fixed $\mathbb{R}^{d}$-valued
semimartingale and recall from Section 2.2 the family $\mathcal{V}(S)$ of
control processes $V$ for $S$.

######  Definition 4.1.

We denote by $\Phi$ the family of all $\mathbb{M}^{d}$-valued weak*
predictable processes $\varphi$ such that for some control process
$V\in\mathcal{V}(S)$,

$\mathcal{D}\left(\|\varphi\|_{\text{var
}};V\right)=\int\|\|\varphi_{r}\|\left.{}_{\text{var
}}\right|^{2}\mathrm{~{}d}V_{r}\text{ is a finite-valued process (i.e., in
}\mathcal{R}^{0}\text{ ). }$ (4.1)

######  Remark 4.2.

As seen at the end of Section 2.2, (4.1) is equivalent to saying that the
predictable $\mathbb{R}^{d}$-valued process $\|\varphi\|_{\text{var
}}=\left(\left\|\varphi_{t}\right\|_{\text{var }}\right)_{t\geq 0}$ is
integrable with respect to $S$.

For any $\varphi\in\Phi$, the processes $V$ and
$\mathcal{D}\left(\|\varphi\|_{\text{var }};V\right)$ are both in
$\mathcal{R}^{0}$. So they are prelocally bounded and we can find a localising
sequence $\left(\tau_{M}\right)_{M\in\mathbb{N}}$ of stopping times with
$V_{\tau_{M}-}\in L^{2}$ and
$\mathcal{D}_{\tau_{M}-}\left(\|\varphi\|_{\text{var }};V\right)\in L^{2}$ for
each $M\in\mathbb{N}$. Thus $\mu_{\tau_{M},V}$ is a finite measure and we have

$E\left[V_{\tau_{M}-}\int_{0}^{\tau_{M}-}\left|\left\|\varphi_{r}\right\|_{\mathrm{var}}\right|^{2}\mathrm{~{}d}V_{r}\right]<\infty$

which means that $\|\varphi\|_{\text{var }}\in
L^{2}\left(\mu_{\tau_{M},V}\right)$. The estimate
$\left|\varphi_{r}(f)\right|\leq\|f\|_{\infty}\left|\left\|\varphi_{r}\right\|_{\text{var
}}\right|$ for any $f\in C(\mathcal{K})$ then implies that

$\sup_{f\in
U_{1}\backslash\\{0\\}}\frac{\|\varphi(f)\|_{L^{2}\left(\mu_{\tau_{M},V}\right)}}{\|f\|_{\infty}}\leq\|\|\varphi\left\|{}_{\mathrm{var}}\right\|_{L^{2}\left(\mu_{\tau_{M},V}\right)}<\infty$

so that $\varphi\in\Phi\left(\tau_{M},V\right)$ for every $M\in\mathbb{N}$;
see Definition 3.3. In summary, any $\mu_{\tau_{M},V}$ is a finite measure and
any $\varphi\in\Phi$ is in each $\Phi\left(\tau_{M},V\right)$, for a suitable
localizing sequence $\left(\tau_{M}\right)_{M\in\mathbb{N}}$.

With these preparations, we are now ready for our first main result. Our first
new stochastic Fubini theorem is called regular because its test functions $f$
are continuous.

######  Theorem 4.3.

Let $S=\left(S_{t}\right)_{t\geq 0}$ be an $\mathbb{R}^{d}$-valued
semimartingale. For every measure-valued integrand $\varphi\in\Phi$, there
exists a stochastic integral process $\varphi\bullet S=\left(\varphi\bullet
S_{t}\right)_{t\geq 0}$ which is a weak* process in $\mathcal{R}^{0}$ (and
hence prelocally in $\mathcal{R}^{2}$ ). It is a linear and continuous mapping
from $C(\mathcal{K})$ to $\mathcal{R}^{0}$ and satisfies the regular weak*
Fubini property that for all $f\in C(\mathcal{K})$,

$(\varphi\bullet S)(f)=\varphi(f)\cdot S\quad\text{ up to indistinguishability
on }\bar{\Omega}=\Omega\times(0,\infty)\text{. }$ (4.2)

Written in integral form, this means that

$\int_{\mathcal{K}}f(z)\left(\int_{0}^{t}\varphi_{r}\mathrm{~{}d}S_{r}\right)(\mathrm{d}z)=\int_{0}^{t}\left(\int_{\mathcal{K}}f(z)\varphi_{r}(\mathrm{~{}d}z)\right)\mathrm{d}S_{r}\quad\text{
for all }t\geq 0,P-\text{ a.s. }$ (4.3)

The process $\varphi\bullet S$ is uniquely determined on $C(\mathcal{K})$ by
(4.2).

###### Proof.

Fix a countable dense subset
$\underline{u}=\left(u_{k}\right)_{k\in\mathbb{N}}$ of the unit ball $U_{1}$
in $C(\mathcal{K})$ and a sequence
$\underline{\gamma}=\left(\gamma_{k}\right)_{k\in\mathbb{N}}$ in $(0,\infty)$
with $\sum_{k=1}^{\infty}\gamma_{k}=1$. Take a control process
$V\in\mathcal{V}(S)$ for $S$ and a localising sequence
$\left(\tau_{M}\right)_{M\in\mathbb{N}}$ of stopping times with
$V_{\tau_{M-}}\in L^{2}$ and
$\mathcal{D}_{\tau_{M-}}\left(\|\varphi\|_{\text{var }};V\right)\in L^{2}$ for
each $M\in\mathbb{N}$. Fix $M\in\mathbb{N}$ and write $\tau:=\tau_{M}$ for
brevity. Then $\varphi\in\Phi$ is in $\Phi(\tau,V)$ and there exists by Lemma
3.6 a sequence $\left(\varphi^{n}\right)_{n\in\mathbb{N}}\subseteq\mathcal{E}$
converging to $\varphi$ for $q_{\underline{\gamma},\underline{u};\tau,V}$. For
any $n\in\mathbb{N}$, the measure-valued stochastic integral
$\varphi^{n}\bullet S^{\tau-}$ is therefore by Lemma 3.1 well defined, a
(measure-valued) weak* semimartingale in $\mathcal{R}^{2}$, and in
$\mathcal{N}^{2}(\underline{\gamma},\underline{u})$ because
$r_{\underline{\gamma},\underline{u}}\left(\varphi^{n}\bullet
S^{\tau-}\right)\leq
q_{\underline{\gamma},\underline{u};\tau,V}\left(\varphi^{n}\right)$ by (3.12)
in Lemma 3.9. The same inequality shows that the sequence
$\left(N^{n}\right)_{n\in\mathbb{N}}:=\left(\varphi^{n}\bullet
S^{\tau-}\right)_{n\in\mathbb{N}}$ is Cauchy for
$r_{\underline{\gamma},\underline{u}}$ because
$\left(\varphi^{n}\right)_{n\in\mathbb{N}}$ is Cauchy for
$q_{\underline{\gamma},\underline{u};\tau,V}$ as it converges (to $\varphi$ ).
Moreover, (3.3) in Lemma 3.1 gives the inequality

$\left\|\left(\varphi^{n}\bullet
S^{\tau-}\right)(f)\right\|_{\mathcal{R}^{2}}\leq\left\|\varphi^{n}(f)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}\quad\text{
for all }f\in C(\mathcal{K})$ (4.4)

Because $\varphi\in\Phi(\tau,V)$, Corollary 3.7 allows us to choose the
sequence $\left(\varphi^{n}\right)_{n\in\mathbb{N}}$ in such a way that it
satisfies (3.11), i.e.,
$\sup_{n\in\mathbb{N}}\left\|\varphi^{n}(f)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}\leq
c\|f\|_{\infty}$ for all $f\in C(\mathcal{K})$ with some constant
$c\in(0,\infty)$. Combining this with (4.4) gives

$\sup_{n\in\mathbb{N}}\left\|N^{n}(f)\right\|_{\mathcal{R}^{2}}\leq\sup_{n\in\mathbb{N}}\left\|\varphi^{n}(f)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}\leq
c\|f\|_{\infty}\quad\text{ for all }f\in C(\mathcal{K})\text{, }$

which allows us to apply Lemma 3.10 and conclude that
$\left(N^{n}\right)_{n\in\mathbb{N}}$ converges with respect to
$r_{\underline{\gamma},\underline{u}}$ to some
$N^{\infty}\in\mathcal{N}^{2}(\underline{\gamma},\underline{u})$. We set
$\varphi\bullet S^{\tau-}:=N^{\infty}$ and note that this is linear and
continuous as a map from $C(\mathcal{K})$ to $\mathcal{R}^{2}$ by
construction; see Remark 3.11, 1). Moreover, due to (3.14) in Lemma 3.10, we
also have

$\left\|\left(\varphi\bullet S^{\tau-}\right)(f)\right\|_{\mathcal{R}^{2}}\leq
c\|f\|_{\infty}\quad\text{ for all }f\in C(\mathcal{K})$ (4.5)

To prove the regular weak* Fubini property (4.2), we first note that this
holds for each $\varphi^{n}\in\mathcal{E}$ due to (3.2) in Lemma 3.1. By
linearity, it is enough to consider $f\in U_{1}$, and then we can take a
sequence
$\left(u_{\ell}^{\prime}\right)_{\ell\in\mathbb{N}}\subseteq\underline{u}$
with $\lim_{\ell\rightarrow\infty}u_{\ell}^{\prime}=f$ in $C(\mathcal{K})$.
Now we write

$\displaystyle($ $\displaystyle\left.\bullet
S^{\tau-}\right)(f)-\varphi(f)\cdot S^{\tau-}$ $\displaystyle=$
$\displaystyle\left(\left(\varphi\bullet
S^{\tau-}\right)(f)-\left(\varphi\bullet
S^{\tau-}\right)\left(u_{\ell}^{\prime}\right)\right)+\left(\left(\varphi\bullet
S^{\tau-}\right)\left(u_{\ell}^{\prime}\right)-\left(\varphi^{n}\bullet
S^{\tau-}\right)\left(u_{\ell}^{\prime}\right)\right)$
$\displaystyle+\left(\left(\varphi^{n}\bullet
S^{\tau-}\right)\left(u_{\ell}^{\prime}\right)-\varphi^{n}\left(u_{\ell}^{\prime}\right)\cdot
S^{\tau-}\right)$
$\displaystyle+\left(\varphi^{n}\left(u_{\ell}^{\prime}\right)\cdot
S^{\tau-}-\varphi\left(u_{\ell}^{\prime}\right)\cdot
S^{\tau-}\right)+\left(\varphi\left(u_{\ell}^{\prime}\right)\cdot
S^{\tau-}-\varphi(f)\cdot S^{\tau-}\right)$

and consider the five differences on the right-hand side one by one. The third
vanishes by the regular weak* Fubini property (3.2) for
$\varphi^{n}\in\mathcal{E}$. Next, $\varphi^{n}\rightarrow\varphi$ for
$q_{\gamma,\underline{u};\tau,V}$ implies by the definition of
$q_{\underline{\gamma},\underline{u};\tau,V}$ that
$\varphi^{n}\left(u_{k}\right)\rightarrow\varphi\left(u_{k}\right)$ in
$L^{2}\left(\mu_{\tau,V}\right)$ for each $u_{k}\in\underline{u}$. So
$\varphi^{n}\left(u_{\ell}^{\prime}\right)\rightarrow\varphi\left(u_{\ell}^{\prime}\right)$
in $L^{2}\left(\mu_{\tau,V}\right)$ as $n\rightarrow\infty$ and thus the
fourth difference tends to 0 in $\mathcal{R}^{2}$ as $n\rightarrow\infty$ by
(2.4), for each $\ell\in\mathbb{N}$. Next, $\varphi\in\Phi(\tau,V)$ gives
$\left\|\varphi\left(u_{\ell}^{\prime}\right)-\varphi(f)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}\leq
C_{\varphi}\left\|u_{\ell}^{\prime}-f\right\|_{\infty}$ (see Definition 3.3),
and $u_{\ell}^{\prime}\rightarrow f$ in $C(\mathcal{K})$ by construction. So
$\varphi\left(u_{\ell}^{\prime}\right)\rightarrow\varphi(f)$ in
$L^{2}\left(\mu_{\tau,V}\right)$, and the same argument as just above implies
that the fifth difference tends to 0 in $\mathcal{R}^{2}$ as
$\ell\rightarrow\infty$.

Now by construction, $\varphi\bullet S^{\tau-}$ is the limit of the sequence
$\left(\varphi^{n}\bullet S^{\tau-}\right)_{n\in\mathbb{N}}$ for
$r_{\underline{\gamma},\underline{u}}$, and we have $\left(\varphi\bullet
S^{\tau-}\right)(f)=\lim_{\ell\rightarrow\infty}\left(\varphi\bullet
S^{\tau-}\right)\left(u_{\ell}^{\prime}\right)$ in $\mathcal{R}^{2}$ by the
definition (3.15) of $\varphi\bullet S^{\tau-}=N^{\infty}$ from Lemma 3.10. In
consequence, the first difference tends to 0 in $\mathcal{R}^{2}$ as
$\ell\rightarrow\infty$. Finally, the definition of
$r_{\underline{\gamma},\underline{\underline{u}}}$ yields
$\left(\varphi^{n}\bullet
S^{\tau-}\right)\left(u_{k}\right)\rightarrow\left(\varphi\bullet
S^{\tau-}\right)\left(u_{k}\right)$ in $\mathcal{R}^{2}$ as
$n\rightarrow\infty$ for each $u_{k}\in\underline{u}$, and so also the second
difference tends to 0 in $\mathcal{R}^{2}$ as $n\rightarrow\infty$, for each
$\ell\in\mathbb{N}$. By taking first $\ell$ large enough and then $n$ large
enough for fixed $\ell$, this implies that $\left(\varphi\bullet
S^{\tau-}\right)(f)$ and $\varphi(f)\cdot S^{\tau-}$ are indistinguishable,
for each $f\in C(\mathcal{K})$, and proves (4.2).

It is clear that $\varphi\bullet S^{\tau-}$ is uniquely determined on
$C(\mathcal{K})$ by (4.2) when restricting this to $\llbracket
0,\tau\llbracket$, and like the right-hand side of (4.2), $\varphi\bullet
S^{\tau-}$ does not depend on the choice of the control process
$V\in\mathcal{V}(S)$. Moreover, it is also independent of the choice of
$\underline{u}$ and $\underline{\gamma}$. To see this, take another countable
dense subset $\underline{u}^{\prime}$ of $U_{1}$ and another sequence
$\underline{\gamma}^{\prime}$ in $(0,\infty)$ summing to 1 . Denote by
$\underline{u}\cup\underline{u}^{\prime}$ and
$\underline{\gamma}\cup\underline{\gamma}^{\prime}$ the set and sequence
resulting from interlacing $\underline{u},\underline{u}^{\prime}$ and
$\underline{\gamma},\underline{\gamma}^{\prime}$, respectively, by putting the
primed quantities at even and the unprimed ones at odd indices. Then obviously
$\left(r_{\underline{\gamma}\cup\underline{\gamma}^{\prime},\underline{u}\cup\underline{u}^{\prime}}(\cdot)\right)^{2}=\left(r_{\underline{\gamma},\underline{u}}(\cdot)\right)^{2}+\left(r_{\underline{\gamma}^{\prime},\underline{u}^{\prime}}(\cdot)\right)^{2}$
and
$\left(q_{\underline{\gamma}\cup\underline{q}^{\prime},\underline{\underline{u}}\cup\underline{u}^{\prime};V}(\cdot)\right)^{2}=\left(q_{\underline{\gamma},\underline{u};\tau,V}(\cdot)\right)^{2}+\left(q_{\underline{\gamma}^{\prime},\underline{u}^{\prime};V}(\cdot)\right)^{2}$,
and so an approximation with respect to
$q_{\underline{\gamma}\cup\underline{q}^{\prime},\underline{\underline{u}}\cup\underline{u}^{\prime};V}$
automatically implies one with respect to both
$q_{\underline{\gamma},\underline{u};\tau,V}$ and
$q_{\underline{\underline{q}}^{\prime},\underline{\underline{u}};V}$. By
uniqueness, the process $\varphi\bullet S^{\tau-}$ constructed in
$\mathcal{N}^{2}\left(\underline{\gamma}\cup\underline{\gamma}^{\prime},\underline{u}\cup\underline{u}^{\prime}\right)$
therefore coincides with
both the processes constructed in
$\mathcal{N}^{2}(\underline{\gamma},\underline{u})$ and
$\mathcal{N}^{2}\left(\underline{\gamma}^{\prime},\underline{u}^{\prime}\right)$.

Now recall that $\tau$ is shorthand for $\tau_{M}$ with a fixed
$M\in\mathbb{N}$. So $\varphi\bullet S^{\tau_{M}-}$ is defined and unique for
each $M\in\mathbb{N}$, which gives a consistent definition of $\varphi\bullet
S$ on each $\llbracket 0,\tau_{M}\llbracket$. As $\tau_{M}\nearrow\infty$, the
process $\varphi\bullet S$ is therefore well defined and has the stochastic
Fubini property on all of $\bar{\Omega}$. Moreover, $\varphi\bullet S$ is a
continuous mapping on each $\llbracket 0,\tau_{M}\llbracket$ from
$C(\mathcal{K})$ to $\mathcal{R}^{2}\subseteq\mathcal{R}^{0}$ and hence maps
$C(\mathcal{K})$ into $\mathcal{R}^{0}$. For fixed $t\geq 0$ and $f\in
C(\mathcal{K})$, combining $((\varphi\bullet
S)(f))_{t}^{*}\leq\left(\left(\varphi\bullet
S^{\tau_{M}-}\right)(f)\right)_{\infty}^{*}I_{\left\\{t<\tau_{M}\right\\}}+((\varphi\bullet
S)(f))_{t}^{*}I_{\left\\{t\geq\tau_{M}\right\\}}$ with Markov’s inequality and
(4.5) gives

$\displaystyle P\left[((\varphi\bullet S)(f))_{t}^{*}>2\varepsilon\right]$
$\displaystyle\leq P\left[\left(\left(\varphi\bullet
S^{\tau_{M}-}\right)(f)\right)_{\infty}^{*}>\varepsilon\right]+P\left[\tau_{M}\leq
t\right]$
$\displaystyle\leq\frac{1}{\varepsilon^{2}}\left\|\left(\varphi\bullet
S^{\tau_{M}-}\right)(f)\right\|_{\mathcal{R}^{2}}^{2}+P\left[\tau_{M}\leq
t\right]$
$\displaystyle\leq\frac{1}{\varepsilon^{2}}c^{2}\|f\|_{\infty}^{2}+P\left[\tau_{M}\leq
t\right].$

As $M\rightarrow\infty$, the second summand goes to 0 because
$\tau_{M}\nearrow\infty$, and so $\varphi\bullet
S:C(\mathcal{K})\rightarrow\mathcal{R}^{0}$ is continuous for the ucp-
topology. This completes the proof..∎

In some situations, one needs a stochastic Fubini theorem for less regular
test functions than $f\in C(\mathcal{K})$. The next result goes in that
direction.

######  Theorem 4.4.

Let $S=\left(S_{t}\right)_{t\geq 0}$ be an $\mathbb{R}^{d}$-valued
semimartingale. For every measure-valued integrand $\varphi\in\Phi$, the
stochastic integral process $\varphi\bullet S=\left(\varphi\bullet
S_{t}\right)_{t\geq 0}$ from Theorem 4.3 also satisfies the general weak*
Fubini property that for all $g\in bB_{1}(\mathcal{K})$,

$(\varphi\bullet S)(g)=\varphi(g)\cdot S\quad\text{ up to indistinguishability
on }\bar{\Omega}=\Omega\times(0,\infty)\text{. }$ (4.6)

This implies in particular that $(\varphi\bullet S)(g)$ is in
$\mathcal{R}^{0}$. Moreover, for every closed subset $D$ of $\mathcal{K}$, we
have $(\varphi\bullet S)(D)=\varphi(D)\cdot S$, i.e.,

$\int_{D}\left(\int_{0}^{t}\varphi_{r}\mathrm{~{}d}S_{r}\right)(\mathrm{d}z)=\int_{0}^{t}\left(\int_{D}\varphi_{r}(\mathrm{~{}d}z)\right)\mathrm{d}S_{r}\quad\text{
for all }t\geq 0,P\text{-a.s. }$ (4.7)

###### Proof.

Because $\mathcal{K}$ is compact and Hausdorff, Urysohn’s lemma implies that
for every closed $D\subseteq\mathcal{K}$, we can write
$I_{D}=\lim_{n\rightarrow\infty}f_{n}$ pointwise for a sequence
$\left(f_{n}\right)_{n\in\mathbb{N}}$ in $U_{1}\subseteq C(\mathcal{K})$.
Hence $I_{D}$ is in $bB_{1}(\mathcal{K})$ and (4.7) follows from (4.6).
Moreover, the right-hand side of (4.6) is in $\mathcal{R}^{0}$ if it is well
defined, and so is then the left-hand side.
The proof of (4.6) is similar to the proof of Theorem 4.3 in that we first
prelocalise and argue on stochastic intervals $\llbracket 0,\tau\llbracket$.
Fix $\varphi\in\Phi$ and a control process $V$ for $S$, and let
$\left(\tau_{M}\right)_{M\in\mathbb{N}}$ be a localizing sequence of stopping
times such that for every $M\in\mathbb{N}$,

$V_{\tau_{M}-}\in
L^{2},\quad\mathcal{D}_{\tau_{M}-}\left(\|\varphi\|_{\text{var }};V\right)\in
L^{2}$ (4.8)

Fix $M\in\mathbb{N}$ and write $\tau:=\tau_{M}$ for brevity. By the proof of
Theorem 4.3, $\varphi\bullet S^{\tau-}$ is well defined and a linear mapping
from $C(\mathcal{K})$ to $\mathcal{R}^{2}$, and so each $\varphi\bullet
S_{t}^{\tau-}(\omega)$ can be identified with a finite charge on
$\mathcal{B}(\mathcal{K})$. As a consequence, $\left(\varphi\bullet
S^{\tau-}\right)(g)$ is a well-defined process, even for any Borel-measurable
bounded function $g$ on $\mathcal{K}$.
Now fix $g\in bB_{1}(\mathcal{K})$ and write
$g=\lim_{n\rightarrow\infty}f_{n}$ pointwise for a sequence
$\left(f_{n}\right)_{n\in\mathbb{N}}\subseteq C(\mathcal{K})$. As $g$ is
bounded by $c$, say, we may assume that $\left\|f_{n}\right\|_{\infty}\leq 2c$
for all $n$, and so the sequence $h_{n}:=g-f_{n},n\in\mathbb{N}$, tends to 0
pointwise on $\mathcal{K}$ and is bounded by $4c$. We first argue that
$\varphi(g)$ is predictable and in $L^{2}\left(\mu_{\tau,V}\right)$, so that
$\varphi(g)\cdot S^{\tau-}$ is well defined and in $\mathcal{R}^{2}$ due to
(2.4). By (4.8), we have $\left|\|\varphi\|_{\text{var
}}\right|<\infty\mu_{\tau,V}$-a.e., and so dominated convergence yields
$\varphi\left(f_{n}\right)\rightarrow\varphi(g)\mu_{\tau,V^{-}}$a.e. as
$n\rightarrow\infty$. Fatou’s lemma for $\mu_{\tau,V}$ together with
$\varphi\in\Phi(\tau,V)$ yields via (3.7) that

$\|\varphi(g)\|_{L^{2}\left(\mu_{\tau,V}\right)}\leq\liminf_{n\rightarrow\infty}\left\|\varphi\left(f_{n}\right)\right\|_{L^{2}\left(\mu_{\tau,V}\right)}\leq
C_{\varphi}\sup_{n\in\mathbb{N}}\left\|f_{n}\right\|_{\infty}<\infty$

and so $\varphi(g)$ satisfies the required integrability for being in
$L^{2}\left(\mu_{\tau,V}\right)$. To check predictability (up to
indistinguishability), we note that
$\left|\varphi\left(f_{n}\right)\right|\leq\left\|f_{n}\right\|_{\infty}\left|\|\varphi\|_{\text{var
}}\right|$ leads via (4.8) to

$\mathcal{D}_{\tau-}\left(\varphi\left(f_{n}\right);V\right)=\int_{0}^{\tau-}\left|\varphi_{r}\left(f_{n}\right)\right|^{2}\mathrm{~{}d}V_{r}\leq\left\|f_{n}\right\|_{\infty}^{2}\mathcal{D}_{\tau-}\left(\|\varphi\|_{\text{var
}};V\right)<\infty\quad P\text{-a.s. }$

so that dominated convergence first gives
$\mathcal{D}_{\tau-}\left(\varphi\left(f_{n}\right);V\right)\rightarrow\mathcal{D}_{\tau-}(\varphi(g);V)P$-a.s.
Thanks to (4.8), we can again use dominated convergence to obtain
$\varphi(g)=\lim_{n\rightarrow\infty}\varphi\left(f_{n}\right)$ in
$L^{2}\left(\mu_{\tau,V}\right)$ and hence also $\mu_{\tau,V}$-a.e. along a
subsequence. As each $\varphi\left(f_{n}\right)$ is predictable, so is
$\varphi(g)$ (up to indistinguishability). Thus $\varphi(g)\cdot S^{\tau-}$ is
well defined and in $\mathcal{R}^{2}$, and so is then $\left(\varphi\bullet
S^{\tau-}\right)(g)$ if we can prove (4.6) on $\llbracket 0,\tau\llbracket$.
To achieve that last point, we write

$\displaystyle\left(\varphi\bullet S^{\tau-}\right)(g)-\varphi(g)\cdot
S^{\tau-}=$ $\displaystyle\left(\left(\varphi\bullet
S^{\tau-}\right)(g)-\left(\varphi\bullet
S^{\tau-}\right)\left(f_{n}\right)\right)$
$\displaystyle+\left(\left(\varphi\bullet
S^{\tau-}\right)\left(f_{n}\right)-\varphi\left(f_{n}\right)\cdot
S^{\tau-}\right)$ $\displaystyle+\left(\varphi\left(f_{n}\right)\cdot
S^{\tau-}-\varphi(g)\cdot S^{\tau-}\right).$

The second summand on the right-hand side is 0 by the regular Fubini property
(4.2). We have just seen that $\varphi\left(f_{n}\right)\rightarrow\varphi(g)$
in $L^{2}\left(\mu_{\tau,V}\right)$ so that the third summand converges to 0
in $\mathcal{R}^{2}$ by (2.4). For the first summand, we have
$h_{n}=g-f_{n}\rightarrow 0$ pointwise and $\left|h_{n}\right|\leq 4c$ for all
$n$. As each $\varphi\bullet S_{t}^{\tau-}(\omega)$ can be identified with a
finite charge on $\mathcal{B}(\mathcal{K})$, we get $\left(\varphi\bullet
S^{\tau-}\right)\left(f_{n}\right)\rightarrow\left(\varphi\bullet
S^{\tau-}\right)(g)$ for $\mu_{\tau,V^{-}}$almost all $(\omega,t)$ by
dominated convergence; see Bhaskara Rao/Bhaskara Rao [6, Theorem 4.6.14]. This
completes the proof.∎

######  Remark 4.5.

Apart from being more general then Theorem 4.3, Theorem 4.4 also has the
advantage that it allows us to replace $\varphi$ by $I_{D}\varphi$ for a
closed set $D\subseteq\mathcal{K}$ and hence obtain (4.3) with an integral
over $D$ instead of over $\mathcal{K}$. This cannot be done by simply
replacing $f$ by $I_{D}f$ in Theorem 4.3 as $I_{D}f$ is no longer in
$C(\mathcal{K})$. That extra generality will be used in Section 6 below.

##  5 An application to Volterra semimartingales

Let $S=\left(S_{t}\right)_{t\geq 0}$ be an $\mathbb{R}^{d}$-valued
semimartingale. In this section, we study stochastic processes
$X=\left(X_{t}\right)_{t\geq 0}$ of the form

$X_{t}=\int_{0}^{t}\Psi_{t,s}\mathrm{~{}d}S_{s},\quad t\geq 0$ (5.1)

for a two-parameter process $\Psi=\left(\Psi_{t,s}\right)_{t\geq 0,0\leq s\leq
t}$ with suitable measurability and integrability properties (made precise
later). We look for conditions on $\Psi$ which ensure that $X$ is again a
semimartingale, and we also want to say more about its decomposition in that
case.
Processes of the form (5.1) are often called Volterra(-type) processes, and
there is a large and growing body of literature around them. Forward or
backward stochastic Volterra integral equations look for a process $X$ solving
(in the forward case, say) an equation of the form
$X_{t}=X_{0}+\int_{0}^{t}a\left(t,s,\left(X_{r}\right)_{r\leq
s}\right)\mathrm{d}s+\int_{0}^{t}b\left(t,s,\left(X_{r}\right)_{r\leq
s}\right)\mathrm{d}W_{s};$ see e.g. Hernández [14] for a recent
article with many more references. More specifically, stochastic Volterra
equations take the form

$X_{t}=x_{0}(t)+\int_{0}^{t}K_{\mu}(s,t)\mu\left(s,X_{s}\right)\mathrm{d}s+\int_{0}^{t}K_{\sigma}(s,t)\sigma\left(s,X_{s}\right)\mathrm{d}W_{s},$

and one then studies existence, uniqueness and properties of solutions; see
for instance Prömel/Scheffels [27]. If one specifies further to

$X_{t}=X_{0}+\int_{0}^{t}K(t-s)\mu\left(X_{s}\right)\mathrm{d}s+\int_{0}^{t}K(t-s)\sigma\left(X_{s}\right)\mathrm{d}W_{s}$

with $\mu,\sigma\sigma^{\top}$ affine, there is a large literature on such
affine Volterra processes; see e.g. Abi Jaber et al. [1] for an important
contribution.
In a different direction, processes of the form
$X_{t}=\int_{0}^{t}K(t,s)\mathrm{d}S_{s}$ with $S$ a semimartingale, or $S=L$
a Lévy process, or $S=W$ a Brownian motion, have been studied for path
properties, stochastic calculus, support properties and other topics; see e.g.
Neuman [26], Bender et al. [5], Di Nunno et al. [11], Kalinin [17],
Baudoin/Nualart [4], to name just a few (diverse, but not necessarily
representative) contributions. A key difference between the above two
directions is whether $X$ is given endogenously as the solution of some
equation or exogenously from given quantities $\Psi,S$ not depending on $X$.
The question whether $X$ in (5.1) is a semimartingale seems to have two main
lines of approach. If $X_{t}=\int_{0}^{t}g(t-s)\mathrm{d}L_{s}$, with a
deterministic function $g$ and a Lévy process $L$, is a convolution-type
integral or moving average process, Basse/Pedersen [3] have given necessary
and sufficient condition, on $g$ and the Lévy triplet of $L$, for $X$ to be a
semimartingale. An extension to
$X_{t}^{\vartheta}=\int_{0}^{t}K^{\vartheta}(t,s)\mathrm{d}W_{s}$ with a
Brownian motion $W$ and $K^{\vartheta}(t,s)$ of a particular form has recently
been presented by El Omari [12]. At the other end of the scale is the work by
Protter [28] who assumes that for all $s$ and $\omega$, (5.2)
$t\mapsto\Psi_{t,s}(\omega)=\Psi(t,s,\omega)$ is in $C^{1}$ with
$t\mapsto\psi(t,s,\omega):=\frac{\partial\Psi}{\partial t}(t,s,\omega)$
locally Lipschitz. He then writes $X$ as

$X_{t}=\int_{0}^{t}\Psi_{t,s}\mathrm{~{}d}S_{s}=\int_{0}^{t}\Psi_{s,s}\mathrm{~{}d}S_{s}+\int_{0}^{t}\left(\Psi_{t,s}-\Psi_{s,s}\right)\mathrm{d}S_{s}$
(5.3)

uses (5.2) to get

$\Psi_{t,s}(\omega)-\Psi_{s,s}(\omega)=\int_{s}^{t}\psi(r,s,\omega)\mathrm{d}r$
(5.4)

and handles the last term in (5.3) by writing

$\int_{0}^{t}\left(\Psi_{t,s}-\Psi_{s,s}\right)\mathrm{d}S_{s}=\int_{0}^{t}\left(\int_{s}^{t}\psi(r,s,\cdot)\mathrm{d}r\right)\mathrm{d}S_{s}=\int_{0}^{t}\left(\int_{0}^{r}\psi(r,s,\cdot)\mathrm{d}S_{s}\right)\mathrm{d}r.$
(5.5)

Note that (5.2) implies via (5.4) that all the $t\mapsto\Psi_{t,s}$ are
absolutely continuous with respect to one fixed (namely Lebesgue) measure, and
so the last step in (5.5) follows by a standard stochastic Fubini theorem.
Thus the last term in (5.3) is absolutely continuous with respect to Lebesgue
measure and hence of finite variation and a semimartingale.

As we discuss later in Section 6, the assumption (5.2) or (5.4) means that we
are in the so-called dominated case. Let us now see what we can do if we relax
this restrictive condition. So let $\Psi=\left(\Psi_{t,s}\right)_{t\geq
0,0\leq s\leq t}$ be an $\mathbb{R}^{d}$-valued two-parameter process such
that for each $t\geq 0$, the process $\left(\Psi_{t,s}\right)_{0\leq s\leq t}$
is predictable and $S$-integrable on $[0,t]$, and such that the diagonal
$\left(\Psi_{s,s}\right)_{s\geq 0}$ is also predictable and $S$-integrable.
Then we can write, as in (5.3),

$X_{t}=\int_{0}^{t}\Psi_{t,s}\mathrm{~{}d}S_{s}=\int_{0}^{t}\Psi_{s,s}\mathrm{~{}d}S_{s}+\int_{0}^{t}\left(\Psi_{t,s}-\Psi_{s,s}\right)\mathrm{d}S_{s}=:\int_{0}^{t}\Psi_{s,s}\mathrm{~{}d}S_{s}+Y_{t}$
(5.6)

and the first $\mathrm{d}S$-integral on the right-hand side is well defined
and a semimartingale. Note that if $S=M$ is a local martingale, that integral
need not be a local martingale unless we know more about the diagonal
$\left(\Psi_{s,s}\right)_{s\geq 0}$; see the classic example due to Emery
[13]. We return to this point in Remark 5.2 below.

Now let us agree, as is natural in view of the problem, that $\Psi_{t,s}:=0$
for $s>t$, and also assume that for each $s\geq 0$, the process
$\left(\Psi_{t,s}\right)_{t\geq 0}$ is right-continuous and of finite
variation. This clearly generalises the assumption (5.2) or (5.4) in [28]. To
apply our results from Section 4, we need to work with signed measures on a
compact metric space. For each fixed $T\in(0,\infty)$, we therefore consider
$\mathcal{K}_{T}:=[0,T]$ and associate to $\Psi$ for each $s\geq 0$ a "random
signed $\mathbb{R}^{d}$-valued distribution function" $\bar{\varphi}_{s}$ on
$[0,\infty)$ via

$t\mapsto\bar{\varphi}_{s}(t)(\omega):=I_{\\{t\geq
s\\}}\left(\Psi_{t,s}(\omega)-\Psi_{s,s}(\omega)\right)\quad\text{ for }s\geq
0\text{ and }t\geq 0\text{. }$ (5.7)

Restricted to $t\in\mathcal{K}_{T}=[0,T]$ (and hence also $s\leq T$ as
$\left.s\leq t\right)$, the process
$\bar{\varphi}^{(T)}=\left(\bar{\varphi}_{s}\right)_{0\leq s\leq T}$ then
induces an $\mathbb{M}^{d}\left(\mathcal{K}_{T}\right)$-valued process
$\varphi^{(T)}=\left(\varphi_{s}^{(T)}\right)_{0\leq s\leq T}$ which is weak*
predictable because $s\mapsto\Psi_{t,s}$ is predictable. In fact,

$\varphi_{s}^{(T)}([0,t]):=\bar{\varphi}_{s}^{(T)}(t)=I_{\\{t\geq
s\\}}\left(\Psi_{t,s}-\Psi_{s,s}\right)\quad\text{ for }0\leq t\leq T\text{
and }0\leq s\leq t$ (5.8)

so we first get $\varphi_{s}^{(T)}(\\{0\\})=I_{\\{0\geq
s\\}}\left(\Psi_{0,s}-\Psi_{s,s}\right)=I_{\\{s=0\\}}\left(\Psi_{0,0}-\Psi_{0,0}\right)=0$
as $s\geq 0$, and then
$s\mapsto\varphi_{s}^{(T)}((0,t])=\varphi_{s}^{(T)}([0,t])=I_{\\{t\geq
s\\}}\left(\Psi_{t,s}-\Psi_{s,s}\right)$ is predictable by our assumptions. By
measure-theoretic induction, $\varphi^{(T)}(g)$ is then predictable for $g$
bounded and measurable on $\mathcal{K}_{T}$ and in particular for $g=f\in
C\left(\mathcal{K}_{T}\right)$. Note that the variation norm of
$\varphi_{s}^{(T)}$ is given by

$\left\|\varphi_{s}^{(T),i}\right\|_{\mathrm{var}}=\operatorname{var}_{t}\left(\left.\Psi_{\cdot,s}^{i}\right|_{[0,T]}\right)=\int_{s}^{T}\left|\Psi_{\mathrm{d}t,s}^{i}\right|\quad\text{
for }i=1,\ldots,d$

With these preparations, we obtain the following result.

######  Theorem 5.1.

Suppose for each $T\in(0,\infty)$ and each $s\in[0,T]$ that the process
$t\mapsto\Psi_{t,s}$ is of finite variation on $[0,T]$ and its
$\left(\mathbb{R}^{d}\right.$-valued, componentwise) total variation process

$\left(\operatorname{var}_{t}\left(\Psi_{\cdot,s}\mid[0,T]\right)\right)_{0\leq
s\leq T}=\left(\int_{s}^{T}\left|\Psi_{\mathrm{d}t,s}^{i}\right|\right)_{0\leq
s\leq T}^{i=1,\ldots,d}\text{ is }S\text{-integrable on }[0,T]\text{. }$ (5.9)

Then the process $X$ from (5.1) can be written as

$\displaystyle X_{t}=\int_{0}^{t}\Psi_{s,s}\mathrm{~{}d}S_{s}+Y_{t}$
$\displaystyle=\int_{0}^{t}\Psi_{s,s}\mathrm{~{}d}S_{s}+\int_{0}^{t}\left(\Psi_{t,s}-\Psi_{s,s}\right)\mathrm{d}S_{s}$
(5.10)
$\displaystyle=\int_{0}^{t}\Psi_{s,s}\mathrm{~{}d}S_{s}+\left(\varphi^{(T)}\bullet
S_{T}\right)([0,t]),\quad 0\leq t\leq T,$

for the $\mathbb{M}^{d}\left(\mathcal{K}_{T}\right)$-valued process
$\varphi^{(T)}$ associated to $\Psi$. If each process $\varphi^{(T)}\bullet S$
is measurevalued (as opposed to only charge-valued), then $X$ is a
semimartingale and $Y$ is predictable and of finite variation.

###### Proof.

Proof. For each $T\in(0,\infty)$, we prove (5.10) by working with the compact
set $\mathcal{K}_{T}:=[0,T]$. Results on $[0,\infty)$ are then obtained by
pasting together.
1) Fix $T\in(0,\infty)$ and consider $\varphi^{(T)}$ as in (5.8). Using the
definition of $Y$ in (5.6) and $\Psi_{t,s}=0$ for $s>t$ as well as (5.8), we
can write, for $0\leq t\leq T$,

$\displaystyle Y_{t}$ $\displaystyle=\int_{0}^{t}I_{\\{t\geq
s\\}}\left(\Psi_{t,s}-\Psi_{s,s}\right)\mathrm{d}S_{s}=\varphi^{(T)}\left(I_{[0,t]}\right)\cdot
S_{t}$ (5.11)
$\displaystyle=\int_{0}^{T}\varphi_{s}^{(T)}([0,t])\mathrm{d}S_{s}=\varphi^{(T)}\left(I_{[0,t]}\right)\cdot
S_{T}.$ (5.12)

Thanks to the assumption (5.9), we can now invoke the weak* Fubini property
(4.7) in Theorem 4.4 to obtain with the closed set
$D:=[0,t]\subseteq\mathcal{K}_{T}$ that

$\left(\varphi^{(T)}\left(I_{[0,t]}\right)\right)\cdot
S_{s}=\left(\varphi^{(T)}\bullet S_{s}\right)\left(I_{[0,t]}\right)\quad\text{
for all }s\in[0,T]$ (5.13)

giving for $s=T$ via (5.12) that

$Y_{t}=\left(\varphi^{(T)}\bullet
S_{T}\right)\left(I_{[0,t]}\right)\quad\text{ for }0\leq t\leq T$ (5.14)

This proves (5.10). If $\varphi^{(T)}\bullet S$ has values in
$\mathbb{M}\left(\mathcal{K}_{T}\right)$, then $Y=\left(Y_{t}\right)_{0\leq
t\leq T}$ is adapted by (5.11), and by (5.14) RCLL and of finite variation
with respect to $t$, and hence in particular a semimartingale on $[0,T]$. So
is then $X$ by (5.10) as $\left(\Psi_{s,s}\right)_{s\geq 0}$ is
$S$-integrable.
2) To show that $Y$ is even predictable, we first note that (5.8) shows that
we have
$\varphi_{s}^{(T)}\left(I_{[0,t]}\right)=\bar{\varphi}_{s}^{(T)}(t)=\varphi_{s}^{(T)}\left(I_{[0,t]}\right)I_{\\{s<t\\}}$
and therefore, by $(5.11)$,

$Y_{t}=\int_{0}^{t}\varphi_{s}^{(T)}\left(I_{[0,t]}\right)I_{\\{s<t\\}}\mathrm{d}S_{s}=\int_{0}^{t-}\varphi_{s}^{(T)}\left(I_{[0,t]}\right)\mathrm{d}S_{s}=\varphi^{(T)}\left(I_{[0,t]}\right)\cdot
S_{t-\cdot}$ (5.15)

On the other hand, the process $U(s,t):=\left(\varphi^{(T)}\bullet
S_{t-}\right)\left(I_{[0,s]}\right)$ on $\mathcal{K}_{T}\times\bar{\Omega}$ is
well defined, because $t\mapsto\left(\varphi^{(T)}\bullet
S_{t}\right)\left(I_{[0,s]}\right)$ is by Theorem 4.4 in $\mathcal{R}^{0}$ and
hence RCLL (with respect to $t$ ), and $U$ is
$\mathcal{B}\left(\mathcal{K}_{T}\right)\otimes\mathcal{P}$-measurable as it
is right-continuous in $s$ and left-continuous in $t$ and adapted to
$\mathbb{F}$. Note that right-continuity in $s$ uses that
$\varphi^{(T)}\bullet S$ is measure-valued. This implies by a monotone class
argument that the diagonal of $U$ is predictable. Finally, (5.15) and (5.13)
also yield that

$Y_{t}=\varphi^{(T)}\left(I_{[0,t]}\right)\cdot
S_{t-}=\left(\varphi^{(T)}\bullet S_{t-}\right)\left(I_{[0,t]}\right)=U(t,t)$

so $Y$ is the diagonal of $U$ and therefore predictable. This completes the
proof.∎

######  Remark 5.2.

The decomposition $X_{t}=\int_{0}^{t}\Psi_{s,s}\mathrm{~{}d}S_{s}+Y_{t}$ in
(5.6) also allows us to say more about the structure and decomposition of the
semimartingale $X$, again under all the assumptions of Theorem 5.1. We have
just seen that $Y$ is predictable and of finite variation. The first summand
in (5.6) is a stochastic integral, with respect to $S$, of the predictable
$S$-integrable process $\left(\Psi_{s,s}\right)_{s\geq 0}$. Thus it is always
a semimartingale, and we can work out its characteristics from the behaviour
of $\Psi$ on the diagonal $\left(\Psi_{s,s}\right)_{s\geq 0}$ and from the
characteristics of $S$ itself. If $S=M$ is a local martingale, we can also ask
whether the integral is even a local martingale. (In that case,
$X_{t}=\int_{0}^{t}\Psi_{t,s}\mathrm{~{}d}M_{s},t\geq 0$, is a special
semimartingale.) This depends on the interaction between the diagonal of
$\Psi$ and the jumps of $M$; we need to check whether the increasing process
$\left(\sum_{0\leq s\leq}\left(\Psi_{s,s}^{\top}\Delta
M_{s}\right)^{2}\right)^{1/2}$ is locally integrable. (Of course, this is
trivially satisfied if $M$ is continuous.) For more details, see
Jacod/Shiryaev [16, Remark III.6.28].

##  6 Connections to the classic case

In this section, we relate our results to some of the classic versions of
stochastic Fubini theorems available in the literature. We do not aim for
maximal generality, but rather explain the main conceptual differences.
We start again with an $\mathbb{R}^{d}$-valued semimartingale
$S=\left(S_{t}\right)_{t\geq 0}$ on $(\Omega,\mathcal{F},\mathbb{F},P)$ and a
compact metric space $\mathcal{K}$. We consider a fixed weak* predictable
process $\varphi=\left(\varphi_{t}\right)_{t\geq 0}$ taking values in
$\mathbb{M}^{d}(\mathcal{K})$ and write it in the form

$\varphi_{t}(\mathrm{~{}d}z)(\omega)=\psi_{t}(z)(\omega)\rho(\omega,t,\mathrm{~{}d}z)$
(6.1)

for an $\mathbb{R}^{d}$-valued process $\psi$ on
$\Omega\times(0,\infty)\times\mathcal{K}$ which is product-measurable and such
that $\psi(z)$ is for each $z\in\mathcal{K}$ an $\mathbb{R}^{d}$-valued
predictable process on $\bar{\Omega}$. In (6.1),
$\rho(\omega,t,\mathrm{~{}d}z)$ is a transition kernel from
$(\bar{\Omega},\mathcal{P})$ to $(\mathcal{K},\mathcal{B}(\mathcal{K}))$,
i.e., $\rho(\cdot,\cdot,D)$ is a predictable process for each
$D\in\mathcal{B}(\mathcal{K})$ and $\rho(\omega,t,\cdot)$ is a finite
(nonnegative) measure on $\mathcal{B}(\mathcal{K})$ for each
$(\omega,t)\in\bar{\Omega}$. This is not a restriction in generality for
$\varphi$ as we can always take $\rho:=\sum_{i=1}^{d}\left|\varphi^{i}\right|$
and $\psi:=\frac{d\varphi}{d\rho}$. From (6.1) and using Cauchy-Schwarz, we
obtain that $\|\varphi\|_{\text{var }}$ has the coordinates

$\left\|\varphi_{t}^{i}\right\|_{\mathrm{var}}=\int_{\mathcal{K}}\left|\psi_{t}^{i}(z)\right|\rho_{t}(\mathrm{~{}d}z)\leq\left(\rho_{t}(\mathcal{K})\int_{\mathcal{K}}\left|\psi_{t}^{i}(z)\right|^{2}\rho_{t}(\mathrm{~{}d}z)\right)^{1/2}.$
(6.2)

So $\|\varphi\|_{\text{var }}$ is $S$-integrable if (and only if) for some
control process $V$ for $S$,

$\text{ the process
}\int\sum_{i=1}^{d}\left(\int_{\mathcal{K}}\left|\psi_{r}^{i}(z)\right|\rho_{r}(\mathrm{~{}d}z)\right)^{2}\mathrm{~{}d}V_{r}\text{
is finite-valued, }$ (6.3)

which is implied by the stronger condition that

$\text{ the process
}\int\rho_{r}(\mathcal{K})\left(\int_{\mathcal{K}}\left|\psi_{r}(z)\right|^{2}\rho_{r}(\mathrm{~{}d}z)\right)\mathrm{d}V_{r}\text{
is finite-valued. }$ (6.4)

A very special case of (6.1) arises if we assume that
$\rho(\omega,t,\mathrm{~{}d}z)\equiv\eta(\mathrm{d}z)$ for some finite
(nonnegative) measure $\eta$ on $\mathcal{B}(\mathcal{K})$. This means that
the $(\omega,t)$-indexed family of measures
$\varphi_{t}(\mathrm{~{}d}z)(\omega)$ is dominated by one fixed measure and we
have

$\varphi_{t}(\mathrm{~{}d}z)(\omega)=\psi_{t}(z)(\omega)\eta(\mathrm{d}z)$
(6.5)

where $\psi$ is a $z$-indexed family of predictable processes on
$\bar{\Omega}$. Apart from slightly different conditions on $\mathcal{K}$ and
$\eta$, this is the framework in which classic stochastic Fubini theorems are
cast. More precisely, Protter [29, Section IV.6] and Veraar [31] both assume
that ( $\mathcal{K},\mathbb{K}$ ) is a measurable space and $\eta$ is a
measure on $\mathbb{K}$, finite in [29] and $\sigma$-finite in [31]. Next, $S$
is in both $[29,31]$ a real-valued semimartingale, assumed continuous in [31],
and $\psi$ is $\mathcal{P}\otimes\mathbb{K}$-measurable in [29] and
$\rm{Prog}\otimes\mathbb{K}$-measurable in [31], where Prog is the progressive
$\sigma$-field on $\bar{\Omega}$. The integrability condition on $\psi$ in
[29] is that

$\text{ the process
}\left(\int_{\mathcal{K}}\left|\psi_{t}(z)\right|^{2}\eta(\mathrm{d}z)\right)_{t\geq
0}^{1/2}\text{ is }S\text{-integrable, }$ (6.6)

which by finiteness of $\eta$ is equivalent to the sufficient condition (6.4)
above. If $S$ is continuous with canonical decomposition $S=S_{0}+M+A$, a
sufficient condition for our weaker assumption (6.3) is by Shiryaev/Cherny
[30, Definition 3.9 and part (ii) of the subsequent remark] that

$\text{ the processes
}\int\left(\int_{\mathcal{K}}\left|\psi_{r}(z)\right|\eta(\mathrm{d}z)\right)\mathrm{d}A_{r}\text{
and
}\int\left(\int_{\mathcal{K}}\left|\psi_{r}(z)\right|\eta(\mathrm{d}z)\right)^{2}\mathrm{~{}d}[M]_{r}$
(6.7)

are both finite-valued.
The condition (6.7) is slightly different from the assumption in [31, Theorem
2.2, conditions (2.1) and (2.2)] that

$\text{ the processes
}\int_{\mathcal{K}}\left(\int\left|\psi_{r}(z)\right|\mathrm{d}A_{r}\right)\eta(\mathrm{d}z)\text{
and
}\int_{\mathcal{K}}\left(\int\left|\psi_{r}(z)\right|^{2}\mathrm{~{}d}[M]_{r}\right)^{1/2}\eta(\mathrm{d}z)$

are both finite-valued.
In the frameworks of either Protter [29, Section IV.6] or Veraar [31], the
classic stochastic Fubini theorem says that in the dominated case (6.5), we
have $P$-a.s. for all $t\geq 0$ that

$\int_{0}^{t}\left(\int_{\mathcal{K}}\psi_{r}(z)\eta(\mathrm{d}z)\right)\mathrm{d}S_{r}=\int_{\mathcal{K}}\left(\int_{0}^{t}\psi_{r}(z)\mathrm{d}S_{r}\right)\eta(\mathrm{d}z)$
(6.8)

If we replace $\psi$ by $\psi I_{D}$ for a measurable subset $D$ of
$\mathcal{K}$, this can be rewritten as

$\int_{0}^{t}\varphi_{r}(D)\mathrm{d}S_{r}=\int_{D}\left(\int_{0}^{t}\psi_{r}(z)\mathrm{d}S_{r}\right)\eta(\mathrm{d}z)$
(6.9)

On the other hand, in our general setting and under (6.3), we obtain from
Theorem 4.4 that for every closed $D\subseteq\mathcal{K}$, we get $P$-a.s. for
all $t\geq 0$ that

$\int_{0}^{t}\varphi_{r}(D)\mathrm{d}S_{r}=\left(\varphi\bullet
S_{t}\right)(D)$ (6.10)

As the left-hand sides of (6.9) and (6.10) agree, so must the right-hand
sides, whenever all the required assumptions are satisfied. Now by
construction, the stochastic integral $\varphi\bullet S$ is in general only a
charge-valued process; so the right-hand side of (6.10) need not be a measure
in the argument $D$. On the other hand, the right-hand side of (6.9) can be
expected to be a measure in the argument $D$, provided that the inner
$(\mathrm{d}S$-)integral is an $\eta$-integrable function of $z$. So we can
exploit the connection between (6.9) and (6.10) to get more information about
$\varphi\bullet S$. The next result makes these ideas precise. Note that
except for $\mathcal{K}$, its assumptions are precisely those of Protter [29,
Theorem IV.65].

######  Proposition 6.1.

Suppose $S=\left(S_{t}\right)_{t\geq 0}$ is a real-valued semimartingale and
$\mathcal{K}$ is a compact metric space with Borel $\sigma$-field
$\mathcal{B}(\mathcal{K})$. Let $\varphi=\left(\varphi_{t}\right)_{t\geq 0}$
be a charge-valued process which satisfies the domination condition (6.5) so
that
$\varphi_{t}(\mathrm{~{}d}z)(\omega)=\psi_{t}(z)(\omega)\eta(\mathrm{d}z)$ for
a finite (nonnegative) measure $\eta$ on $\mathcal{B}(\mathcal{K})$. Suppose
that $\psi$ is $\mathcal{P}\otimes\mathcal{B}(\mathcal{K})$-measurable and
satisfies (6.6). Then the stochastic integral process $\varphi\bullet S$ is
well defined and takes values in $\mathbb{M}$. In other words, our stochastic
integral is then not only charge-valued, but measure-valued.

###### Proof.

1) We first argue that $\varphi$ is $\mathbb{M}$-valued, weak∗ predictable and
in $\Phi$. As in (6.2),

$\left\|\varphi_{t}\right\|_{\text{var
}}=\left|\varphi_{t}\right|(\mathcal{K})=\int_{\mathcal{K}}\left|\psi_{t}(z)\right|\eta(\mathrm{d}z)\leq\left(\eta(\mathcal{K})\int_{\mathcal{K}}\left|\psi_{t}(z)\right|^{2}\eta(\mathrm{d}z)\right)^{1/2}.$
(6.11)

Thanks to (6.6), the right-hand side has a finite integral with respect to
some control process $V$ for $S$, and if we choose (as we can) $V$ strictly
increasing, the left-hand side of (6.11) must be finite $P$-a.s. for all
$t\geq 0$ like the right-hand side. So $\varphi$ is $\mathbb{M}$-valued, and
$\|\varphi\|_{\text{var }}$ is $S$-integrable again due to (6.6). For each
$z$, the $z$-section $\psi(z)$ is predictable because $\psi$ is
$\mathcal{P}\otimes\mathcal{B}(\mathcal{K})$-measurable, and for any $f\in
C(\mathcal{K})$,

$\int\left|f(z)\psi_{t}(z)\right|\eta(\mathrm{d}z)\leq\|f\|_{\infty}\int_{\mathcal{K}}\left|\psi_{t}(z)\right|\eta(\mathrm{d}z)<\infty\quad
P\text{-a.s. for all }t\geq 0$

shows that $\varphi(f)$ is well defined and hence a predictable process. So
$\varphi$ is also weak* predictable and therefore in $\Phi$.
2) Thanks to 1), Theorem 4.4 tells us that $\varphi\bullet S$ is well defined
and that for every closed set $D\subseteq\mathcal{K}$,

$(\varphi\bullet S)(D)=\varphi(D)\cdot
S=\int\varphi_{r}(D)\mathrm{d}S_{r}=\int\left(\int_{D}\psi_{r}(z)\eta(\mathrm{d}z)\right)\mathrm{d}S_{r}$
(6.12)

On the other hand, the classic stochastic Fubini theorem in Protter [29,
Theorem IV.65] (applied to $\psi I_{D}$ ) tells us that

$\int\left(\int_{D}\psi_{r}(z)\eta(\mathrm{d}z)\right)\mathrm{d}S_{r}=\int_{D}\left(\int\psi_{r}(z)\mathrm{d}S_{r}\right)\eta(\mathrm{d}z)$
(6.13)

and that $\int\psi_{r}(z)\mathrm{d}S_{r}$ can be chosen product-measurable on
$\bar{\Omega}\times\mathcal{K}$ so that
$z\mapsto\int\psi_{r}(z)\mathrm{d}S_{r}$ is then
$\mathcal{B}(\mathcal{K})$-measurable. So the right-hand side of (6.13), and
hence by $(6.12)$ also $(\varphi\bullet S)(D)$, is a finite signed measure as
a function of $D$ if we can show that

$\int_{\mathcal{K}}\left|\int_{0}^{t}\psi_{r}(z)\mathrm{d}S_{r}\right|\eta(\mathrm{d}z)<\infty\quad
P\text{-a.s. for each }t\geq 0$ (6.14)

3) By the assumption (6.6), the process
$U:=\left(\int_{\mathcal{K}}|\psi(z)|^{2}\eta(\mathrm{d}z)\right)^{1/2}$ is
$S$-integrable. By Shiryaev/Cherny [30, Definition 3.9; see also Lemma 4.11],
there exists a decomposition $S=S_{0}+M+A$ into a local martingale $M$ and an
adapted RCLL process $A$ of finite variation $|A|$, both null at 0 , such that
$U$ is in both $L_{\mathrm{var}}(A)$ and $L_{\mathrm{loc}}^{1}(M)$. By the
definition of $L_{\mathrm{var}}(A)$ in [30], this implies via Fubini-Tonelli
and Cauchy-Schwarz that

$\displaystyle\int_{\mathcal{K}}\left(\left|\int_{0}^{t}\psi_{r}(z)\mathrm{d}A_{r}\right|\right)\eta(\mathrm{d}z)$
$\displaystyle\leq\int_{\mathcal{K}}\left(\int_{0}^{t}\left|\psi_{r}(z)\right|\mathrm{d}|A|_{r}\right)\eta(\mathrm{d}z)$
(6.15)
$\displaystyle=\int_{0}^{t}\left(\int_{\mathcal{K}}\left|\psi_{r}(z)\right|\eta(\mathrm{d}z)\right)\mathrm{d}|A|_{r}$
$\displaystyle\leq\int_{0}^{t}(\eta(\mathcal{K}))^{1/2}U_{r}\mathrm{~{}d}|A|_{r}<\infty\quad
P\text{-a.s. }$

Next, the property $U\in L_{\text{loc }}^{1}(M)$ means that $\left(\int
U^{2}\mathrm{~{}d}[M]\right)^{1/2}$ is locally integrable so that
along a localizing sequence $\left(\tau_{n}\right)_{n\in\mathbb{N}}$ of
stopping times, we have by Fubini-Tonelli that

$\displaystyle
E\left[\left(\int_{\mathcal{K}}\left(\int_{0}^{\tau_{n}}\left|\psi_{r}(z)\right|^{2}\mathrm{~{}d}[M]_{r}\right)\eta(\mathrm{d}z)\right)^{1/2}\right]$
$\displaystyle=E\left[\left(\int_{0}^{\tau_{n}}\left(\int_{\mathcal{K}}\left|\psi_{r}(z)\right|^{2}\eta(\mathrm{d}z)\right)\mathrm{d}[M]_{r}\right)^{1/2}\right]$
$\displaystyle=E\left[\left(\int_{0}^{\tau_{n}}U_{r}^{2}\mathrm{~{}d}[M]_{r}\right)^{1/2}\right]<\infty.$

By Cauchy-Schwarz,

$\int_{\mathcal{K}}\left(\int\left|\psi_{r}(z)\right|^{2}\mathrm{~{}d}[M]_{r}\right)^{1/2}\eta(\mathrm{d}z)\leq(\eta(\mathcal{K}))^{1/2}\left(\int_{\mathcal{K}}\left(\int\left|\psi_{r}(z)\right|^{2}\mathrm{~{}d}[M]_{r}\right)\eta(\mathrm{d}z)\right)^{1/2}$

and so using Fubini-Tonelli again yields that also

$\int_{\mathcal{K}}E\left[\left(\int_{0}^{\tau_{n}}\left|\psi_{r}(z)\right|^{2}\mathrm{~{}d}[M]_{r}\right)^{1/2}\right]\eta(\mathrm{d}z)=E\left[\int_{\mathcal{K}}\left(\int_{0}^{\tau_{n}}\left|\psi_{r}(z)\right|^{2}\mathrm{~{}d}[M]_{r}\right)^{1/2}\eta(\mathrm{d}z)\right]<\infty$

for each $n$. By the Burkholder-Davis-Gundy inequality,

$E\left[\left(\int\psi(z)\mathrm{d}M\right)_{\tau_{n}}^{*}\right]\leq\text{
const.
}E\left[\left(\int_{0}^{\tau_{n}}\left|\psi_{r}(z)\right|^{2}\mathrm{~{}d}[M]_{r}\right)^{1/2}\right]$

for each $z$, where the constant does not depend on $z$, and so

$\displaystyle
E\left[\int_{\mathcal{K}}\left|\int_{0}^{\tau_{n}}\psi_{r}(z)\mathrm{d}M_{r}\right|\eta(\mathrm{d}z)\right]$
$\displaystyle\leq
E\left[\int_{\mathcal{K}}\left(\int\psi(z)\mathrm{d}M\right)_{\tau_{n}}^{*}\eta(\mathrm{d}z)\right]$
$\displaystyle=\int_{\mathcal{K}}E\left[\left(\int\psi(z)\mathrm{d}M\right)_{\tau_{n}}^{*}\right]\eta(\mathrm{d}z)$
$\displaystyle\leq\text{ const.
}\int_{\mathcal{K}}E\left[\left(\int_{0}^{\tau_{n}}\left|\psi_{r}(z)\right|^{2}\mathrm{~{}d}[M]_{r}\right)^{1/2}\right]\eta(\mathrm{d}z)<\infty$

for each $n$. This implies that

$\int_{\mathcal{K}}\left|\int_{0}^{t}\psi_{r}(z)\mathrm{d}M_{r}\right|\eta(\mathrm{d}z)<\infty\quad
P\text{-a.s. on each }\llbracket 0,\tau_{n}\rrbracket\text{, }$

and so (6.14) follows by combining this with (6.15). This completes the
proof.∎

If we compare the classic stochastic Fubini theorem to our new result in
Theorem 4.4, we can see several differences. First, the former needs stronger
assumptions on the integrand $\varphi($ or $\psi)$ than the latter; this can
also be seen in the proof of Proposition 6.1. Second, if we represent
$\varphi$ as in (6.1) as
$\varphi_{t}(\mathrm{~{}d}z)=\psi_{t}(z)\rho(\omega,t,\mathrm{~{}d}z)$, the
general weak* Fubini property (4.7) in Theorem 4.4 for $D=\mathcal{K}$ (with
the two sides interchanged) takes the form

$\int_{0}^{t}\left(\int_{\mathcal{K}}\psi_{r}(z)\rho_{r}(\mathrm{~{}d}z)\right)\mathrm{d}S_{r}=\int_{\mathcal{K}}\left(\int_{0}^{t}\psi_{r}(z)\rho_{r}(\cdot)\mathrm{d}S_{r}\right)(\mathrm{d}z)$
(6.16)

In comparison to the classic result (6.8), the left-hand side above is the
obvious generalisation where we now mix the integrands $\psi_{r}(z)$ with
$\rho_{r}(\mathrm{~{}d}z)$ instead of only $\eta(\mathrm{d}z)$. However, the
right-hand side of (6.16) is more complicated than in (6.8); we really need
the stochastic integral with respect to $S$ of the measure-valued process
$\psi\rho$, and it is no longer directly possible to rewrite or construct this
explicitly as a mixture (over $z$ ) of stochastic integrals
$\int\psi_{r}(z)\mathrm{d}S_{r}$. This is because in contrast to the fixed
measure $\eta$, the kernel $\rho$ depends on both $\omega$ and $t$ in general.
In comparison to the existing literature on stochastic Fubini theorems, our
main contribution is twofold. At the technical level, we deal with a
parametric family of integrands that is not dominated, with respect to its
parameter, by a single and nonrandom measure. At the conceptual level, we
achieve our results via the idea of viewing integrands as measure-valued
processes and developing and using a corresponding stochastic integration
theory. The latter viewpoint is similar in spirit to the approach of Bichteler
and co-authors and has also been used in van Neerven/Veraar [23]; but its
implementation must be done here more generally because we do not have a
dominating measure $\eta$, and we also need to develop a concept of stochastic
integration of measure-valued integrands with respect to general
semimartingales.

######  Remark 6.2.

It is an interesting question whether our approach allows to obtain the
classic stochastic Fubini theorem as a special case, and maybe even under
weaker assumptions. Purely formally, one could start from (6.16), plug in
$\eta$ for $\rho_{r}$ and then hope that one could move the deterministic
$\eta$ outside to $\mathrm{d}z$ on the right-hand side of (6.16). This would
also imply that in the dominated setting (6.5), our stochastic integral
process $\varphi\bullet S$ is dominated
by the measure $\eta$. If $\varphi\in\mathcal{E}$ is an elementary integrand,
the above results can be proved easily because all integrals then reduce to
finite sums. But in general, one would probably need a similar approximation
as in the key Lemma 3.6, now at the level of densities $\psi$, and it seems
very likely that this would require extra conditions on $\psi$, maybe similar
to the condition (6.6) in Protter [29, Theorem IV.65]. We leave this issue as
an open problem and refer to Section 7 for a caveat against too much optimism.

##  7 An illustrative example

On the filtered probability space $(\Omega,\mathcal{F},\mathbb{F},P)$, let
$S=W=\left(W_{t}\right)_{t\geq 0}$ be a Brownian motion. Fix $T\in(0,\infty)$
and take the compact set $\mathcal{K}=\mathcal{K}_{T}=[0,T]$. For $\alpha>0$,
define

$\varphi_{t}(\mathrm{~{}d}z):=\alpha(z-t)^{\alpha-1}I_{\\{z>t\\}}\mathrm{d}z=:\psi_{t}(z)\mathrm{d}z\quad\text{
for }t\geq 0\text{ and }z\in\mathcal{K}$ (7.1)

Then each $\varphi_{t}$ is a (nonnegative) measure on
$\mathcal{B}(\mathcal{K})$ with

$\left\|\varphi_{t}\right\|_{\mathrm{var}}=\varphi_{t}(\mathcal{K})=\int_{t}^{T}\alpha(z-t)^{\alpha-1}\mathrm{~{}d}z=(T-t)^{\alpha}<\infty$
(7.2)

and so $\varphi=\left(\varphi_{t}\right)_{t\geq 0}$ is an $\mathbb{M}$-valued
process. Moreover, as it is nonrandom, the process $\varphi$ is clearly weak*
predictable. Note that $\psi_{t}(z)=0$ for $z\in\mathcal{K}=[0,T]$ and $t>T$;
so it is enough to look only at $\left(\varphi_{t}\right)_{0\leq t\leq T}$ and
$\left(W_{t}\right)_{0\leq t\leq T}$. If we take for $W$ the control process
$V_{t}=t$, then (7.2) gives

$\mathcal{D}_{t}\left(\|\varphi\|_{\mathrm{var}};V\right)=\int_{0}^{t}\left\|\varphi_{r}\right\|_{\mathrm{var}}^{2}\mathrm{~{}d}V_{r}=\int_{0}^{t}(T-r)^{2\alpha}\mathrm{d}r=\frac{1}{2\alpha+1}\left(T^{2\alpha+1}-(T-t)^{2\alpha+1}\right)$

This is finite for any $t\in[0,T]$ and $\alpha>0$ so that $\varphi$ is in
$\Phi$. Thus $\varphi\bullet W$ is well defined (on $[0,T]$ ) and we can use
the general weak* Fubini property (4.6) from Theorem 4.4. In particular,
taking $D=[0,u]$ and computing
$\varphi_{r}([0,u])=(u-r)^{\alpha}I_{\\{u>r\\}}$ as in (7.2), we get

$\left(\varphi\bullet S_{T}\right)([0,u])=\varphi([0,u])\cdot
S_{T}=\int_{0}^{T}I_{\\{u>r\\}}(u-r)^{\alpha}\mathrm{d}S_{r}=\int_{0}^{u}(u-r)^{\alpha}\mathrm{d}W_{r}$

To check whether $\varphi\bullet S$ is a measure-valued process, we want to
use Proposition 6.1. From (7.1), it is clear that we are in the dominated case
(6.5), with $\eta$ being Lebesgue
measure on $\mathcal{K}=[0,T]$, and $\varphi$ is clearly
$\mathcal{P}\otimes\mathcal{B}(\mathcal{K})$-measurable. For the integrability
condition (6.6), we need to look at

$\displaystyle\int_{0}^{t}\left(\int_{\mathcal{K}}\left|\psi_{r}(z)\right|\eta(\mathrm{d}z)\right)\mathrm{d}V_{r}$
$\displaystyle=\int_{0}^{t}\int_{r}^{T}\alpha^{2}(z-r)^{2\alpha-2}\mathrm{~{}d}z\mathrm{~{}d}r$
(7.3)
$\displaystyle=\frac{\alpha^{2}}{2\alpha-1}\int_{0}^{t}(T-r)^{2\alpha-1}\mathrm{~{}d}r$
$\displaystyle=\frac{\alpha^{2}}{2\alpha(2\alpha-1)}\left(T^{2\alpha}-(T-t)^{2\alpha}\right)$

where the second equality requires $2\alpha-2>-1$ or $\alpha>\frac{1}{2}$ to
ensure that the inner integral does not diverge at $z=r$. For
$\alpha>\frac{1}{2}$, the quantity in (7.3) is always finite; so (6.6) is then
satisfied and Proposition 6.1 tells us that $\varphi\bullet S$ is for
$\alpha>\frac{1}{2}$ a measure-valued process.
To get more information about $\varphi\bullet S$, we now want to use the
results on Volterra-type semimartingales from Section 5. As in (5.7) and
(5.4), we look at $\Psi$ given by $\Psi_{s,s}=0$ and $\Psi_{t,s}I_{\\{t\geq
s\\}}=I_{\\{t\geq s\\}}\varphi_{s}((0,t])=I_{\\{t\geq
s\\}}\int_{0}^{t}\psi_{s}(z)\mathrm{d}z=I_{\\{t\geq
s\\}}\int_{s}^{t}\alpha(z-s)^{\alpha-1}\mathrm{~{}d}z=I_{\\{t\geq
s\\}}(t-s)^{\alpha}$. Clearly $t\mapsto\Psi_{t,s}$ is of finite variation, and
it also satisfies (5.9) due to (7.2) and because $\varphi\in\Phi$. We then
obtain as in (5.6) that for any $\alpha>0$,

$Y_{u}=\int_{0}^{u}\left(\Psi_{u,s}-\Psi_{s,s}\right)\mathrm{d}S_{s}=\int_{0}^{u}(u-s)^{\alpha}\mathrm{d}W_{s}$
(7.4)

and Theorem 5.1 tells us that $Y=\left(Y_{u}\right)_{0\leq u\leq T}$ is a
semimartingale if $\varphi\bullet S$ is measurevalued. But the process $Y$ in
(7.4) has been studied (even for $W$ replaced by a Lévy process) in detail in
Basse/Pedersen [3], and [3, Theorem 3.1], with $\sigma=1$, shows that $Y$ is a
semimartingale (in the filtration of $W$ ) if and only if the function
$x\mapsto f(x)=x^{\alpha}$ has $f^{\prime}$ locally square-integrable on
$[0,\infty)$, which holds as in (7.3) if and only if $\alpha>\frac{1}{2}$.

######  Remark 7.1.

In this very specific example, we see that $\varphi\bullet S$ is a measure-
valued process if and only if a Volterra-type process
$\left(\int_{0}^{t}\Psi_{t,s}\mathrm{~{}d}S_{s}\right)_{t\geq 0}$ with $\Psi$
naturally associated to $\varphi$ is a semimartingale. It is very tempting to
conjecture that such a result could hold more generally, maybe even beyond the
dominated setting (6.5). But even formulating this precisely, let alone
proving it, is beyond the scope of the present paper.

Acknowledgement: Martin Schweizer gratefully acknowledges financial support by
the Swiss Finance Institute (SFI). Tahir Choulli gratefully acknowledges
financial support by NSERC (the Natural Sciences and Engineering Research
Council of Canada, Grant G121210818). Moreover, financial support by the
Forschungsinstitut für Mathematik (FIM) at ETH Zûrich is gratefully
acknowledged. It is a pleasure to thank Thomas Hille for a number of useful
technical remarks and discussions on measurability, Alex Schied for some very
useful comments, and Mark Veraar for promptly and helpfully answering
questions on the related literature.

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