# First Order Linear Marcus SPDEs

Lena–Susanne Hartmann111Institute of Mathematics, Friedrich Schiller
University Jena, Ernst–Abbe–Platz 2, 07743 Jena, Germany; lena.hartmann@uni-
jena.de and Ilya Pavlyukevich222Institute of Mathematics, Friedrich Schiller
University Jena, Ernst–Abbe–Platz 2, 07743 Jena, Germany;
<EMAIL_ADDRESS>
###### Abstract

In this paper we solve a Lévy driven linear stochastic first order partial
differential equation (transport equation) understood in the canonical
(Marcus) form. The solution can be obtained with the help of the method of
stochastic characteristics. It has the same form as a solution of a
deterministic PDE or a solution of a stochastic PDE driven by a Brownian
motion studied by Kunita (1984, 1997).

Keywords: Canonical (Marcus) SDE; linear first order partial differential
equation; transport equation; stochastic characteristics.

AMS Subject Classification: 35F10, 35R60, 60G51, 60H05 60H15

##  1 Introduction

The transport of a contaminant in an incompressible fluid can be described
with the help of the first order linear partial differential equation
(transport equation)

$\displaystyle\partial_{t}u(x,t)$ $\displaystyle=\nabla^{T}u(x,t)a(x,t),\quad
x\in\mathbb{R}^{d},\ t>0,$ (1) $\displaystyle u(x,0)$
$\displaystyle=u_{0}(x),$

where $u(x,t)$ is the concentration of the contaminant at time instant $t$ at
the position $x\in\mathbb{R}^{d}$, and $-a(x,t)$ is the flow instant velocity
at the position $x$ at time $t$, see e.g. Debnath [7] and Van der Perk [38].

The equation (1) can be solved with the help of the method of characteristics.
Indeed, consider the so-called characteristics ordinary differential equation

$\displaystyle\mathrm{d}\varphi_{0,t}(x)$
$\displaystyle=-a(\varphi_{0,t}(x),t)\,\mathrm{d}t,\quad\varphi_{0,0}(x)=x,\quad
t\geq 0.$ (2)

Under usual smoothness and boundedness assumptions on the function $a$, the
ODE (2) has a unique global solution and $t\mapsto\varphi_{0,t}(\cdot)$ is a
flow of diffemorphisms on $\mathbb{R}^{d}$. Then it is easy to check that the
unique solution of the transport equation (1) is given by the formula

$\displaystyle u(x,t)=u_{0}(\varphi_{t,0}(x)),$ (3)

where $x\mapsto\varphi_{t,0}(x)=\varphi^{-1}_{0,t}(x)$ is the inverse of the
map $x\mapsto\varphi_{0,t}(x)$, see e.g. Chapter 6 in Perthame [32].

Analogously one can show that the more general equation with the “sediment
deposition” term $ub$ (if $b\leq 0$) and the “sink/source” term $c$

$\displaystyle\partial_{t}u(x,t)$
$\displaystyle=\nabla^{T}u(x,t)a(x,t)+u(x,t)b(x,t)+c(x,t),\quad
x\in\mathbb{R}^{d},\ t>0,$ (4) $\displaystyle u(x,0)$
$\displaystyle=u_{0}(x),$

has the solution

$\displaystyle
u(x,t)=\mathrm{e}^{\int_{0}^{t}b(\varphi_{t,s}(x),s)\,\mathrm{d}s}u_{0}(\varphi_{t,0}(x))+\int_{0}^{t}\mathrm{e}^{\int_{s}^{t}b(\varphi_{t,r}(x),r)\,\mathrm{d}r}c(\varphi_{t,s}(x),s)\,\mathrm{d}s,$
(5)

see Section 6 for detail.

In a turbulent flow, the fluid velocity is given by a turbulent solution of
Navier–Stokes equations that is very difficult to analyze. Instead, one can
consider a plausible significant simplification, namely one can assume that
the fluid velocity is represented as a sum of the mean velocity field and a
spatially dependent noise, see Man and Tsai [25], Wang and Zheng [39]. This
yields the stochastic equation

$\displaystyle\partial_{t}u(x,t)$
$\displaystyle=\nabla^{T}u(x,t)a(x,t)+\nabla^{T}u(x,t)A(x,t)\circ\dot{W}(t),$
$\displaystyle u(x,0)$ $\displaystyle=u_{0}(x).$

Here $W$ is a Brownian motion, and $\circ\,\dot{W}$ denotes the noise term in
the Stratonovich sense. Introducing the characteristics SDE in the
Stratonovich form

$\displaystyle\mathrm{d}\varphi_{0,t}(x)$
$\displaystyle=a(\varphi_{0,t}(x),t)\,\mathrm{d}t+A(\varphi_{0,t}(x),t)\circ\mathrm{d}W_{t},\quad\varphi_{0,0}(x)=x,$

we obtain the same representation (3) for the solution where
$t\mapsto\varphi_{t,0}(\cdot)$ is now the inverse flow of stochastic
diffemorphisms. The general theory can be found in Kunita [19, 21] as well as
in Chow [6], Lototsky and Rozovsky [24], and Flandoli [12, 13].

The goal of this paper is to show that a Lévy driven first order equation can
be also solved by the method of stochastic characteristics and hence to extend
the results by Kunita [19, 21] to the case of Lévy driven equations. Apart
from the mathematical interest, this study is motivated by the questions
appearing in the modelling of contaminant transport. Considering Lev́y noise
allows one to incorporate such effects as asymmetry of flow fluctuations and
presence of extreme and large flow perturbations, see, e.g. Kallianpur and
Xiong [18], Birnir [3, Section 1.7], Oh and Tsai [30], and Tsai and Huang
[37].

A general theory for Lévy-driven stochastic partial differential equations can
be found in Peszat and Zabczyk [33], Mandrekar and Rüdiger [26] (evolution
equation approach), and Holden et al. [17] (white noise theory approach). It
should be noted that in order to have the formulae (3) and (5) valid in the
case of a Lévy-driven equation, we have to understand the stochastic term in
the canonical (Marcus) form. This approach is different from above mentioned
approaches, in particular from those by Proske [34] who obtained a solution of
a transport equation driven by a multiparameter Lévy process within the frame
of the white noise theory.

In the present paper we will assume that all the coefficients of the transport
equation are sufficiently smooth. There is vast literature devoted to the
analysis of (stochastic) transport equations with irregular coefficients in
the countinous setting, see, e.g. DiPerna and Lions [8], Ambrosio [1],
Flandoli [12, 13], Fedrizzi and Flandoli [10], Fedrizzi et al. [11], Catuogno
and Olivera [5], Olivera and Tudor [31], Mollinedo and Olivera [29]. To our
best knowledge, Lévy-driven transport equations with irregular coefficients
are still to be studied.

Notation. By $C^{n}$ we denote $n$ times continuously differentiable functions
and by $C^{n}_{b}$ we denote bounded $n$ times continuously differentiable
functions with bounded derivatives. For a mapping
$F\colon\mathbb{R}^{d}\mapsto\mathbb{R}^{m}$, $DF$ is the Jacobian (gradient)
matrix, namely,

$\displaystyle DF=\begin{pmatrix}\frac{\partial F_{1}}{\partial
x_{1}}&\cdots&\frac{\partial F_{1}}{\partial x_{d}}\\\ \cdots&\cdots&\cdots\\\
\frac{\partial F_{m}}{\partial x_{1}}&\cdots&\frac{\partial F_{m}}{\partial
x_{d}}\\\ \end{pmatrix},$

and in particular, for $F\colon\mathbb{R}^{d}\mapsto\mathbb{R}$,
$DF=(\frac{\partial F}{\partial x_{1}},\cdots,\frac{\partial F}{\partial
x_{d}})=\nabla^{T}F$. The exponential function is denoted by
$x\mapsto\mathrm{e}^{x}$ whereas the exponential map in the sense of (8),
(10), (26) and (26) is denoted by the bold symbol $\mathbf{e}$.

Acknowledgements. The authors acknowledge support by the DFG project PA
2123/4-1. The authors are grateful to the referees for their valuable
comments.

##  2 Setting and the main results

On a filtered probability space $(\Omega,\mathscr{F},\mathbb{F},\mathbf{P})$
that satisfies the usual hypotheses we consider an $m$-dimensional Brownian
motion $W$ and an $m$-dimensional pure jump Lévy process $Z$,

$\displaystyle Z_{t}=\int_{0}^{t}\int_{\|z\|\leq
1}z\tilde{N}(\mathrm{d}z,\mathrm{d}s)+\int_{0}^{t}\int_{\|z\|>1}zN(\mathrm{d}z,\mathrm{d}s),$

where $N$ is a Poisson random measure with the intensity Lévy measure $\nu$,
$\nu(\\{0\\})=0$, $\int_{\mathbb{R}^{m}}(|z|^{2}\wedge
1)\,\nu(\mathrm{d}z)<\infty$ and $\tilde{N}$ is the compensated Poisson random
measure,
$\tilde{N}(\mathrm{d}z,\mathrm{d}t)=N(\mathrm{d}z,\mathrm{d}t)-\mathbb{I}(|z|\leq
1)\nu(\mathrm{d}z)\mathrm{d}t$. For the exposition of the theory of Lévy
processes and Lévy driven stochastic calculus we refer the reader to e.g. Sato
[36], Kunita [22] and Applebaum [2].

Let

$\displaystyle a$ $\displaystyle\colon\mathbb{R}^{d}\to\mathbb{R}^{d},$
$\displaystyle b,c$ $\displaystyle\colon\mathbb{R}^{d}\to\mathbb{R},$
$\displaystyle A,\alpha$
$\displaystyle\colon\mathbb{R}^{d}\to\mathbb{R}^{d\times m},$ $\displaystyle
B,C,\beta,\sigma$ $\displaystyle\colon\mathbb{R}^{d}\to\mathbb{R}^{m},$

be measurable functions. For $u\in C^{1}(\mathbb{R}^{d},\mathbb{R})$, consider
first order operators

$\displaystyle\mathcal{P}u(x)$
$\displaystyle=\nabla^{T}u(x)a(x)+u(x)b(x)+c(x),$
$\displaystyle\mathcal{R}u(x)$
$\displaystyle=\nabla^{T}u(x)A(x)+u(x)B(x)+C(x),$
$\displaystyle\mathcal{Q}u(x)$
$\displaystyle=\nabla^{T}u(x)\alpha(x)+u(x)\beta(x)+\sigma(x),$

and the first order linear equation written in the compact differential form
as

$\displaystyle\mathrm{d}u(t,x)$
$\displaystyle=\mathcal{P}u(t,x)\,\mathrm{d}t+\mathcal{R}u(t,x)\circ\mathrm{d}W_{t}+\mathcal{Q}u(t,x)\diamond\mathrm{d}Z_{t},$
(6) $\displaystyle u(0,x)$ $\displaystyle=u_{0}(x),\quad x\in\mathbb{R}^{d},\
t\in[0,T],$

with some initial condition $u_{0}\colon\mathbb{R}^{d}\to\mathbb{R}$. More
precisely, this equation is understood as the integral equation as follows:

$\displaystyle u(t,x)$
$\displaystyle=u_{0}(x)+\int_{0}^{t}\Big{[}\nabla^{T}u(s,x)a(x)+u(s,x)b(x)+c(x)\Big{]}\,\mathrm{d}s$
(7)
$\displaystyle+\int_{0}^{t}\Big{[}\nabla^{T}u_{x}(s,x)A(x)+u(s,x)B(x)+C(x)\Big{]}\circ\mathrm{d}W_{s}$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\mathbf{e}^{\mathcal{Q}z}u(s-,x)-u(s-,x)\Big{)}\tilde{N}(\mathrm{d}z,\mathrm{d}s)$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\mathbf{e}^{\mathcal{Q}z}u(s-,x)-u(s-,x)-\mathcal{Q}u(s-,x)z\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}s$
$\displaystyle+\int_{0}^{t}\int_{\|z\|>1}\Big{(}\mathbf{e}^{\mathcal{Q}z}u(s-,x)-u(s-,x)\Big{)}\tilde{N}(\mathrm{d}z,\mathrm{d}s),$

where for each $z\in\mathbb{R}^{m}$ the mapping

$\displaystyle u(\cdot)\mapsto\mathbf{e}^{\mathcal{Q}z}u(\cdot)$ (8)

is defined with the help of the solution of the first order linear time
autonomous partial differential equation

$\displaystyle\partial_{r}g(r,x)$
$\displaystyle=\nabla^{T}g(r,x)\alpha(x)z+g(r,x)\beta(x)z+\sigma(x)z,\quad
r\in[0,1],$ (9) $\displaystyle g(0,x)$ $\displaystyle=u(x),$

and

$\displaystyle\mathbf{e}^{\mathcal{Q}z}u(x):=g(1;x,z).$ (10)

It is possible to write down the solution of the equation (9) explicitly. We
will do this in Section 6.

The term $\circ\,\mathrm{d}W$ denotes the Stratonovich integral w.r.t. the
Brownian motion.

######  Definition 2.1.

A random field
$u\colon\mathbb{R}^{d}\times\mathbb{R}_{+}\times\Omega\to\mathbb{R}$ is a
solution of equation (6) if $u$ is a càdlàg $C^{2}$-semimartingale and (7) is
satisfied a.s.

The goal of this paper is to solve (6) with the help of the method of
characteristics. The main result is as follows. Assume that the functions
satisfy the following conditions.

H${}_{\text{coeff}}$

$\displaystyle a$ $\displaystyle\in
C^{3}_{b}(\mathbb{R}^{d},\mathbb{R}^{d})\quad\text{and}\quad b,c\in
C^{3}_{b}(\mathbb{R}^{d},\mathbb{R})$ (11) $\displaystyle A,\alpha$
$\displaystyle\in C^{4}_{b}(\mathbb{R}^{d},\mathbb{R}^{d\times
m})\quad\text{and}\quad B,C,\beta,\sigma\in
C^{4}_{b}(\mathbb{R}^{d},\mathbb{R}^{m}).$

Consider a $(d+2)$-dimensional system of Marcus SDEs (characteristics
equations)

$\displaystyle\begin{aligned}
\varphi_{0,t}(x)=x-\int_{0}^{t}a(\varphi_{0,r}(x))\,\mathrm{d}r&-\int_{0}^{t}A(\varphi_{0,r}(x))\circ\mathrm{d}W_{r}\\\
&-\int_{0}^{t}\alpha(\varphi_{0,r}(x))\diamond\mathrm{d}Z_{r},\end{aligned}$
(12) $\displaystyle\begin{aligned}
\xi_{0,t}(x,\xi_{0})=\xi_{0}-\int_{0}^{t}\xi_{0,r}(x,\xi_{0})b(\varphi_{0,r}(x))\,\mathrm{d}r&-\int_{0}^{t}\xi_{0,r}(x,\xi_{0})B(\varphi_{0,r}(x))\circ\mathrm{d}W_{r}\\\
&-\int_{0}^{t}\xi_{0,r}(x,\xi_{0})\beta(\varphi_{0,r}(x))\diamond\mathrm{d}Z_{r},\end{aligned}$
(13) $\displaystyle\begin{aligned}
\zeta_{0,t}(x,\xi_{0},\zeta_{0})=\zeta_{0}-\int_{0}^{t}\xi_{0,r}(x,\xi_{0})c(\varphi_{0,r}(x))\,\mathrm{d}r&-\int_{0}^{t}\xi_{0,r}(x,\xi_{0})C(\varphi_{0,r}(x))\circ\mathrm{d}W_{r}\\\
&-\int_{0}^{t}\xi_{0,r}(x,\xi_{0})\sigma(\varphi_{0,r}(x))\diamond\mathrm{d}Z_{r}.\end{aligned}$
(14)

Under the assumptions H${}_{\text{coeff}}$, there exists a unique strong
solution to the SDE (12), (13), (14). Furthermore the associated solution flow
is a $C^{2}$-flow of diffeomorphisms of $\mathbb{R}^{d+2}$, see Applebaum [2,
Theorems 6.10.5 and 6.10.10].

Let $(\varphi_{t,0},\xi_{t,0},\zeta_{t,0})$ be the inverse flow of
$(\varphi_{0,t},\xi_{0,t},\zeta_{0,t})$.

######  Theorem 2.1.

Let assumptions _H ${}_{\text{coeff}}$_ hold true and let $u_{0}\in
C^{2}_{b}(\mathbb{R}^{d},\mathbb{R})$. Then the function

$\displaystyle
u(t,x):=\xi_{t,0}(x,1)u_{0}(\varphi_{t,0}(x))+\zeta_{t,0}(x,1,0)$ (15)

is the unique solution of (6).

The intuition behind the formula (15) is quite clear. In the deterministic
case, $A=\alpha=0$, $B=C=\beta=\sigma=0$, the formula (15) is exactly the
formula (5) and is well known. In the continuous case of Gaussian noise,
$\alpha=0$, $B=\beta=0$, the formula (15) was derived by Kunita, see [19, 21].
The important feature of the equation (6) in the continuous case is that the
stochastic integrals have to be considered in the Stratonovich sense.
Informally this can be justified by the following consideration. It is well-
known that the Stratonovich stochastic integral can be approximated pathwise
by Stieltjes integrals w.r.t. to approximations of a Brownian motion by random
continuous functions of bounded variation, e.g. by polygonal lines (the so-
called, Wong–Zakai approximations). Each of these approximations can be
treated path-wise as a deterministic first order PDE that has a solution (15)
(or (5)), and hence the limit should have the same form. In the case of jump
noise, the role of the Stratonovich stochastic differential equations is
played by the Marcus (canonical) stochastic differential equations. These
equations can be also seen as a limit of continuous Wong–Zakai approximations
and enjoy the Newton–Leibniz change of variables formula of conventional
calculus. Hence it is intuitively clear that the equation (6) has to be
considered as a Marcus equation.

######  Example 2.2 (transport equation).

Consider the transport equation

$\displaystyle u(t,x)=u_{0}(x)$
$\displaystyle+\int_{0}^{t}\nabla^{T}u(r,x)a(x)\,\mathrm{d}r+\int_{0}^{t}\nabla^{T}u(r,x)A(x)\circ\mathrm{d}W_{r}$
(16)
$\displaystyle+\int_{0}^{t}\nabla^{T}u(r,x)\alpha(x)\diamond\mathrm{d}Z_{r}.$

In this case, the characteristics equation is a $d$-dimensional Marcus SDE

$\displaystyle\varphi_{0,t}(x)$
$\displaystyle=x-\int_{0}^{t}a(\varphi_{0,r}(x))\,\mathrm{d}r-\int_{0}^{t}A(\varphi_{0,r}(x))\circ\mathrm{d}W_{r}-\int_{0}^{t}\alpha(\varphi_{0,r}(x))\diamond\mathrm{d}Z_{r}.$

Then the solution has the form

$\displaystyle u(t,x)=u_{0}(\varphi_{t,0}(x)).$

######  Example 2.3 (explicit one-dimensional solution).

In dimension $m=d=1$ if $a(x)=A(x)=\alpha(x)$, the equation (16) can be solved
explicitly with the help of the Itô formula for Marcus SDEs. Indeed, let $Z$
be a general (not necessarily pure jump) one-dimensional Lévy process, and
consider the equation

$\displaystyle u(t,x)=u_{0}(x)+\int_{0}^{t}\nabla
u(r,x)\alpha(x)\diamond\mathrm{d}Z_{r}.$ (17)

Assume that $\alpha(x)>0$ and denote

$\displaystyle H(x)=\int_{0}^{x}\frac{\mathrm{d}y}{\alpha(y)},\quad
x\in\mathbb{R}.$

Then the characteristics equation

$\displaystyle\varphi_{0,t}(x)$
$\displaystyle=x-\int_{0}^{t}\alpha(\varphi_{0,r}(x))\diamond\mathrm{d}Z_{r}$

has the solution

$\displaystyle\varphi_{0,t}(x)=H^{-1}(H(x)-Z_{t})$

and the inverse flow $\varphi_{t,0}$ can be found by a straightforward
calculation as

$\displaystyle\varphi_{t,0}(x)=H^{-1}(H(x)+Z_{t}).$

Let us show that

$\displaystyle u(t,x):=u_{0}(H^{-1}(H(x)+Z_{t}))$ (18)

satisfies (17). Indeed, $u(0,x)=u_{0}(x)$, and

$\displaystyle\nabla u(t,x)=\nabla
u_{0}(\varphi_{t,0}(x))\nabla\varphi_{t,0}(x)=\nabla
u_{0}(\varphi_{t,0}(x))\frac{\alpha(H(x)+Z_{t})}{\alpha(x)}.$

On the other hand, the Itô formula for Marcus SDEs (see, e.g. Section 4 in
Kurtz et al. [23]) yields

$\displaystyle u(t,x)=u_{0}(H^{-1}(H(x)+Z_{t}))$
$\displaystyle=u_{0}(x)+\int_{0}^{t}\frac{\partial}{\partial
Z}u_{0}(H^{-1}(H(x)+Z_{r}))\diamond\mathrm{d}Z_{r}$
$\displaystyle=u_{0}(x)+\int_{0}^{t}\nabla
u_{0}(\varphi_{0,r}(x))\alpha(H(x)+Z_{r})\diamond\mathrm{d}Z_{r}$
$\displaystyle=u_{0}(x)+\int_{0}^{t}\nabla
u(r,x)\alpha(x)\diamond\mathrm{d}Z_{r}.$

######  Example 2.4.

In this example we apply formula (18) to the first order equation

$\displaystyle\partial_{t}u(t,x)=\nabla
u(t,x)\sqrt{x^{2}+1}\diamond\mathrm{d}Z_{t},\quad u(0,x)=u_{0}(x).$ (19)

Note that although $\alpha(x)=\sqrt{x^{2}+1}$ is not bounded, the formula (15)
still holds true. Indeed, in this case,

$\displaystyle H(x)$
$\displaystyle=\operatorname{arcsinh}(x)=\ln(x+\sqrt{x^{2}+1})\quad\text{and}\quad$
$\displaystyle H^{-1}(x)$
$\displaystyle=\sinh(x)=\frac{\mathrm{e}^{x}-\mathrm{e}^{-x}}{2},\quad
x\in\mathbb{R}.$

Hence, (19) has the explicit solution

$\displaystyle
u(t,x)=u_{0}\Big{(}\sinh(\operatorname{arcsinh}(x)+Z_{t})\Big{)},\quad t\geq
0,\ x\in\mathbb{R}.$

Sample paths of a symmetric $\alpha$-stable Lévy process $Z$ and the solution
$u$ are presented on Fig. 1.

Figure 1:  A sample path of a symmetric $\alpha$-stable Lévy process $Z$ with
$\mathbf{E}\mathrm{e}^{\mathrm{i}\lambda
Z_{1}}=\mathrm{e}^{-0.1|\lambda|^{\alpha}}$ for $\alpha=1.75$ (left); the
solution $u(t,x)$ of equation (19) with the initial condition
$u_{0}(x)=1/(1+x^{2})$ sampled at $t=0,10,20,\dots,100$ (right).

The paper is organized as follows. Since the Marcus integrals generalize the
Stratonovich integrals in the jump setting, we will inspect Kunita’s arguments
in the continuous case and adapt them to the case of Lévy noise. One of the
central auxiliary results will be the generalized Itô formula for Marcus SDEs
(Itô–Wentzell formula).

##  3 Canonical (Marcus) SDEs

For the proof of the main theorem we have to recall several facts about
canonical SDEs in the semimartingale setting. Canonical SDEs were introduced
by Marcus in [27]. They were studied by Marcus [28], Fujiwara [14], Kurtz et
al. [23] in the semimartingale setting and by Fujiwara and Kunita [15, 16] and
Kunita [22] in the setting of semimartingales with a spatial parameter.

Let $(\Omega,\mathscr{F},\mathbb{F},\mathbf{P})$ be a filtered probability
space which satisfies the usual hypotheses and let $W$ and $N$ be an
independent $m$-dimensional Brownian motion, and a Poisson random measure on
$\mathbb{R}^{m}\backslash\\{0\\}$ with the compensator
$\nu(\mathrm{d}z)\,\mathrm{d}t$ as defined in the previous section.

For $d\geq 1$, let

$\displaystyle f(x,r,\omega)$
$\displaystyle\colon\mathbb{R}^{d}\times\mathbb{R}_{+}\times\Omega\to\mathbb{R}^{d},$
$\displaystyle F(x,r,\omega)$
$\displaystyle\colon\mathbb{R}^{d}\times\mathbb{R}_{+}\times\Omega\to\mathbb{R}^{d\times
m},$ $\displaystyle\varphi(x,r,z,\omega)$
$\displaystyle\colon\mathbb{R}^{d}\times\mathbb{R}_{+}\times\mathbb{R}^{m}\times\Omega\to\mathbb{R}^{d}$

be predictable processes with parameters $x$ and $(x,z)$ respectively. In what
follows, we will often omit the dependence on $\omega\in\Omega$.

We make the following assumptions. There are constants $\delta\in(0,1)$,
$L>0$, $K>0$ such that

Hf:

$\displaystyle f(\cdot,r)\in C^{2+\delta}(\mathbb{R}^{d},\mathbb{R}^{d}),$
(20) $\displaystyle\sup_{x}\frac{|f(x,r)|}{1+|x|}\leq K,$
$\displaystyle\|\partial^{\alpha}f^{i}(\cdot,r)\|\leq K,\quad 1\leq i\leq
d,\quad|\alpha|=1,2,$
$\displaystyle\|\partial^{\alpha}f^{i}(x,r)-\partial^{\alpha}f^{i}(y,r)\|\leq
L\|x-y\|^{\delta},\quad 1\leq i\leq d,\quad|\alpha|=2.$

HF:

$\displaystyle F(\cdot,r)\in C^{3+\delta}(\mathbb{R}^{d},\mathbb{R}^{d\times
m}),$ (21) $\displaystyle\sup_{x}\frac{|F^{i}_{j}(x,r)|}{1+|x|}\leq K,\quad
1\leq i\leq d,\ 1\leq j\leq m,$
$\displaystyle\|\partial^{\alpha}F^{i}_{j}(\cdot,r)\|\leq K,\quad 1\leq i\leq
d,\ 1\leq j\leq m,\quad|\alpha|=1,2,$
$\displaystyle\|\partial^{\alpha}F^{i}_{j}(\cdot,r)F^{k}_{l}(\cdot,r)\|\leq
K,\quad 1\leq i,k\leq d,\ 1\leq j,l\leq m,\quad|\alpha|=2,3,$
$\displaystyle\|\partial^{\alpha}F^{i}_{j}(x,r)-\partial^{\alpha}F^{i}_{j}(y,r)\|\leq
L\|x-y\|^{\delta},\quad 1\leq i\leq d,\ 1\leq j\leq m,\quad|\alpha|=3.$

Hφ: there are non-negative functions $K_{0}(z)$, $K_{1}(z)$ and $L_{1}(z)$ and
$L_{2}(z)$, such that

$\displaystyle\int_{\|z\|\leq
1}\Big{(}K_{0}(z)^{2}+K_{1}(z)^{2}+L_{1}(z)+L_{2}(z)^{2}\Big{)}\,\nu(\mathrm{d}z)<\infty$
(22)

and

$\displaystyle\varphi(\cdot,r,z)\in
C^{2+\delta}(\mathbb{R}^{d},\mathbb{R}^{d\times m}),$
$\displaystyle\sup_{x}\frac{|\varphi^{i}_{j}(\cdot,r,z)|}{1+|x|}\leq
K_{0}(z),\quad 1\leq i\leq d,\ 1\leq j\leq m,$
$\displaystyle\|\partial^{\alpha}\varphi^{i}_{j}(\cdot,r,z)\|\leq
K_{1}(z),\quad 1\leq i,j\leq d,\quad 1\leq|\alpha|\leq 2,$
$\displaystyle|\partial^{\alpha}\varphi^{i}_{j}(x,r,z)\varphi^{k}_{l}(x,r,z)-\partial^{\alpha}\varphi^{i}_{j}(y,r,z)\varphi^{k}_{l}(y,r,z)|\leq
L_{1}(z)\|x-y\|,$ $\displaystyle\hskip 187.78531pt1\leq i,k\leq d,\ 1\leq
j,l\leq m,\quad|\alpha|=1,$
$\displaystyle|\partial^{\alpha}\varphi^{i}_{j}(x,r,z)-\partial^{\alpha}\varphi^{i}_{j}(y,r,z)|\leq
L_{2}(z)\|x-y\|^{\delta},$ $\displaystyle\hskip 187.78531pt1\leq i\leq d,\
1\leq j\leq m,\quad|\alpha|=2.$

Consider a semimartingale $\Phi$ with parameter $x$ given by

$\displaystyle\Phi(x,t)$
$\displaystyle=\int_{0}^{t}f(x,r)\,\mathrm{d}r+\int_{0}^{t}F(x,r)\,\mathrm{d}W_{r}$
(23) $\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\varphi(x,r,z)\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)+\int_{0}^{t}\int_{\|z\|>1}\varphi(x,r,z)\,N(\mathrm{d}z,\mathrm{d}r).$

We want to give sense to the following Marcus SDE with the generator
$\Phi(\cdot,\cdot)$, which we formally write as

$\displaystyle X_{t}(x)$
$\displaystyle=x+\int_{0}^{t}\Phi(X_{r}(x),\diamond\,\mathrm{d}r).$ (24)

This writing has the following meaning:

For each $r\geq 0$, $x\in\mathbb{R}^{d}$, $z\in\mathbb{R}^{m}$ and
$\omega\in\Omega$ consider an ODE (the Marcus ODE) for the function
$h=h(u)=h(u;x,r,z)$

$\displaystyle\begin{cases}&\displaystyle\frac{\mathrm{d}}{\mathrm{d}u}h(u)=\varphi(h(u),r,z),\quad
u\in[0,1],\\\ &\displaystyle h(0)=x.\end{cases}$ (25)

Under Assumption Hφ there is a global solution $h$ for $u\in\mathbb{R}$ which
we denote by

$\displaystyle\mathbf{e}^{u\varphi(\cdot;r,z)}(x):=h(u;x,r,z),\quad
u\in\mathbb{R}.$ (26)

and in particular we define the exponential mapping

$\displaystyle\mathbf{e}^{\varphi(\cdot;r,z)}(x):=h(1;x,r,z).$ (27)

For almost all $r$, $\omega$ and $z$ the mapping
$x\mapsto\mathbf{e}^{\varphi(\cdot;r,z)}(x)$ is a $C^{2}$-diffeomorphism.

Consider the following functions which are well-defined due to Assumptions Hf,
HF and Hφ:

$\displaystyle\hat{f}(x,r)$
$\displaystyle=f(x,r)+\frac{1}{2}\sum_{j=1}^{d}DF_{j}(x,r)F_{j}(x,r)$
$\displaystyle\qquad+\int_{\|z\|\leq
1}\Big{(}\mathbf{e}^{\varphi(\cdot,r,z)}(x)-x-\varphi(x,r,z)\Big{)}\,\nu(\mathrm{d}z),$
$\displaystyle\hat{F}(x,r)$ $\displaystyle=F(x,r),$
$\displaystyle\hat{\varphi}(x,r,z)$
$\displaystyle=\mathbf{e}^{\varphi(\cdot,r,z)}(x)-x.$

Consider an Itô semimartingale with parameter

$\displaystyle\hat{\Phi}(x,t)$
$\displaystyle=\int_{0}^{t}\hat{f}(x,r)\,\mathrm{d}r+\int_{0}^{t}\hat{F}(x,r)\,\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\hat{\varphi}(x,r,z)\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)+\int_{0}^{t}\int_{\|z\|>1}\hat{\varphi}(x,r,z)\,N(\mathrm{d}z,\mathrm{d}r).$

For an adapted càdlàg process $\eta=(\eta_{t})_{t\in[0,T]}$ we set

$\displaystyle\int_{0}^{t}\hat{\Phi}(\eta_{r-},\mathrm{d}r)$
$\displaystyle=\int_{0}^{t}\hat{f}(\eta_{r-},r)\,\mathrm{d}r+\int_{0}^{t}\hat{F}(\eta_{r-},r)\,\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\hat{\varphi}(\eta_{r-},r,z)\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)+\int_{0}^{t}\int_{\|z\|>1}\hat{\varphi}(\eta_{r-},r,z)\,N(\mathrm{d}z,\mathrm{d}r)$

and consider the Itô SDE

$\displaystyle X_{t}(x)$
$\displaystyle=x+\int_{0}^{t}\hat{\Phi}(X_{r-}(x),\,\mathrm{d}r).$ (28)

Then for any $x$, there is a unique strong solution $X$ of (28), see Kunita
[22, Theorem 3.1, Section 3.1]. There is a modification of $X$ such that for
each $x\in\mathbb{R}^{d}$ the mapping $t\mapsto X_{t}(x)$ is càdlàg, and for
each $t\in[0,T]$ the mappings $x\mapsto X_{t}(x)$ are $C^{2}$-diffemorphisms,
see Kunita [22, Section 3.5] and [20, Section 4].

The functions $f$, $F$ and $\varphi$ given, we will understand the canonical
(Marcus) SDE (24) as the Itô SDE (28).

##  4 Generalized Itô formula for canonical SDEs

For the proof of formula (15) we will need the so-called generalized Itô
formula for solutions of canonical SDEs. Let us first recall the conventional
formula for the canonical Marcus SDEs which formally reminds of a conventional
Newton–Leibniz formula known in calculus.

######  Theorem 4.1 (Itô’s formula for solutions of canonical SDEs).

Let $X$ be the solution of the SDE (24) and let $\Theta\in
C^{2}(\mathbb{R}^{d},\mathbb{R})$. Then

$\displaystyle\Theta(X_{t})$
$\displaystyle=\Theta(x)+\int_{0}^{t}\nabla^{T}\Theta\,\Phi(X_{r},\diamond\,\mathrm{d}r),$
(29)

where the canonical integral in the r.h.s. of (29) equals

$\displaystyle\int_{0}^{t}$
$\displaystyle\nabla^{T}\Theta(X_{r-})f(X_{r-},r)\,\mathrm{d}r+\int_{0}^{t}\nabla^{T}\Theta(X_{r-})F(X_{r-},r)\circ\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\Theta(\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}))-\Theta(X_{r-})\Big{)}\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\Theta(\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}))-\Theta(X_{r-})-\varphi(X_{r-},r,z)\Big{)}\nu(\mathrm{d}z)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{\|z\|>1}\Big{(}\Theta(\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}))-\Theta(X_{r-})\Big{)}N(\mathrm{d}z,\mathrm{d}r).$

######  Remark 4.2.

Note that the process $\Theta(X)$ has the jumps

$\displaystyle\Theta(X_{r})-\Theta(X_{r-})$
$\displaystyle=\Theta(\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}))-\Theta(X_{r-})$
$\displaystyle=\int_{0}^{1}\nabla^{T}\Theta(\mathbf{e}^{u\varphi(\cdot,r,z)}(X_{r-}))\varphi(\mathbf{e}^{u\varphi(\cdot,r,z)}(X_{r-}),r,z)\,\mathrm{d}u,$

which justifies the formal writing (29).

Now we prove the generalized Itô formula for a pair of canonical SDEs driven
by the same Brownian motion and the Poisson random measure.

######  Theorem 4.3 (generalized Itô formula for canonical SDEs).

Consider solutions of canonical SDEs with generators $\Phi$ and $\Psi$ such
that the functions $f,F,\varphi$ and $g,G,\psi$ satisfy assumptions of the
Section 3 respectively,

$\displaystyle X_{t}(x)$
$\displaystyle=x+\int_{0}^{t}\Phi(X_{r}(x),\diamond\,\mathrm{d}r),$ (30)
$\displaystyle Y_{t}$
$\displaystyle=y+\int_{0}^{t}\Psi(Y_{r},\diamond\,\mathrm{d}r).$ (31)

Then the following formula holds true

$\displaystyle X_{t}(Y_{t})$
$\displaystyle=y+\int_{0}^{t}\Phi(X_{r}(Y_{r}),\diamond\,\mathrm{d}r)+\int_{0}^{t}DX\Psi(Y_{r},\diamond\,\mathrm{d}r),$
(32)

where the latter integrals are understood as

$\displaystyle\int_{0}^{t}\Phi(X_{r}(Y_{r}),\diamond\,\mathrm{d}r)=\int_{0}^{t}f(X_{r-}(Y_{r-}),r)\,\mathrm{d}r+\int_{0}^{t}F(X_{r-}(Y_{r-}),r)\,\mathrm{d}W_{r}$
(33)
$\displaystyle+\frac{1}{2}\sum_{j=1}^{m}\int_{0}^{t}DF_{j}(X_{r-}(Y_{r-}),r)F_{j}(X_{r-}(Y_{r-}),r)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\mathrm{e}^{\varphi(\cdot,r,z)}(X_{r-}(Y_{r-}))-X_{r-}(Y_{r-})\Big{)}\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\mathrm{e}^{\varphi(\cdot,r,z)}(X_{r-}(Y_{r-}))-X_{r-}(Y_{r-})-\varphi(X_{r-}(Y_{r-}),r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{\|z\|>1}\Big{(}\mathrm{e}^{\varphi(\cdot,r,z)}(X_{r-}(Y_{r-}))-X_{r-}(Y_{r-})\Big{)}N(\mathrm{d}z,\mathrm{d}r)$

and

$\displaystyle\int_{0}^{t}$ $\displaystyle
DX\Psi(Y_{r},\diamond\,\mathrm{d}r)=\int_{0}^{t}DX_{r-}(Y_{r-})g(Y_{r-},r)\,\mathrm{d}r$
(34)
$\displaystyle+\sum_{j=1}^{m}\frac{1}{2}\int_{0}^{t}DX_{r-}(Y_{r-})DG_{j}(Y_{r-},r)G_{j}(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle+\frac{1}{2}\sum_{j,k=1}^{m}\int_{0}^{t}(D(DX_{r-})_{j})_{k}(Y_{r-})G_{j}(Y_{r-},r)G_{k}(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle+\frac{1}{2}\sum_{j=1}^{m}\int_{0}^{t}DX_{r-}(Y_{r-})DF_{j}(X_{r-}(Y_{r-}),r)G_{j}(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}DX_{r-}(Y_{r-})G(Y_{r-},r)\,\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{[}\mathrm{e}^{\varphi(\cdot,r,z)}\Big{(}X_{r-}\Big{(}\mathrm{e}^{\psi(\cdot,r,z)}(Y_{r-})\Big{)}-\mathrm{e}^{\varphi(\cdot,r,z)}(X_{r-}(Y_{r-}))\Big{]}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{[}\mathbf{e}^{\varphi(\cdot,r,z)}\Big{(}X_{r-}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})\Big{)}\Big{)}-\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}(Y_{r-}))$
$\displaystyle\hskip 211.26027pt-
DX_{r-}(Y_{r-})\psi(Y_{r-},r,z)\Big{]}\,\nu(\mathrm{d}z)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{\|z\|>1}\Big{[}\mathrm{e}^{\varphi(\cdot,r,z)}\Big{(}X_{r-}\Big{(}\mathrm{e}^{\psi(\cdot,r,z)}(Y_{r-})\Big{)}-\mathrm{e}^{\varphi(\cdot,r,z)}(X_{r-}(Y_{r-}))\Big{]}\,N(\mathrm{d}z,\mathrm{d}r).$

This Theorem will be proven in Section 7. We will also need the following
formula which is proved analogously, see Section 8.

######  Theorem 4.4.

Let $\Phi$ be a one-dimensional semimartingale given by (23) with a
$d$-dimensional parameter $x$ and let $Y$ be a solution of the $d$-dimensional
canonical SDE (31). Then

$\displaystyle\Phi(Y_{t},t)=\int_{0}^{t}\Phi(Y_{r-},\mathrm{d}r)+\int_{0}^{t}\nabla^{T}\Phi(Y_{r},\diamond\,\mathrm{d}r),$
(35)

where

$\displaystyle\int_{0}^{t}\Phi(Y_{r-},\mathrm{d}r)=\int_{0}^{t}f(Y_{r-},r)\,\mathrm{d}r+\int_{0}^{t}F(Y_{r-},r)\,\mathrm{d}W_{r}$
(36) $\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\varphi(Y_{r-},r,z)\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)+\int_{0}^{t}\int_{\|z\|>1}\varphi(Y_{r-},r,z)\,N(\mathrm{d}z,\mathrm{d}r)$

and

$\displaystyle\int_{0}^{t}\nabla^{T}\Phi(Y_{r},\diamond\,\mathrm{d}r)=\int_{0}^{t}\nabla^{T}\Phi(Y_{r-},r)g(Y_{r-},r)\,\mathrm{d}r$
(37)
$\displaystyle+\int_{0}^{t}\nabla^{T}\Phi(Y_{r-},r)G(Y_{r-},r)\circ\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\int_{0}^{1}\Big{(}\nabla^{T}\Phi(\mathrm{e}^{\psi(\cdot,r,z)}(Y_{r-}),r-)+\nabla^{T}\varphi(\mathrm{e}^{\psi(\cdot,r,z)}(Y_{r-}),r,z))\Big{)}\times$
$\displaystyle\hskip
234.73523pt\times\psi(\mathrm{e}^{\psi(\cdot,r,z)}(Y_{r-}),r,z)\,\mathrm{d}u\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Big{[}\int_{0}^{1}\Big{(}\nabla^{T}\Phi(\mathrm{e}^{\psi(\cdot,r,z)}(Y_{r-}),r-)+\nabla^{T}\varphi(\mathrm{e}^{\psi(\cdot,r,z)}(Y_{r-}),r,z))\Big{)}\times$
$\displaystyle\hskip
79.80911pt\times\psi(\mathrm{e}^{\psi(\cdot,r,z)}(Y_{r-}),r,z)\,\mathrm{d}u-\nabla^{T}\Phi(Y_{r-},r-)\psi(Y_{r-},r,z)\Big{]}\,\nu(\mathrm{d}z)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{|z|>1}\int_{0}^{1}\Big{(}\nabla^{T}\Phi(\mathrm{e}^{\psi(\cdot,r,z)}(Y_{r-}),r-)+\nabla^{T}\varphi(\mathrm{e}^{\psi(\cdot,r,z)}(Y_{r-}),r,z))\Big{)}\times$
$\displaystyle\hskip
234.73523pt\times\psi(\mathrm{e}^{\psi(\cdot,r,z)}(Y_{r-}),r,z)\,\mathrm{d}u\,N(\mathrm{d}z,\mathrm{d}r).$

##  5 Equations for the inverse flows of the canonical SDEs

We know from Kunita [20, 22] that the solution $x\mapsto X_{t}(x)$, $t\geq 0$,
maps $\mathbb{R}^{d}$ onto itself diffeomorfically, and there exists a
modification such that $X_{s,t}:=X_{t}\circ X^{-1}_{s}$ defines the stochastic
flow of diffeomorphisms. Denote by $DX_{t}(x)$ its gradient (Jacobi) matrix,
that satisfies the so-called variational SDE and is a (right) stochastic
exponent. Let $(DX_{t}(x))^{-1}$ be its matrix inverse.

Consider the inverse flow $X_{t,0}:=X_{0,t}^{-1}$, $t\geq 0$. We show that the
inverse flow satisfies the following formula.

######  Theorem 5.1.

The inverse flow $t\mapsto X_{t,0}$, $t\geq 0$ satisfies the canonical SDE

$\displaystyle
X_{t,0}(x)=x-\int_{0}^{t}\Big{(}DX_{0,r}(X_{r,0}(x))\Big{)}^{-1}\Phi(x,\diamond\,\mathrm{d}r)$
(38)

which is understood in the following sense:

$\displaystyle X_{t,0}(x)$
$\displaystyle=x-\int_{0}^{t}\Big{(}DX_{0,r-}(X_{r-,0}(x))\Big{)}^{-1}f(x,r)\,\mathrm{d}r$
(39)
$\displaystyle-\int_{0}^{t}\Big{(}DX_{0,r-}(X_{r-,0}(x))\Big{)}^{-1}F(x,r)\circ\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(X_{r-,0}(x))-X_{r-,0}(x)\Big{)}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(X_{r-,0}(x))-X_{r-,0}(x)$
$\displaystyle\hskip
140.84256pt-(DX_{0,r-}(X_{r-,0}(x)))^{-1}\varphi(x,r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{\|z\|>1}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(X_{r-,0}(x))-X_{r-,0}(x)\Big{)}\,N(\mathrm{d}z,\mathrm{d}r).$

where $\mathbf{e}^{\psi(\cdot,r,z)}$ is the exponential mapping defined with
the help of the solution $w=w(u;y)=w(u;y,r,z)$ of the ODE

$\displaystyle\begin{cases}\displaystyle\frac{\mathrm{d}}{\mathrm{d}u}w(u;y)&=-\Big{(}D\mathbf{e}^{u\varphi(\cdot,r,z)}(X_{r-}(\cdot))\Big{)}^{-1}\varphi(\mathbf{e}^{u\varphi(\cdot,r,z)}(X_{r-}(\cdot)),r,z)\circ
w(u;y),\\\ w(0;y)&=y,\quad u\in[0,1],\end{cases}$ (40)

i.e. $\mathbf{e}^{\psi(\cdot,r,z)}(y):=\mathbf{e}^{\psi(r,z)}(y):=w(1;y)$.

###### Proof.

For brevity we assume that $\nu(\|z\|>1)=0$ and denote $X_{0,t}=X_{t}$, and
$DX_{0,t}=DX_{t}$. Recall that $DX_{t}(x)$ is a right stochastic exponent, see
Section V.9 of Protter [35] and Section 4 of Fujiwara and Kunita [15] for more
detail. It is well defined and is invertible.

Define the following drift, diffusion and jump coefficients:

$\displaystyle g(y,r)$
$\displaystyle=-\Big{(}DX_{r-}(y)\Big{)}^{-1}f(X_{r-}(y),r),$ $\displaystyle
G(y,r)$ $\displaystyle=-\Big{(}DX_{r-}(y)\Big{)}^{-1}F(X_{r-}(y),r),$
$\displaystyle\psi(y,r,z,u)$
$\displaystyle=-\Big{(}D\mathbf{e}^{u\varphi(\cdot,r,z)}(X_{r-}(y))\Big{)}^{-1}\varphi(\mathbf{e}^{u\varphi(\cdot,r,z)}(X_{r-}(y)),r,z).$

In particular,

$\displaystyle\psi(y,r,z,0)$
$\displaystyle=-\Big{(}DX_{r-}(y)\Big{)}^{-1}\varphi(X_{r-}(y),r,z).$

The functions $g$, $G$ and $\psi$ are predictable and with the help of
localization we can assume that they satisfy Assumptions of Section 3. Then
equation (40) has a unique global solution.

Consider the supplementary SDE

$\displaystyle Y_{t}$
$\displaystyle=x+\int_{0}^{t}g(Y_{r-},r)\,\mathrm{d}r+\int_{0}^{t}G(Y_{r-},r)\circ\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\mathbf{e}^{\psi(r,z)}(Y_{r-})-Y_{r-}\Big{)}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\mathbf{e}^{\psi(r,z)}(Y_{r-})-Y_{r-}-\Big{(}DX_{r-}(Y_{r-})\Big{)}^{-1}\varphi(X_{r-}(Y_{r-}),r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r,$

where $\mathbf{e}^{\psi(r,z)}$ is defined in (40).

We show that for each $T>0$ and for any localized solution we have
$X_{t}(Y_{t})\equiv x$ on $[0,T]$. Let us again consider the one-dimensional
case. We apply the generalized Itô formula and show that all the integral
terms vanish. Indeed, for the drift term we get

$\displaystyle f(X_{r-}$
$\displaystyle(Y_{r-}),r,z)+DX_{r-}{Y_{r-}}g(Y_{r-},r,z)$
$\displaystyle=f(X_{r-}(Y_{r-}),r,z)-DX_{r-}{Y_{r-}}\Big{(}DX_{r-}(Y_{r-})\Big{)}^{-1}f(X_{r-}(Y_{r-}),r)\equiv
0.$

The other Lebesgue and Itô stochastic integrals w.r.t. $W$ vanish analogously.
To treat the jump terms we consider the function
$h(u;x):=\mathbf{e}^{u\varphi(\cdot,r,z)}(X_{r-}(x))$ where the mapping
$(u,x)\mapsto\mathbf{e}^{u\varphi}(x)$ has been defined in (25), (26), (27),
so that $h(0;x):=X_{r-}(x)$ and $h(1;x):=X_{r}(x)$. Then taking into account
(40) we obtain that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}u}h(u;w(u;y))$
$\displaystyle=\frac{\partial}{\partial u}h(u;w(u;y))+\frac{\partial}{\partial
x}h(u;w(u;y))\frac{\mathrm{d}}{\mathrm{d}u}w(u;y)$ (41)
$\displaystyle=\varphi(h(u;w(u;y));r,z)$
$\displaystyle-\frac{\partial}{\partial
x}h(u;w(u;y))\Big{(}\frac{\partial}{\partial
x}h(u;w(u;y))\Big{)}^{-1}\cdot\varphi(h(u;w(u;y));r,z)$ $\displaystyle\equiv
0.$

In other words, we have

$\displaystyle\mathbf{e}^{\varphi(r,z)}(X_{r-}(\mathbf{e}^{\psi(r,z)}(Y_{r-}))-X_{r-}(Y_{r-})$
$\displaystyle=\Big{(}h(1;w(1;y))-h(0;w(0,y))\Big{)}\Big{|}_{y=Y_{r-}}$
$\displaystyle=\int_{0}^{1}\frac{\mathrm{d}}{\mathrm{d}u}h(u;w(u;y))\,\mathrm{d}u\Big{|}_{y=Y_{r-}}$
$\displaystyle\equiv 0.$

Furthermore, putting together the compensated terms in the generalized Itô
formula we get

$\displaystyle\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}(Y_{r-}))-X_{r-}(Y_{r-})-\varphi(X_{r-}(Y_{r-}),r,z)$
$\displaystyle+\mathbf{e}^{\varphi(\cdot,r,z)}\Big{(}X_{r-}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})\Big{)}\Big{)}-\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}(Y_{r-}))$
$\displaystyle-DX_{r-}(Y_{r-})\psi(Y_{r-},r,z)\equiv 0.$

Hence $Y_{t}(y)=X_{t}^{-1}(y)$ for each localized solution $Y$. Since $X$
exists on $[0,T]$, passing to the limit in the localization sequence we get
that the $Y$ is the inverse flow and satisfies the SDE (39). ∎

######  Theorem 5.2 (Itô’s formula for the inverse flow w.r.t. the first
variable).

Let $\Theta\in C^{2}(\mathbb{R}^{d},\mathbb{R})$. Then the inverse flow
$t\mapsto X_{t,0}$ satisfies the canonical SDE

$\displaystyle\Theta(X_{t,0}(x))$
$\displaystyle=\Theta(x)-\int_{0}^{t}\nabla^{T}\Theta(X_{r,0}(x))DX_{r,0}(x)\Phi(x,\diamond\,\mathrm{d}r)$
(42) $\displaystyle=\Theta(x)-\int_{0}^{t}\nabla^{T}\Big{(}\Theta\circ
X_{r,0}(x)\Big{)}\Phi(x,\diamond\,\mathrm{d}r),$

which is understood as follows:

$\displaystyle\Theta(X_{t,0}(x))$
$\displaystyle=\Theta(x)-\int_{0}^{t}\nabla^{T}\Theta(X_{r-,0}(x))DX_{r-,0}(x)f(x,r)\,\mathrm{d}r$
(43)
$\displaystyle-\int_{0}^{t}\nabla^{T}\Theta(X_{r-,0}(x))DX_{r-,0}(x)F(x,r)\circ\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\Theta(\mathbf{e}^{-\varphi(\cdot,r,z)}(X_{r-,0}(x)))-\Theta(X_{r-,s}(x))\Big{)}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\Theta(\mathbf{e}^{-\varphi(\cdot,r,z)}(X_{r-,0}(x)))-\Theta(X_{r-,0}(x))$
$\displaystyle\hskip
93.89265pt+\nabla^{T}\Theta(X_{r-,0}(x))DX_{r-,0}(x)\varphi(x,r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{\|z\|>1}\Big{(}\Theta(\mathbf{e}^{-\varphi(\cdot,r,z)}(X_{r-,0}(x)))-\Theta(X_{r-,0}(x))\Big{)}\,N(\mathrm{d}z,\mathrm{d}r).$

###### Proof.

The proof goes along the lines of the proof of Theorem 4.4.5 in Kunita [21].
For brevity we denote the forward flow by $X_{t}:=X_{0,t}$ and the inverse
flow by $Y_{t}:=X_{t,0}$, $t\in[0,T]$. We have shown that the inverse flow $Y$
satisfies the SDE (38).

First we note that since $X_{t}(Y_{t}(x))\equiv x$, the gradient matrices
$DX_{t}(x)$ and $DY_{t}(x)$ satisfy the relation

$\displaystyle DX_{t}(Y_{t}(x))\cdot DY_{t}(x)=\text{Id}$

or equivalently,

$\displaystyle\Big{(}DX_{t}(Y_{t}(x))\Big{)}^{-1}=DY_{t}(x).$

Second, taking into account (41) we get that

$\displaystyle h(u;w(u;x))\equiv x,\quad u\in[0,1],$

and hence

$\displaystyle Dh(u;w(u;x))Dw(u;x)=\text{Id}$ (44)

or equivalently,

$\displaystyle\Big{(}Dh(u;w(u;x))\Big{)}^{-1}=Dw(u;x),\quad u\in[0,1].$

Thus the equation (40) for $w$ takes the form

$\displaystyle\begin{cases}\displaystyle\frac{\mathrm{d}}{\mathrm{d}u}w(u;x)&=-Dw(u;x)\varphi(x,r,z),\quad
u\in[0,1],\\\ w(0;x)&=x.\end{cases}$

This is the first order transport equation, its solution is given by
$\mathbf{e}^{-u\varphi(\cdot,r,z)}(x)$ and hence

$\displaystyle\mathbf{e}^{\psi(\cdot,r,z)}(\cdot)=\mathbf{e}^{-\varphi(\cdot,r,z)}(\cdot).$
(45)

Hence applying the Itô formula to the equation (39) and taking into account
(44) and (45) yields

$\displaystyle\Theta(Y_{t}(x))$
$\displaystyle=x-\int_{0}^{t}\nabla^{T}\Theta(Y_{r-}(x))DY_{r-}(x)f(x,r)\,\mathrm{d}r$
$\displaystyle-\int_{0}^{t}\nabla^{T}\Theta(Y_{r-}(x))DY_{r-}(x)F(x,r)\circ\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\Theta(\mathbf{e}^{-\varphi(\cdot,r,z)}(Y_{r-}))-\Theta(Y_{r-})\Big{)}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\Theta(\mathbf{e}^{-\varphi(\cdot,r,z)}(Y_{r-}))-\Theta(Y_{r-})$
$\displaystyle\hskip
93.89265pt+\nabla^{T}\Theta(Y_{r-}(x))DY_{r-}(x)\varphi(x,r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r,$

or equivalently

$\displaystyle\Theta(Y_{t}(x))$
$\displaystyle=x-\int_{0}^{t}\nabla^{T}\Big{(}\Theta\circ
Y_{r-}(x)\Big{)}f(x,r)\,\mathrm{d}r$
$\displaystyle-\int_{0}^{t}\nabla^{T}\Big{(}\Theta\circ
Y_{r-}(x)\Big{)}F(x,r)\circ\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\Theta(\mathbf{e}^{-\varphi(\cdot,r,z)}(Y_{r-}))-\Theta(Y_{r-})\Big{)}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\Theta(\mathbf{e}^{-\varphi(\cdot,r,z)}(Y_{r-}))-\Theta(Y_{r-})$
$\displaystyle\hskip 93.89265pt+\nabla^{T}\Big{(}\Theta\circ
Y_{r-}(x)\Big{)}\varphi(x,r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r,$

The latter formula can be formally written in the canonical form (42). ∎

##  6 Proof of Theorem 2.1

Let $W$ be a $m$-dimensional Brownian motion and $Z$ be $m$-dimensional
compensated pure jump Lévy process. For simplicity assume that
$\nu(\|z\|>0)=0$. Consider the linear equation (6). To solve it, we consider a
$(d+2)$-dimensional system of characteristics Marcus SDEs (12), (13), (14).
Note that $\varphi$ is a $d$-dimensional process whereas $\xi$ and $\zeta$ are
one-dimensional. Denote
$X:=(\varphi^{1},\dots,\varphi^{d},\xi,\zeta)=(X^{1},\dots,X^{d},X^{d+1},X^{d+2})\in\mathbb{R}^{d+2}$
(the column vector), and consider the functions

$\displaystyle f(X)$
$\displaystyle=-\begin{pmatrix}a^{1}(X^{1},\dots,X^{d})\\\ \vdots\\\
a^{d}(X^{1},\dots,X^{d})\\\ X^{d+1}b(X^{1},\dots,X^{d})\\\
X^{d+1}c(X^{1},\dots,X^{d})\end{pmatrix},$ $\displaystyle F(X)$
$\displaystyle=-\begin{pmatrix}A^{1}_{1}(X^{1},\dots,X^{d})&\cdots&A^{m}_{1}(X^{1},\dots,X^{d})\\\
\vdots&\ddots&\vdots\\\
A^{1}_{d}(X^{1},\dots,X^{d})&\cdots&A^{m}_{d}(X^{1},\dots,X^{d})\\\
X^{d+1}B_{1}(X^{1},\dots,X^{d})&\cdots&X^{d+1}B_{m}(X^{1},\dots,X^{d})\\\
X^{d+1}C_{1}(X^{1},\dots,X^{d})&\cdots&X^{d+1}C_{m}(X^{1},\dots,X^{d})\end{pmatrix},$
$\displaystyle\Sigma(X)$
$\displaystyle=-\begin{pmatrix}\alpha^{1}_{1}(X^{1},\dots,X^{d})&\cdots&\alpha^{m}_{1}(X^{1},\dots,X^{d})\\\
\vdots&\ddots&\vdots\\\
\alpha^{1}_{d}(X^{1},\dots,X^{d})&\cdots&\alpha^{m}_{d}(X^{1},\dots,X^{d})\\\
X^{d+1}\beta_{1}(X^{1},\dots,X^{d})&\cdots&X^{d+1}\beta_{m}(X^{1},\dots,X^{d})\\\
X^{d+1}\sigma_{1}(X^{1},\dots,X^{d})&\cdots&X^{d+1}\sigma_{m}(X^{1},\dots,X^{d})\end{pmatrix}.$
(46)

In the matrix form, the system (12), (13), (14) reads as a canonical equation
of the type (24)

$\displaystyle X_{0,t}=X_{0}+\int_{0}^{t}\Phi(X_{0,r},\diamond\,\mathrm{d}r)$

with $\varphi(X,r,z)=\Sigma(X)z$ (here we allow an abuse of notation). There
is a unique solution $X=(X_{0,t}(x,\xi_{0},\zeta_{0}))_{t\geq 0}$ which is a
$C^{2}$-flow on $\mathbb{R}^{d+2}$. Denote
$Y_{t}=X_{t,0}=X^{-1}_{0,t}=(\varphi^{1}_{t,0},\dots,\varphi^{d}_{t,0},\xi_{t,0},\zeta_{t,0})$
its inverse flow.

Consider a function

$\displaystyle\Theta(x,\xi_{0},\zeta_{0})=\xi_{0}u_{0}(x)+\zeta_{0}$

and define a process

$\displaystyle u(t;x,\xi_{0},\zeta_{0})$
$\displaystyle=\Theta(Y_{t}(x,\xi_{0},\zeta_{0}))$
$\displaystyle=\xi_{t,0}(x,\xi_{0})u_{0}(\varphi_{t,0}(x))+\zeta_{t,0}(x,\xi_{0},\zeta_{0}),\quad
x\in\mathbb{R}^{d},\ \xi_{0},\zeta_{0}\in\mathbb{R}.$

Then by Theorem 5.2 we get that

$\displaystyle u(t,x,$
$\displaystyle\xi_{0},\zeta_{0})=\Theta(Y_{t})=\Theta(x,\xi_{0},\zeta_{0})-\int_{0}^{t}\nabla^{T}\Big{(}\Theta\circ
Y_{r}(x,\xi_{0},\zeta_{0})\Big{)}\Phi(x,\xi_{0},\zeta_{0},\diamond\,\mathrm{d}r)$
$\displaystyle=u_{0}(0,x,\xi_{0},\zeta_{0})+\int_{0}^{t}\nabla^{T}_{x}u(r-,x,\xi_{0},\zeta_{0})a(x)\,\mathrm{d}r$
$\displaystyle+\xi_{0}\int_{0}^{t}\partial_{\xi_{0}}u(r-,x,\xi_{0},\zeta_{0})b(x)\,\mathrm{d}r+\xi_{0}\int_{0}^{t}\partial_{\zeta_{0}}u(r-,x,\xi_{0},\zeta_{0})c(x)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\nabla^{T}_{x}u(r-,x,\xi_{0},\zeta_{0})A(x)\circ\mathrm{d}W_{r}+\xi_{0}\int_{0}^{t}\partial_{\xi_{0}}u(r-,x,\xi_{0},\zeta_{0})B(x)\circ\mathrm{d}W_{r}$
$\displaystyle+\xi_{0}\int_{0}^{t}\partial_{\zeta_{0}}u(r-,x,\xi_{0},\zeta_{0})C(x)\circ\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\Theta(\mathbf{e}^{-\Sigma(\cdot)z}(Y_{r-}(x,\xi_{0},\zeta_{0})))-u(r-,x,\xi_{0},\zeta_{0})\Big{)}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\Theta(\mathbf{e}^{-\Sigma(\cdot)z}(Y_{r-}(x,\xi_{0},\zeta_{0})))-u(r-,x,\xi_{0},\zeta_{0})$
$\displaystyle\hskip
93.89265pt+\nabla^{T}_{x}u(r-,x,\xi_{0},\zeta_{0})\alpha(x)z+\xi_{0}\partial_{\xi_{0}}u(r-,x,\xi_{0},\zeta_{0})\beta(x)z$
$\displaystyle\hskip
93.89265pt+\xi_{0}\partial_{\zeta_{0}}u(r-,x,\xi_{0},\zeta_{0})\sigma(x)z\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r.$

Let us study the derivatives $\partial_{\xi_{0}}u$ and
$\partial_{\zeta_{0}}u$. First we note that the process $\xi$ is found
explicitly as an exponential

$\displaystyle\xi_{0,t}$ $\displaystyle(x,\xi_{0})$
$\displaystyle=\xi_{0}\exp\Big{(}-\int_{0}^{t}b(\varphi_{0,r}(x))\,\mathrm{d}r-\int_{0}^{t}B(\varphi_{0,r}(x))\circ\mathrm{d}W_{r}-\int_{0}^{t}\beta(\varphi_{0,r}(x))\diamond\mathrm{d}Z_{r}\Big{)}.$

The derivative equals to

$\displaystyle\partial_{\xi_{0}}\xi_{0,t}$ $\displaystyle(x,\xi_{0})$
$\displaystyle=\exp\Big{(}-\int_{0}^{t}b(\varphi_{0,r}(x))\,\mathrm{d}r-\int_{0}^{t}B(\varphi_{0,r}(x))\circ\mathrm{d}W_{r}-\int_{0}^{t}\beta(\varphi_{0,r}(x))\diamond\mathrm{d}Z_{r}\Big{)}$

and it satisfies (13) with the initial values $(x,\xi_{0})=(x,1)$. By the
formula of the derivative of the inverse function we get

$\displaystyle\partial_{\xi_{0}}\xi_{t,0}(x,\xi_{0})$
$\displaystyle=\Big{(}\partial_{\xi_{0}}\xi_{0,t}(\varphi_{t,0}(x),\xi_{t,0}(x,\xi_{0}))\Big{)}^{-1}$
(47)
$\displaystyle=\Big{(}\xi_{0,t}(\varphi_{t,0}(x),1)\Big{)}^{-1}=\xi_{t,0}(x,1).$

Analogously,

$\displaystyle\partial_{\xi_{0}}\zeta_{t,0}(x,\xi_{0},\zeta_{0})=\zeta_{t,0}(x,1,0),$
(48) $\displaystyle\partial_{\zeta_{0}}\zeta_{t,0}(x,\xi_{0},\zeta_{0})=1.$

Thus taking into account (47) and (LABEL:e:dz) we can write

$\displaystyle\partial_{\xi_{0}}u(r,x,\xi_{0},\zeta_{0})$
$\displaystyle=\partial_{\xi_{0}}\Big{(}\Theta(\varphi_{t,0}(x),\xi_{t,0}(x,\xi_{0}),\zeta_{t,0}(x,\xi_{0},\zeta_{0}))\Big{)}$
$\displaystyle=u_{0}(\varphi_{t,0}(x))\partial_{\xi_{0}}\xi_{t,0}(x,\xi_{0})+\partial_{\xi_{0}}\zeta_{t,0}(x,\xi_{0},\zeta_{0})$
$\displaystyle=u_{0}(\varphi_{t,0}(x))\xi_{t,0}(x,1)+\zeta_{t,0}(x,1,0)=u(r,x,1,0)$

and

$\displaystyle\partial_{\zeta_{0}}u(r,x,\xi_{0},\zeta_{0})$
$\displaystyle=\partial_{\zeta_{0}}\Big{(}\Theta(\varphi_{t,0}(x),\xi_{t,0}(x,\xi_{0}),\zeta_{t,0}(x,\xi_{0},\zeta_{0}))\Big{)}$
$\displaystyle=\partial_{\zeta_{0}}\zeta_{t,0}(x,\xi_{0},\zeta_{0})=1.$

Inspecting the structure of the matrix function $\Sigma$ in (46) we get that
the mapping
$\mathbf{e}^{\Sigma(\cdot)z}\colon\mathbb{R}^{d}\to\mathbb{R}^{d+2}$ has the
following form:

$\displaystyle\mathbf{e}^{\Sigma(\cdot)z}\begin{pmatrix}x\\\ \xi_{0}\\\
\zeta_{0}\end{pmatrix}=\begin{pmatrix}\mathbf{e}^{-\alpha(\cdot)z}(x)\\\
\displaystyle\xi_{0}\exp\Big{(}-\int_{0}^{1}\beta(\mathbf{e}^{-r\alpha(\cdot)z}(x))z\,\mathrm{d}r\Big{)}\\\
\displaystyle\zeta_{0}-\xi_{0}\int_{0}^{1}\exp\Big{(}-\int_{0}^{s}\beta(\mathbf{e}^{-r\alpha(\cdot)z}(x))z\,\mathrm{d}r\Big{)}\sigma(\mathbf{e}^{-r\alpha(\cdot)z}(x))\,\mathrm{d}s\end{pmatrix}.$

Recalling (45), namely that

$\displaystyle\mathbf{e}^{\mathcal{Q}z}(\Theta(\cdot))=\Theta(\mathbf{e}^{-\Sigma(\cdot)z}(\cdot)),$

we get the equality

$\displaystyle
u(t,x,1,0)=u_{0}(x)+\int_{0}^{t}\nabla^{T}u(r-,x,1,0)a(x)\,\mathrm{d}r+\int_{0}^{t}u(r-,x,1,0)b(x)\,\mathrm{d}r+c(x)t$
$\displaystyle+\int_{0}^{t}\nabla^{T}u(r-,x,1,0)A(x)\circ\mathrm{d}W_{r}+\int_{0}^{t}u(r-,x,1,0)B(x)\circ\mathrm{d}W_{r}+C(x)W_{t}$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\mathbf{e}^{\mathcal{Q}z}(u(r-,x,1,0))-u(r-,x,1,0)\Big{)}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\mathbf{e}^{\mathcal{Q}z}(u(r-,x,1,0))-u(r-,x,1,0)-\mathcal{Q}u(r-,x,1,0)z\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r.$

This means that $u(t,x)=u(t,x,1,0)$ is the solution of (7).

To show uniqueness, let us first assume that $b(x)=0$, $B(x)=0$, $\beta(x)=0$,
and $c(x)=0$, $C(x)=0$, $\sigma(x)=0$. In this case, the solution defined by
the characteristics has the form

$\displaystyle u(t,x)=u_{0}(\varphi_{t,0}(x)).$ (49)

Let $v$ be another semimartingale solution of the form

$\displaystyle v(t,x)=u_{0}(x)$
$\displaystyle+\int_{0}^{t}f(x,r)\,\mathrm{d}r+\int_{0}^{t}F(x,r)\,\mathrm{d}W_{r}+\int_{0}^{t}\int_{\|z\|\leq
1}\varphi(x,r,z)\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$

with some $f,F$ and $\varphi$. Then Theorem 4.4 yields that

$\displaystyle
v(t,\varphi_{0,t}(x))=u_{0}(x)+\int_{0}^{t}f(\varphi_{0,r-}(x),r)\,\mathrm{d}r+\int_{0}^{t}F(\varphi_{0,r-}(x),r)\,\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\varphi(\varphi_{0,r-}(x),r,z)\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle-\int_{0}^{t}\partial_{x}v(r-,\varphi_{0,r-}(x))a(\varphi_{0,r-}(x))\,\mathrm{d}r-\int_{0}^{t}\partial_{x}v(r-,\varphi_{0,r-}(x))A(\varphi_{0,r-}(x))\circ\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}v(r-,\mathbf{e}^{-\alpha}(\varphi_{0,r-}))+\varphi(\mathbf{e}^{-\alpha}(\varphi_{0,r-})),r,z)$
$\displaystyle\hskip
187.78531pt-v(r-,\varphi_{0,r-})-\varphi(r-,\varphi_{0,r-},r,z)\Big{)}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}v(r-,\mathbf{e}^{-\alpha}(\varphi_{0,r-}))+\varphi(\mathbf{e}^{-\alpha}(\varphi_{0,r-})),r,z)$
$\displaystyle\hskip
234.73523pt+\partial_{x}v(r-,\varphi_{0,r-})\alpha(\varphi_{0,r-})\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r,$
$\displaystyle-\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}v(r-,\varphi_{0,r-})+\varphi(r-,\varphi_{0,r-},r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r,$

where we know that $\varphi$ is given by

$\displaystyle\varphi(x,r,z)=\mathbf{e}^{\mathcal{Q}z}(v(r-,x))-v(r-,x).$

Let us take a closer look at the jump terms. On the one hand we know that

$\displaystyle
v(r,\mathbf{e}^{\alpha(\cdot)z}(\varphi_{0,r-}(x)))=\mathbf{e}^{\mathcal{Q}z}v(r,\varphi_{0,r-}(x)).$

On the other hand, inverting the sign of the jump size $z$ is equivalent to
the reversion of the fictitious time in the Marcus ODE for
$\mathrm{e}^{\alpha(\cdot)z}$. Hence we obtain that

$\displaystyle\mathbf{e}^{\mathcal{Q}z}(v(r,\mathbf{e}^{-\alpha(\cdot)z}(\varphi_{0,r-}(x))))=v(r,\varphi_{0,r-}(x)).$

We also see that

$\displaystyle\partial_{x}v(r-,\varphi_{0,r-}(x))\alpha(\varphi_{0,r-}(x))=\mathcal{Q}v(r-,\varphi_{0,r-}(x)),$

and thus we get

$\displaystyle
v(t,\varphi_{0,t}(x))=u_{0}(x)+\int_{0}^{t}\partial_{t}v(r-,\varphi_{0,r-}(x))\diamond\mathrm{d}\varphi_{0,r}(x)$
$\displaystyle-\int_{0}^{t}\partial_{x}v(r-,\varphi_{0,r-}(x))a(\varphi_{0,r-}(x))\,\mathrm{d}r-\int_{0}^{t}\partial_{x}v(r-,\varphi_{0,r-}(x))A(\varphi_{0,r-}(x))\circ\mathrm{d}W_{r}$
$\displaystyle-\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\mathbf{e}^{\mathcal{Q}z}(v(r-,\varphi_{0,r-}(x))-v(r-,\varphi_{0,r-}(x))\Big{)}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle-\int_{0}^{t}\int_{\|z\|\leq
1}\Big{(}\mathbf{e}^{\mathcal{Q}z}(v(r-,\varphi_{0,r-}(x)))-v(r-,\varphi_{0,r-}(x))-\mathcal{Q}v(r-,\varphi_{0,r-}(x))z\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r$
$\displaystyle=u_{0}(x)$

and hence $v(t,x)$ coincides with the solution $u$ given by (49).

In the presence of linear terms $b$, $B$ and $\beta$, the process
$u(t,\varphi_{t,0}(x))$ given by the characteristics solution has the form

$\displaystyle u$
$\displaystyle(t,\varphi_{t,0}(x))=u_{0}(\varphi_{t,0}(x))\times$
$\displaystyle\times\exp\Big{(}\int_{0}^{t}b(\varphi_{0,r}(x))\,\mathrm{d}r+\int_{0}^{t}B(\varphi_{0,r}(x))\circ\mathrm{d}W_{r}+\int_{0}^{t}\beta(\varphi_{0,r}(x))\diamond\mathrm{d}Z_{r}\Big{)}.$

and the difference $d(t,x):=v(t,\varphi_{0,t}(x))-u(t,\varphi_{t,0}(x))$
satisfies the linear equation

$\displaystyle d(t,x)=\int_{0}^{t}d(r-,x)b(\varphi_{r-}(x))\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}d(r-,x)B(\varphi_{r-}(x))\circ\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}d(r-,x)\beta(\varphi_{r-}(x))\diamond\mathrm{d}Z_{r},$

and thus $d\equiv 0$. The same relation holds for the difference of the non-
homogeneous equations.

##  7 Proof of Theorem 4.3

To simplify the notation, we assume that $X$ and $Y$, as well as $W$ and $Z$
are one-dimensional processes, i.e. $d=m=1$. We also assume that
$\nu([-1,1]^{c})=0$. Adding the large jumps is straightforward.

Let us write the SDEs for $X$ and $Y$ in the Itô form:

$\displaystyle X_{t}(x)$
$\displaystyle=x+\int_{0}^{t}f(X_{r-}(x),r)\,\mathrm{d}r+\int_{0}^{t}F(X_{r-}(x),r)\,\mathrm{d}W_{r}$
$\displaystyle+\frac{1}{2}\int_{0}^{t}F^{\prime}(X_{r-}(x),r)F(X_{r-}(x),r)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Big{(}\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}(x))-X_{r-}(x)\Big{)}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Big{(}\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}(x))-X_{r-}(x)-\varphi(X_{r-}(x),r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r$
$\displaystyle Y_{t}$
$\displaystyle=y+\int_{0}^{t}g(Y_{r-},r)\,\mathrm{d}r+\int_{0}^{t}G(Y_{r-},r)\,\mathrm{d}W_{r}+\frac{1}{2}\int_{0}^{t}G^{\prime}(Y_{r-},r)G(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-Y_{r-}\Big{)}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-Y_{r-}-\psi(Y_{r-},r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r.$

First we perform localization of the semimartingales $X(x)$ and $Y$. We assume
that the semimartingales $\Phi$ and $\Psi$ satisfy the assumptions in Section
3 so that $X$ and $Y$ are well defined and sufficiently smooth. Since the
jumps of $Y$ are assumed to be bounded, for any initial value $y$ we can
localise $Y$ so that the stopped process belongs to a certain ball. Similarly,
we can stop $X$ such that it is also bounded with all its derivatives up to
order 2 uniformly for all values $x$ in the ball defined above. Therefore from
now on we will work with the stopped semimartingales and also assume that all
the coefficients $f$, $F$, $\varphi$, $g$, $G$, $\psi$ as well as their
derivatives have compact support.

For the proof of the generalized Itô formula we apply the method by Carmona
and Nualart [4], Theorem III.3.3.

Consider a sequence of mollifiers $h_{n}\in
C^{\infty}_{c}(\mathbb{R},\mathbb{R})$ given by $h_{n}(x)=nh(nx)$, where $h\in
C^{\infty}_{c}(\mathbb{R}^{d},\mathbb{R})$ supported on a unit ball $|x|\leq
1$, $h(x)\geq 0$, and such that $\int_{\mathbb{R}}h(x)\,\mathrm{d}x=1$. For
the smoothing properties of mollifiers see, e.g. Evans [9, Appendix C.4].

Then for each $x\in\mathbb{R}$, the classical Itô formula applied to the
semimartingale $Y$ yields

$\displaystyle
h_{n}(Y_{t}-x)=h_{n}(y-x)+\int_{0}^{t}h^{\prime}_{n}(Y_{r-}-x)g(Y_{r-},r)\,\mathrm{d}r$
(50)
$\displaystyle+\int_{0}^{t}h^{\prime}_{n}(Y_{r-}-x)G(Y_{r-},r)\,\mathrm{d}W_{r}$
$\displaystyle+\frac{1}{2}\int_{0}^{t}h^{\prime}_{n}(Y_{r-}-x)G^{\prime}(Y_{r-},r)G(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle+\frac{1}{2}\int_{0}^{t}h^{\prime\prime}_{n}(Y_{r-}-x)G^{2}(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}h^{\prime}_{n}(Y_{r-}-x)\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-Y_{r-}-\psi(Y_{r-},r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Big{[}h_{n}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-x\Big{)}-h_{n}(Y_{r-}-x)\Big{]}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Big{[}h_{n}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-x\Big{)}-h_{n}(Y_{r-}-x)$
$\displaystyle\hskip
140.84256pt-h^{\prime}_{n}(Y_{r-}-x)\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-Y_{r-}\Big{)}\Big{]}\,\nu(\mathrm{d}z)\,\mathrm{d}r.$

Next we apply the Itô product formula to $X(x)h(Y-x)$ to get

$\displaystyle X_{t}(x)$ $\displaystyle h_{n}(Y_{t}-x)=xh_{n}(y-x)$
$\displaystyle I_{1}$
$\displaystyle=\begin{cases}\displaystyle+\int_{0}^{t}h_{n}(Y_{r-}-x)f(X_{r-}(x),r)\,\mathrm{d}r\\\
\end{cases}$ $\displaystyle I_{2}$
$\displaystyle=\begin{cases}\displaystyle+\frac{1}{2}\int_{0}^{t}h_{n}(Y_{r-}-x)F^{\prime}(X_{r-}(x),r)F(X_{r-}(x),r)\,\mathrm{d}r\\\
\end{cases}$ $\displaystyle I_{3}$
$\displaystyle=\begin{cases}\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}h_{n}(Y_{r-}-x)\Big{(}\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}(x))-X_{r-}(x)\\\
\hskip
234.73523pt\displaystyle-\varphi(X_{r-}(x),r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r\end{cases}$
$\displaystyle I_{4}$
$\displaystyle=\begin{cases}\displaystyle+\int_{0}^{t}h_{n}(Y_{r-}-x)F(X_{r-}(x),r)\,\mathrm{d}W_{r}\\\
\end{cases}$ $\displaystyle J_{1}$
$\displaystyle=\begin{cases}\displaystyle+\int_{0}^{t}X_{r-}(x)h^{\prime}_{n}(Y_{r}-x)g(Y_{r-},r)\,\mathrm{d}r\\\
\end{cases}$ (51) $\displaystyle J_{2}$
$\displaystyle=\begin{cases}\displaystyle+\frac{1}{2}\int_{0}^{t}X_{r-}(x)h^{\prime}_{n}(Y_{r}-x)G^{\prime}(Y_{r-},r)G(Y_{r-},r)\,\mathrm{d}r\\\
\end{cases}$ $\displaystyle J_{3}$
$\displaystyle=\begin{cases}\displaystyle+\frac{1}{2}\int_{0}^{t}X_{r-}(x)h^{\prime\prime}_{n}(Y_{r}-x)G^{2}(Y_{r-},r)\,\mathrm{d}r\\\
\end{cases}$ $\displaystyle J_{4}$
$\displaystyle=\begin{cases}\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}X_{r-}(x)\Big{[}h_{n}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-x\Big{)}-h_{n}(Y_{r-}-x)\\\
\displaystyle\hskip
140.84256pt-h^{\prime}_{n}(Y_{r-}-x)\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-Y_{r-}\Big{)}\Big{]}\,\nu(\mathrm{d}z)\,\mathrm{d}r\\\
\end{cases}$ $\displaystyle J_{5}$
$\displaystyle=\begin{cases}\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}X_{r-}(x)h^{\prime}_{n}(Y_{r-}-x)\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})\\\
\hskip 234.73523pt-
Y_{r-}-\psi(Y_{r-},r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r\\\ \end{cases}$
$\displaystyle J_{6}$
$\displaystyle=\begin{cases}\displaystyle+\int_{0}^{t}X_{r-}(x)h^{\prime}_{n}(Y_{r}-x)G(Y_{r-},r)\,\mathrm{d}W_{r}\\\
\end{cases}$ $\displaystyle J_{7}$
$\displaystyle=\begin{cases}\displaystyle+\frac{1}{2}\int_{0}^{t}F(X_{r-}(x),r)h^{\prime}_{n}(Y_{r}-x)G(Y_{r-},r)\,\mathrm{d}r\\\
\end{cases}$ $\displaystyle J_{8}$
$\displaystyle=\begin{cases}\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Big{[}\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}(x))-X_{r-}(x)\Big{]}\times\\\
\hskip
117.36761pt\displaystyle\times\Big{[}h_{n}\big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-x\big{)}-h_{n}(Y_{r-}-x)\Big{]}\,\nu(\mathrm{d}z)\,\mathrm{d}r\\\
\end{cases}$ $\displaystyle K_{1}$
$\displaystyle=\begin{cases}\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Big{[}\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}(x))h_{n}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-x\Big{)}\\\
\displaystyle\hskip 211.26027pt-
X_{r-}(x)h_{n}(Y_{r-}-x)\Big{]}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r).\end{cases}$

We decompose the term $K_{1}$ further into the sum

$\displaystyle I_{5}$
$\displaystyle=\begin{cases}\displaystyle\int_{0}^{t}\int_{|z|\leq
1}\Big{(}\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}(x))-X_{r-}(x)\Big{)}h_{n}(Y_{r-}-x)\,\tilde{N}(\mathrm{d}z,\mathrm{d}r)\\\
\end{cases}$ $\displaystyle J_{9}$
$\displaystyle=\begin{cases}+\displaystyle\int_{0}^{t}\int_{|z|\leq
1}\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}(x))\times\\\ \displaystyle\hskip
117.36761pt\times\Big{[}h_{n}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-x\Big{)}-h_{n}(Y_{r-}-x)\Big{]}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r).\end{cases}$

All the (stochastic) integrals exist due to the integrability assumptions on
the functions $\varphi$ and $\psi$ and the properties of the exponential
mappings.

The proof of the generalized Itô formula will consist in integration of the
equality (51) w.r.t. $x$ and passing to the limit as $n\to\infty$.

We distinguish between the Lebesgue integrals w.r.t. $\mathrm{d}r$, the Itô
integrals w.r.t. $\mathrm{d}W$ and the compensated Poissonian random measure
$\tilde{N}$, and the terms containing the derivatives $h^{\prime}_{n}$ and
$h^{\prime\prime}_{n}$.

We start with the terms coming from the integral
$\int_{0}^{t}h_{n}(Y_{r-}-x)\,\mathrm{d}X_{r}$ which will give us the first
integral in (32).

Initial and end points. It follows from the properties of the mollifiers and
the continuity of $x\mapsto X_{t}(x)$ that

$\displaystyle\lim_{n\to\infty}\int_{\mathbb{R}}X_{t}(x)$ $\displaystyle
h_{n}(Y_{t}-x)\,\mathrm{d}x=X_{t}(Y_{t})\quad\text{and}\quad\lim_{n\to\infty}\int_{\mathbb{R}}x$
$\displaystyle h_{n}(y-x)\,\mathrm{d}x=y.$

Terms $I_{1}$ and $I_{2}$. We start with the Lebesgue integrals coming from
the drift part and the noise-induced drift appearing in the Stratonovich
integrals w.r.t. $W$. For each $\omega\in\Omega$,

$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}|h_{n}(Y_{r-}-x)f(X_{r-}(x),r)|\,\mathrm{d}x\,\mathrm{d}r$
$\displaystyle\leq\|f\|\int_{0}^{t}\int_{\mathbb{R}}h_{n}(Y_{r-}-x)\,\mathrm{d}x\,\mathrm{d}r$
$\displaystyle=T\cdot\|f\|<\infty$

and the Fubini theorem yields

$\displaystyle\int_{\mathbb{R}}\int_{0}^{t}h_{n}(Y_{r-}-x)$ $\displaystyle
f(X_{r-}(x),r)\,\mathrm{d}r\,\mathrm{d}x=\int_{0}^{t}\int_{\mathbb{R}^{d}}h_{n}(Y_{r-}-x)f(X_{r-}(x),r)\,\mathrm{d}x\,\mathrm{d}r$
$\displaystyle=\int_{0}^{t}\int_{\|x\|\leq
1/n}h_{n}(x)f(X_{r-}(Y_{r-}-x),r)\,\mathrm{d}x\,\mathrm{d}r.$

For each $r\in[0,T]$, the function $y\mapsto f(X_{r-}(y),r)$ is continuous and
for any $K>0$ by Theorem 6(iii) in Evans [9, Appendix C.4]

$\displaystyle\lim_{n\to\infty}\sup_{|y|\leq K}\Big{|}\int_{|x|\leq
1/n}h_{n}(x)f(X_{r-}(y-x),r)\,\mathrm{d}x-f(X_{r-}(y),r)\Big{|}=0.$

Since for $r\in[0,T]$

$\displaystyle\Big{|}\int_{|x|\leq
1/n}h_{n}(x)f(X_{r-}(Y_{r-}-x),r)\,\mathrm{d}x\Big{|}\leq\|f\|$

the Lebesgue theorem implies that

$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}$ $\displaystyle
h_{n}(Y_{r-}-x)f(X_{r-}(x),r)\,\mathrm{d}x\,\mathrm{d}r\to\int_{0}^{t}f(X_{r-}(Y_{r-}),r)\,\mathrm{d}r.$

Analogously we get the convergence of the term $I_{2}$.

Term $I_{3}$. Consider the function

$\displaystyle H^{I_{3}}(x,r)=\int_{|z|\leq
1}\Big{(}\mathbf{e}^{\varphi(\cdot,r,z)}(x)-x-\varphi(x,r,z)\Big{)}\,\nu(\mathrm{d}z).$

Since

$\displaystyle|\mathbf{e}^{\varphi(\cdot,r,z)}(x)-x-\varphi(x,r,z)|\leq
K_{0}(z)L_{0}(z)\mathrm{e}^{K_{0}(z)}(1+|x|)$

we get that $x\mapsto H^{I_{3}}(x,r)$ is well-defined and continuous.
Recalling that $X$ and $Y$ are assumed to be bounded, the argument of the
previous step applies and

$\displaystyle\int_{0}^{t}$ $\displaystyle h_{n}(Y_{r-}-x)\int_{|z|\leq
1}\Big{(}\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}(x))-X_{r-}(x)-\varphi(X_{r-}(x),r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r$
$\displaystyle\to\int_{0}^{t}\int_{|z|\leq
1}\Big{(}\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}(Y_{r-}))-X_{r-}(Y_{r-})-\varphi(X_{r-}(Y_{r-}),r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r.$

Term $I_{4}$. By Fubini’s theorem for stochastic integrals, see Protter [35,
Theorem 6.64], for each $|y|\leq K$

$\displaystyle\int_{\mathbb{R}}\int_{0}^{t}h_{n}(y-x)F(X_{r-}(x),r)\,\mathrm{d}W_{r}\,\mathrm{d}x=\int_{0}^{t}\int_{\mathbb{R}}h_{n}(y-x)F(X_{r-}(x),r)\,\mathrm{d}x\,\mathrm{d}W_{r}.$

By the properties of mollifiers, see Evans [9, Appendix C.4, Theorem 6], for
each $r\in[0,T]$ and $|y|\leq K$

$\displaystyle\Big{|}\int_{\mathbb{R}}h_{n}(y-x)F(X_{r-}(x),r)\,\mathrm{d}x-F(X_{r-}(y),r)\Big{|}\leq
2\|F\|$

and by the Itô isometry and the Lebesgue theorem

$\displaystyle\mathbf{E}\Big{|}$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}h_{n}(Y_{r-}-x)F(X_{r-}(x),r)\,\mathrm{d}x\,\mathrm{d}W_{r}-\int_{0}^{t}F(X_{r-}(Y_{r-}),r)\,\mathrm{d}W_{r}\Big{|}^{2}$
$\displaystyle=\int_{0}^{t}\mathbf{E}\Big{|}\int_{\mathbb{R}}h_{n}(Y_{r-}-x)F(X_{r-}(x),r)\,\mathrm{d}x-F(X_{r-}(Y_{r-}),r)\Big{|}^{2}\,\mathrm{d}r\to
0.$

Term $I_{5}$. The jump term $I_{5}$ is estimated analogously with the help of
the Itô isometry for stochastic integrals w.r.t. a compensated Poisson random
measure.

Terms $J_{1}$, $J_{2}$ and $J_{7}$. Consider the Lebesgue integral $J_{1}$. We
apply Fubini’s theorem for each $\omega$, integrate by parts, use that $X$ is
a $C^{1}$-diffeomorphism, and apply Lebesgue’s theorem:

$\displaystyle\int_{\mathbb{R}}\int_{0}^{t}X_{r-}(x)h^{\prime}_{n}(Y_{r-}-x)$
$\displaystyle g(Y_{r-},r)\,\mathrm{d}r\,\mathrm{d}x$
$\displaystyle=\int_{0}^{t}\Big{[}\int_{\mathbb{R}}X_{r-}(x)h^{\prime}_{n}(Y_{r-}-x)\,\mathrm{d}x\Big{]}\,g(Y_{r-},r)\mathrm{d}r$
$\displaystyle=\int_{0}^{t}\Big{[}\int_{\mathbb{R}}X^{\prime}_{r-}(x)h_{n}(Y_{r-}-x)\,\mathrm{d}x\Big{]}g(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle\to\int_{0}^{t}X^{\prime}_{r-}(Y_{r-})g(Y_{r-},r)\,\mathrm{d}r.$

The terms $J_{2}$ and $J_{7}$ are treated analogously.

Term $J_{3}$. Analogously, using that $X$ is a $C^{2}$-diffeomorphism, and
applying integration by parts twice we get

$\displaystyle\int_{\mathbb{R}}\int_{0}^{t}X_{r-}(x)h^{\prime\prime}_{n}(Y_{r-}-x)$
$\displaystyle G^{2}(Y_{r-},r)\,\mathrm{d}r\,\mathrm{d}x$
$\displaystyle=\int_{0}^{t}\Big{[}\int_{\mathbb{R}}X_{r-}(x)h^{\prime\prime}_{n}(Y_{r-}-x)\,\mathrm{d}x\Big{]}\,G^{2}(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle=\int_{0}^{t}\Big{[}\int_{\mathbb{R}}X^{\prime\prime}_{r-}(x)h_{n}(Y_{r-}-x)\,\mathrm{d}x\Big{]}\,G^{2}(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle\to\int_{0}^{t}X^{\prime\prime}_{r-}(Y_{r-})G^{2}(Y_{r-},r)\,\mathrm{d}r.$

Terms $J_{4}+J_{5}+J_{8}$. First we note that due to the assumptions, the sum
of the terms $J_{4}$, $J_{5}$ and $J_{8}$ is well defined. Hence the sum
$J_{4}+J_{5}+J_{8}$ simplifies to

$\displaystyle J_{4}+J_{5}+J_{8}=\int_{0}^{t}\int_{|z|\leq 1}$
$\displaystyle\Big{[}\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}(x))\Big{(}h_{n}\big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-x\big{)}-h_{n}(Y_{r-}-x)\Big{)}$
$\displaystyle-
X_{r-}(x)\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-Y_{r-}\Big{)}h^{\prime}_{n}(Y_{r-}-x)\Big{]}\,\nu(\mathrm{d}z)\,\mathrm{d}r$

and by Fubini’s theorem, integration by parts and by the dominated convergence
theorem we get

$\displaystyle\lim_{n\to\infty}$
$\displaystyle\int_{\mathbb{R}}(J_{4}+J_{5}+J_{8})\,\mathrm{d}x$
$\displaystyle=\int_{0}^{t}\int_{|z|\leq
1}\Big{[}\mathbf{e}^{\varphi(\cdot,r,z)}\Big{(}X_{r-}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})\Big{)}\Big{)}-\mathbf{e}^{\varphi(\cdot,r,z)}(X_{r-}(Y_{r-}))$
$\displaystyle\hskip
211.26027pt-X^{\prime}_{r-}(Y_{r-})\psi(Y_{r-},r,z)\Big{]}\,\nu(\mathrm{d}z)\,\mathrm{d}r.$

Term $J_{6}$. The term $J_{6}$ is treated with the help of integration by
parts analogously to the term $I_{4}$.

Term $J_{9}$. The term $J_{9}$ is treated analogously to the term $I_{5}$.

##  8 Proof of Theorem 4.4

The proof of this Theorem is analogous. After localization, and application of
a mollifier $h_{n}$ to $Y$ we obtain again the formula (LABEL:e:hY). The
product formula for $\Phi(x,t)h_{n}(Y_{t}-x)$ takes the form

$\displaystyle\Phi(x,t)h_{n}(Y_{t}-x)=\Phi(x,0)h_{n}(y-x)+\int_{0}^{t}h_{n}(Y_{r-}-x)f(X_{r-}(x),r)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}h_{n}(Y_{r-}-x)F(X_{r-}(x),r)\,\mathrm{d}W_{r}+\int_{0}^{t}\Phi(x,r-)h^{\prime}_{n}(Y_{r-}-x)g(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\Phi(x,r-)h^{\prime}_{n}(Y_{r-}-x)G^{\prime}(Y_{r-},r)G(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\Phi(x,r-)h^{\prime\prime}_{n}(Y_{r-}-x)G^{2}(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Phi(x,r-)\Big{[}h_{n}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-x\Big{)}-h_{n}(Y_{r-}-x)$
$\displaystyle\hskip
164.31752pt-h^{\prime}_{n}(Y_{r-}-x)\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-Y_{r-}\Big{)}\Big{]}\,\nu(\mathrm{d}z)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Phi(x,r-)h^{\prime}_{n}(Y_{r-}-x)\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-Y_{r-}-\psi(Y_{r-},r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\Phi(x,r-)h^{\prime}_{n}(Y_{r}-x)G(Y_{r-},r)\,\mathrm{d}W_{r}$
$\displaystyle+\frac{1}{2}\int_{0}^{t}F(X_{r-}(x),r)h^{\prime}_{n}(Y_{r}-x)G(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\varphi(x,r,z)\Big{(}h_{n}\big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-x\big{)}-h_{n}(Y_{r-}-x)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r$
(52) $\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Big{[}\Big{(}\Phi(x,r-)+\varphi(x,r,z)\Big{)}h_{n}\Big{(}\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-})-x\Big{)}$
$\displaystyle\hskip
234.73523pt-\Phi(x,r-)h_{n}(Y_{r-}-x)\Big{]}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r).$

Integrating w.r.t. $x$ and passing to the limit and $n\to\infty$ get the
formula

$\displaystyle\Phi(Y_{t},t)=\Phi(y,0)+\int_{0}^{t}f(Y_{r-},r)\,\mathrm{d}r+\int_{0}^{t}F(Y_{r-},r)\,\mathrm{d}W_{r}$
$\displaystyle+\int_{0}^{t}\nabla^{T}\Phi(Y_{r-},r-)g(Y_{r-},r)\,\mathrm{d}r+\frac{1}{2}\int_{0}^{t}\nabla^{T}\Phi(Y_{r-},r-)G^{2}(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\nabla^{T}\nabla\Phi(Y_{r-},r-)G^{\prime}(Y_{r-},r)G(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Big{(}\Phi(\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-}),r-))-\Phi(Y_{r-},r-)$
$\displaystyle\hskip
234.73523pt-\nabla^{T}\Phi(Y_{r-},r-)\psi(Y_{r-},r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\nabla^{T}\Phi(Y_{r-},r-)G(Y_{r-},r)\,\mathrm{d}W_{r}+\frac{1}{2}\int_{0}^{t}\nabla^{T}F(Y_{r-},r)G(Y_{r-},r)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Big{(}\varphi(\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-}),r,z)-\varphi(Y_{r-},r,z)\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}r$
$\displaystyle+\int_{0}^{t}\int_{|z|\leq
1}\Big{[}\Phi(\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-}),r-)+\varphi(\mathbf{e}^{\psi(\cdot,r,z)}(Y_{r-}),r,z)-\Phi(Y_{r-},r-)\Big{]}\,\tilde{N}(\mathrm{d}z,\mathrm{d}r).$

which can be transformed to (35), (LABEL:e:Phinabla1), (LABEL:e:Phinabla2).

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