First Order Linear Marcus SPDEs

TL;DR

This paper solves a Levy-driven linear stochastic first-order PDE using stochastic characteristics, providing a solution form similar to deterministic PDEs and Brownian-driven SPDEs, advancing understanding of Levy noise in stochastic transport equations.

Contribution

It introduces a method to solve Levy-driven linear stochastic PDEs in Marcus form, extending classical solutions to include Levy noise.

Findings

Solution expressed via stochastic characteristics

Equivalent form to deterministic PDE solutions

Applicable to Levy-driven stochastic transport equations

Abstract

In this paper we solve a L\'evy 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).

Equations

$${\partial }_{t}u(x,t)$$

$$u(x,0)$$

$$\mathrm{d}{\phi }_{0,t}(x)$$

$$u(x,t)=u_0({\phi }_{t,0}(x)),$$

$${\partial }_{t}u(x,t)$$

$$u(x,0)$$

$$u(x,t)={\mathrm{e}}^{{\int }_0^{t}b({\phi }_{t,s}(x),s)\mathrm{d}s}{u}_0({\phi }_{t,0}(x))+{\int }_0^t{\mathrm{e}}^{{\int }_s^{t}b({\phi }_{t,r}(x),r)\mathrm{d}r}c({\phi }_{t,s}(x),s)\mathrm{d}s,$$

$${\partial }_{t}u(x,t)$$

$$u(x,0)$$

$$\mathrm{d}{\phi }_{0,t}(x)$$

$$DF=\left(\begin{array}{ccc}\frac{\partial F_1}{\partial x_1}& \cdots & \frac{\partial F_1}{\partial x_d}\\ \cdots & \cdots & \cdots \\ \frac{\partial F_m}{\partial x_1}& \cdots & \frac{\partial F_m}{\partial x_d}\end{array}\right),$$

$$Z_t={\int }_0^t{\int }_{\Vert z\Vert \le 1}z\tilde{N}(\mathrm{d}z,\mathrm{d}s)+{\int }_0^t{\int }_{\Vert z\Vert >1}zN(\mathrm{d}z,\mathrm{d}s),$$

$$a$$

$$A,\alpha$$

$$\mathcal{P}u(x)$$

$$\mathcal{R}u(x)$$

$$\mathcal{Q}u(x)$$

$$\mathrm{d}u(t,x)$$

$$u(0,x)$$

$$u(t,x)$$

$$\displaystyle+\int_{0}^{t}\Big{[}\nabla^{T}u_{x}(s,x)A(x)+u(s,x)B(x)+C(x)\Big{]}\circ\mathrm{d}W_{s}$$

$$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq 1}\Big{(}\mathbf{e}^{\mathcal{Q}z}u(s-,x)-u(s-,x)\Big{)}\tilde{N}(\mathrm{d}z,\mathrm{d}s)$$

$$\displaystyle+\int_{0}^{t}\int_{\|z\|\leq 1}\Big{(}\mathbf{e}^{\mathcal{Q}z}u(s-,x)-u(s-,x)-\mathcal{Q}u(s-,x)z\Big{)}\,\nu(\mathrm{d}z)\,\mathrm{d}s$$

$$\displaystyle+\int_{0}^{t}\int_{\|z\|>1}\Big{(}\mathbf{e}^{\mathcal{Q}z}u(s-,x)-u(s-,x)\Big{)}\tilde{N}(\mathrm{d}z,\mathrm{d}s),$$

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

$${\partial }_{r}g(r,x)$$

$$g(0,x)$$

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

$$a$$

$$A,\alpha$$

$$\begin{array}{rl}{\phi }_{0,t}(x)=x-{\int }_0^{t}a({\phi }_{0,r}(x))\mathrm{d}r& -{\int }_0^{t}A({\phi }_{0,r}(x))\circ \mathrm{d}{W}_r\\ & -{\int }_0^t\alpha ({\phi }_{0,r}(x))\diamond \mathrm{d}{Z}_r,\end{array}$$

$$\begin{array}{rl}{\xi }_{0,t}(x,{\xi }_0)={\xi }_0-{\int }_0^t{\xi }_{0,r}(x,{\xi }_0)b({\phi }_{0,r}(x))\mathrm{d}r& -{\int }_0^t{\xi }_{0,r}(x,{\xi }_0)B({\phi }_{0,r}(x))\circ \mathrm{d}{W}_r\\ & -{\int }_0^t{\xi }_{0,r}(x,{\xi }_0)\beta ({\phi }_{0,r}(x))\diamond \mathrm{d}{Z}_r,\end{array}$$

$$\begin{array}{rl}{\zeta }_{0,t}(x,{\xi }_0,{\zeta }_0)={\zeta }_0-{\int }_0^t{\xi }_{0,r}(x,{\xi }_0)c({\phi }_{0,r}(x))\mathrm{d}r& -{\int }_0^t{\xi }_{0,r}(x,{\xi }_0)C({\phi }_{0,r}(x))\circ \mathrm{d}{W}_r\\ & -{\int }_0^t{\xi }_{0,r}(x,{\xi }_0)\sigma ({\phi }_{0,r}(x))\diamond \mathrm{d}{Z}_r.\end{array}$$

$$u(t,x):={\xi }_{t,0}(x,1)u_0({\phi }_{t,0}(x))+{\zeta }_{t,0}(x,1,0)$$

$$u(t,x)=u_0(x)$$

$$+{\int }_0^t{\nabla }^{T}u(r,x)\alpha (x)\diamond \mathrm{d}{Z}_r.$$

$${\phi }_{0,t}(x)$$

$$u(t,x)=u_0({\phi }_{t,0}(x)).$$

Full text

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)

[TABLE]

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

[TABLE]

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

[TABLE]

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 $c$

[TABLE]

has the solution

[TABLE]

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

[TABLE]

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

[TABLE]

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,

[TABLE]

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 $Z$,

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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:

[TABLE]

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

[TABLE]

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

[TABLE]

and

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

Then the solution has the form

[TABLE]

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

[TABLE]

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

[TABLE]

Then the characteristics equation

[TABLE]

has the solution

[TABLE]

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

[TABLE]

Let us show that

[TABLE]

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

[TABLE]

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

[TABLE]

Example 2.4**.**

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

[TABLE]

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

[TABLE]

Hence, (19) has the explicit solution

[TABLE]

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

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

[TABLE]

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:

[TABLE]

HF:

[TABLE]

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

[TABLE]

and

[TABLE]

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

[TABLE]

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

[TABLE]

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 $h=h(u)=h(u;x,r,z)$

[TABLE]

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

[TABLE]

and in particular we define the exponential mapping

[TABLE]

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:

[TABLE]

Consider an Itô semimartingale with parameter

[TABLE]

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

[TABLE]

and consider the Itô SDE

[TABLE]

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

[TABLE]

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

[TABLE]

Remark 4.2**.**

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

[TABLE]

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,

[TABLE]

Then the following formula holds true

[TABLE]

where the latter integrals are understood as

[TABLE]

and

[TABLE]

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

[TABLE]

where

[TABLE]

and

[TABLE]

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

[TABLE]

which is understood in the following sense:

[TABLE]

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

[TABLE]

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:

[TABLE]

In particular,

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

In other words, we have

[TABLE]

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

[TABLE]

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

[TABLE]

which is understood as follows:

[TABLE]

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

[TABLE]

or equivalently,

[TABLE]

Second, taking into account (41) we get that

[TABLE]

and hence

[TABLE]

or equivalently,

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

or equivalently

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

and define a process

[TABLE]

Then by Theorem 5.2 we get that

[TABLE]

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

[TABLE]

The derivative equals to

[TABLE]

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

[TABLE]

Analogously,

[TABLE]

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

[TABLE]

and

[TABLE]

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:

[TABLE]

Recalling (45), namely that

[TABLE]

we get the equality

[TABLE]

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

[TABLE]

Let $v$ be another semimartingale solution of the form

[TABLE]

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

[TABLE]

where we know that $\varphi$ is given by

[TABLE]

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

[TABLE]

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

[TABLE]

We also see that

[TABLE]

and thus we get

[TABLE]

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

[TABLE]

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

[TABLE]

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:

[TABLE]

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

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

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

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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

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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 $\omega\in\Omega$,

[TABLE]

and the Fubini theorem yields

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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]

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

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the Lebesgue theorem implies that

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

Term $I_{3}$. Consider the function

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Since

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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

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

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

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and by the Itô isometry and the Lebesgue theorem

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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:

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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

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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

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

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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

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

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