On Martingale Problems and Feller Processes

TL;DR

This paper links the well-posedness of certain martingale problems to the existence of Feller processes, providing new results for Lévy-driven SDEs and stable-like processes with unbounded coefficients.

Contribution

It establishes a sufficient condition connecting martingale problem well-posedness to Feller processes and applies this to new existence and uniqueness results for complex stochastic models.

Findings

Martingale problem well-posedness implies Feller process existence.

New existence and uniqueness results for Lévy-driven SDEs.

Results applicable to stable-like processes with unbounded coefficients.

Abstract

Let $A$ be a pseudo-differential operator with negative definite symbol $q$. In this paper we establish a sufficient condition such that the well-posedness of the $(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem implies that the unique solution to the martingale problem is a Feller process. This provides a proof of a former claim by van Casteren. As an application we prove new existence and uniqueness results for L\'evy-driven stochastic differential equations and stable-like processes with unbounded coefficients.

Figures

Equations

$$d{X}_t=\sigma (X_{t-})d{L}_t,X_0\sim \mu ,$$

$$Af(x)=-{\int }_{\text{mathds}{R}^d}q(x,\xi )e^{ix\cdot \xi }\hat{f}(\xi )d\xi ,f\in C_c^{\infty }(\text{mathds}{R}^d),x\in \text{mathds}{R}^d,$$

$$\nu (\{y\in \text{mathds}{R}^k;|\sigma (x)\cdot y+x|\le r\})\stackrel{|x|\to \infty }{\to }0\text{for all }r>0$$

$$q(x,\xi )=q(x,0)-ib(x)\cdot \xi +\frac{1}{2}\xi \cdot Q(x)\xi +{\int }_{y\ne 0}(1-e^{iy\cdot \xi }+iy\cdot \xi \text{mathds}{1}_{(0,1)}(|y|))\nu (x,dy),x,\xi \in \text{mathds}{R}^d,$$

$$\underset{|x|\to \infty }{\lim }\underset{|\xi |\le |x{|}^{-1}}{\sup }|q(x,\xi )|<\infty ,$$

$$\underset{|x|\to \infty }{\lim }\underset{|\xi |\le |x{|}^{-1}}{\sup }|q(x,\xi )|<\infty .$$

$$T_{t}f(x):=\text{mathds}{E}^{x}f(X_t),x\in E,f\in {\mathcal{B}}_b(E):=\{f:E\to \text{mathds}R;\text{f bounded, Borel measurable}\}$$

$$Lf(x)=-{\int }_{\text{mathds}{R}^d}{e}^{ix\cdot \xi }q(x,\xi )\hat{f}(\xi )d\xi ,f\in C_c^{\infty }(\text{mathds}{R}^d),x\in \text{mathds}{R}^d$$

$$q(x,\xi )=q(x,0)-ib(x)\cdot \xi +\frac{1}{2}\xi \cdot Q(x)\xi +{\int }_{\text{mathds}{R}^d\backslash \{0\}}(1-e^{iy\cdot \xi }+iy\cdot \xi \text{mathds}{1}_{(0,1)}(|y|))\nu (x,dy).$$

$$\forall K\subseteq \text{mathds}{R}^d\text{cpt.}:\underset{x\in K}{\sup }|q(x,0)|+\underset{x\in K}{\sup }|b(x)|+\underset{x\in K}{\sup }|Q(x)|+\underset{x\in K}{\sup }{\int }_{y\ne 0}(|y{|}^2\wedge 1)\nu (x,dy)<\infty .$$

$$\underset{|x|\to \infty }{\lim }\nu (x,B(-x,r))=0\text{for all }r>0.$$

$$d{X}_t=\sigma (X_{t-})d{L}_t,X_0\sim \mu ,$$

$$M_t^f:=f(X_t)-f(X_0)-{\int }_0^{t}Af(X_s)ds,t\ge 0,$$

$$\underset{|x|\to \infty }{\mathrm{limsup}}\underset{|\xi |\le |x{|}^{-1}}{\sup }|q(x,\xi )|<\infty$$

$$p(x,\xi ):={\int }_{y\ne 0}(1-e^{iy\cdot \xi }+iy\cdot \xi \text{mathds}{1}_{(0,|x|/2)}(|y|))\nu (x,dy)$$

$$|1-e^{iy\cdot \xi }|\le 2\text{and}|1-e^{iy\cdot \xi }+iy\cdot \xi |\le \frac{1}{2}|\xi {|}^2|y{|}^2$$

$$|p(x,\xi )|\le \frac{|\xi {|}^2}{2}{\int }_{0<|y|<|x|/2}|y{|}^2\nu (x,dy)+2{\int }_{|y|\ge |x|/2}\nu (x,dy)$$

$$\underset{|x|\to \infty }{\mathrm{limsup}}\underset{|\xi |\le |x{|}^{-1}}{\sup }|p(x,\xi )|<\infty .$$

$$g(\eta ):=\frac{1}{2}{\int }_{(0,\infty )}\frac{1}{(2\pi r{)}^{d/2}}\exp \left(-\frac{|\eta {|}^2}{2r}-\frac{r}{2}\right)dr,\eta \in \text{mathds}{R}^d,$$

$$\frac{|z{|}^2}{1+|z{|}^2}={\int }_{\text{mathds}{R}^d}(1-\cos (\eta \cdot z))g(\eta )d\eta$$

$$\underset{|z|\ge 1/2}{\inf }\frac{|z{|}^2}{1+|z{|}^2}=\frac{1}{5}>0$$

$$\frac{1}{5}\nu (x;\{y;|y|\ge |x|/2\})\le {\int }_{|y|\ge |x|/2}\frac{{\left(\frac{|y|}{|x|}\right)}^2}{1+{\left(\frac{|y|}{|x|}\right)}^2}\nu (x,dy)$$

$$\le {\int }_{\text{mathds}{R}^d}\mathrm{Re}q\left(x,\frac{\eta }{|x|}\right)d\eta .$$

$$|\psi (\xi )|\le 2\underset{|\zeta |\le 1}{\sup }|\psi (\zeta )|(1+|\xi {|}^2),\xi \in \text{mathds}{R}^d,$$

$$\nu (x;\{y;|y|\ge |x|/2\})$$

$$0\le \xi \cdot Q(x)\xi \le \mathrm{Re}q(x,\xi )\le |q(x,\xi )|$$

$$\underset{|y|\le |x|/2}{\inf }\frac{1}{|x{|}^2+|y{|}^2}\ge \frac{4}{5}\frac{1}{|x{|}^2}$$

$$\frac{4}{5}\frac{1}{|x{|}^2}{\int }_{|y|\le |x|/2}|y{|}^2\nu (x,dy)\le {\int }_{|y|\le |x|/2}\frac{|y{|}^2}{|x{|}^2+|y{|}^2}\nu (x,dy)={\int }_{|y|\le |x|/2}\frac{{\left(\frac{|y|}{|x|}\right)}^2}{1+{\left(\frac{|y|}{|x|}\right)}^2}\nu (x,dy).$$

$${\int }_{|y|\le |x|/2}|y{|}^2\nu (x,dy)$$

$${\int }_{|y|\le |x|/2}|y{|}^2\nu (x,dy)\le \frac{5}{4}|x{|}^2\underset{|\xi |\le 1}{\sup }\left|q\left(x,\frac{\xi }{|x|}\right)\right|{\int }_{\text{mathds}{R}^d}(1+|\eta {|}^2)g(\eta )d\eta$$

$$\underset{|x|\to \infty }{\mathrm{limsup}}\underset{|\xi |\le |x{|}^{-1}}{\sup }\left|q(x,\xi )-i\xi \cdot \left(b(x)+{\int }_{1\le |y|<|x|/2}y\nu (x,dy)\right)\right|<\infty ,$$

Full text

Keywords: Feller process, martingale problem, stochastic differential equation, stable-like process, unbounded coefficients

MSC 2010: Primary: 60J25. Secondary: 60G44, 60J75, 60H10, 60G51.

1 Introduction

Let $(L_{t})_{t\geq 0}$ be a $k$-dimensional Lévy process with characteristic exponent $\psi:\mathds{R}^{d}\to\mathds{C}$ and $\sigma:\mathds{R}^{d}\to\mathds{R}^{d\times k}$ a continuous function which is at most of linear growth. It is known that there is a intimate correspondence between the Lévy-driven stochastic differential equation (SDE)

[TABLE]

and the pseudo-differential operator $A$ with symbol $q(x,\xi):=\psi(\sigma(x)^{T}\xi)$, i. e.

[TABLE]

where $\hat{f}$ denotes the Fourier transform of a smooth function $f$ with compact support. Kurtz [6] proved that the existence of a unique weak solution to the SDE for any initial distribution $\mu$ is equivalent to the well-posedness of the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem. Recently, we have shown in [7] that a unique solution to the martingale problem – or, equivalently, to the SDE (1) – is a Feller process if the Lévy measure $\nu$ satisfies

[TABLE]

which is equivalent to saying that $A$ maps $C_{c}^{\infty}(\mathds{R}^{d})$ into $C_{\infty}(\mathds{R}^{d})$, the space of continuous functions vanishing at infinity.

In this paper, we are interested in the following more general question: Consider a pseudo-differential operator $A$ with continuous negative definite symbol $q$,

[TABLE]

such that the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem is well-posed, i. e. for any initial distribution $\mu$ there exists a unique solution to the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem. Under which assumptions does the well-posedness of the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem imply that the unique solution to the martingale problem is a Feller process? Since the infinitesimal generator of the solution is, when restricted to $C_{c}^{\infty}(\mathds{R}^{d})$, the pseudo-differential operator $A$, it is clear that $A$ has to satisfy $Af\in C_{\infty}(\mathds{R}^{d})$ for all $f\in C_{c}^{\infty}(\mathds{R}^{d})$. In a paper by van Casteren [16] it was claimed that this mapping property of $A$ already implies that the solution is a Feller process; however, this result turned out to be wrong, see [1, Example 2.27(ii)] for a counterexample. Our main result states van Casteren’s claim is correct if the symbol $q$ satisfies a certain growth condition; the required definitions will be explained in Section 2.

Theorem 1.1 ()

Let $A$ be a pseudo-differential operator with continuous negative definite symbol $q$ such that $q(\cdot,0)=0$ and $A$ maps $C_{c}^{\infty}(\mathds{R}^{d})$ into $C_{\infty}(\mathds{R}^{d})$. If the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem is well-posed and

[TABLE]

then the solution $(X_{t})_{t\geq 0}$ to the martingale problem is a conservative rich Feller process with symbol $q$.

1.2 Remark

1. (i).

If the martingale problem is well-posed and $A(C_{c}^{\infty}(\mathds{R}^{d}))\subseteq C_{\infty}(\mathds{R}^{d})$, then the solution is a $C_{b}$-Feller process, i. e. the associated semigroup $(T_{t})_{t\geq 0}$ satisfies $T_{t}:C_{b}(\mathds{R}^{d})\to C_{b}(\mathds{R}^{d})$ for all $t\geq 0$. The growth condition (G) is needed to prove the Feller property; that is, to show that $T_{t}f$ vanishes at infinity for any $f\in C_{\infty}(\mathds{R}^{d})$ and $t\geq 0$. 2. (ii).

There is a partial converse to Theorem 1.1: If $(X_{t})_{t\geq 0}$ is a Feller process and $C_{c}^{\infty}(\mathds{R}^{d})$ is a core for the generator $A$ of $(X_{t})_{t\geq 0}$, then the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem is well-posed, see e. g. [5, Theorem 4.10.3] or [11, Theorem 1.37] for a proof. 3. (iii).

The mapping property $A(C_{c}^{\infty}(\mathds{R}^{d}))\subseteq C_{\infty}(\mathds{R}^{d})$ can be equivalently formulated in terms of the symbol $q$ and its characteristics, cf. Lemma 2.1. 4. (iv).

For the particular case that $A$ is the pseudo-differential operator associated with the SDE (1), i. e. $q(x,\xi)=\psi(\sigma(x)^{T}\xi)$, we recover [7, Theorem 1.1]. Note that the growth condition (G) is automatically satisfied for any function $\sigma$ which is at most of linear growth.

Although it is, in general, hard to prove the well-posedness of a martingale problem, Theorem 1.1 is very useful since it allows us to use localization techniques for martingale problems to establish new existence results for Feller processes with unbounded coefficients.

Corollary 1.3 ()

Let $A$ be a pseudo-differential operator with symbol $q$ such that $q(\cdot,0)=0$, $A(C_{c}^{\infty}(\mathds{R}^{d}))\subseteq C_{\infty}(\mathds{R}^{d})$ and

[TABLE]

Assume that there exists a sequence $(q_{k})_{k\in\mathds{N}}$ of symbols such that $q_{k}(x,\xi)=q(x,\xi)$ for all $|x|<k$, $\xi\in\mathds{R}^{d}$, and the pseudo-differential operator $A_{k}$ with symbol $q_{k}$ maps $C_{c}^{\infty}(\mathds{R}^{d})$ into $C_{\infty}(\mathds{R}^{d})$. If the $(A_{k},C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem is well posed for all $k\geq 1$, then there exists conservative rich Feller process $(X_{t})_{t\geq 0}$ with symbol $q$, and $(X_{t})_{t\geq 0}$ is the unique solution to the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem.

The paper is organized as follows. After introducing basic notation and definitions in Section 2, we prove Theorem 1.1 and Corollary 1.3. In Section 4 we present applications and examples; in particular we obtain new existence and uniqueness results for Lévy-driven stochastic differential equations and stable-like processes with unbounded coefficients.

2 Preliminaries

We consider $\mathds{R}^{d}$ endowed with the Borel $\sigma$-algebra $\mathcal{B}(\mathds{R}^{d})$ and write $B(x,r)$ for the open ball centered at $x\in\mathds{R}^{d}$ with radius $r>0$; $\mathds{R}^{d}_{\Delta}$ is the one-point compactification of $\mathds{R}^{d}$. If a certain statement holds for $x\in\mathds{R}^{d}$ with $|x|$ sufficiently large, we write “for $|x|\gg 1$”. For a metric space $(E,d)$ we denote by $C(E)$ the space of continuous functions $f:E\to\mathds{R}$; $C_{\infty}(E)$ (resp. $C_{b}(E)$) is the space of continuous functions which vanish at infinity (resp. are bounded). A function $f:[0,\infty)\to E$ is in the Skorohod space $D([0,\infty),E)$ if $f$ is right-continuous and has left-hand limits in $E$. We will always consider $E=\mathds{R}^{d}$ or $E=\mathds{R}^{d}_{\Delta}$.

An $E$-valued Markov process $(\Omega,\mathcal{A},\mathds{P}^{x},x\in E,X_{t},t\geq 0)$ with càdlàg (right-continuous with left-hand limits) sample paths is called a Feller process if the associated semigroup $(T_{t})_{t\geq 0}$ defined by

[TABLE]

has the Feller property, i. e. $T_{t}f\in C_{\infty}(E)$ for all $f\in C_{\infty}(E)$, and $(T_{t})_{t\geq 0}$ is strongly continuous at $t=0$, i. e. $\|T_{t}f-f\|_{\infty}\xrightarrow[]{t\to 0}0$ for any $f\in C_{\infty}(E)$. Following [13] we call a Markov process $(X_{t})_{t\geq 0}$ with càdlàg sample paths a $C_{b}$-Feller process if $T_{t}(C_{b}(E))\subseteq C_{b}(E)$ for all $t\geq 0$. An $\mathds{R}^{d}_{\Delta}$-valued Markov process with semigroup $(T_{t})_{t\geq 0}$ is conservative if $T_{t}\mathds{1}_{\mathds{R}^{d}}=\mathds{1}_{\mathds{R}^{d}}$ for all $t\geq 0$.

If the smooth functions with compact support $C_{c}^{\infty}(\mathds{R}^{d})$ are contained in the domain of the generator $(L,\mathcal{D}(L))$ of a Feller process $(X_{t})_{t\geq 0}$, then we speak of a rich Feller process. A result due to von Waldenfels and Courrège, cf. [1, Theorem 2.21], states that the generator $L$ of an $\mathds{R}^{d}$-valued rich Feller process is, when restricted to $C_{c}^{\infty}(\mathds{R}^{d})$, a pseudo-differential operator with negative definite symbol:

[TABLE]

where $\hat{f}(\xi):=\mathcal{F}f(\xi):=(2\pi)^{-d}\int_{\mathds{R}^{d}}e^{-ix\cdot\xi}f(x)\,dx$ denotes the Fourier transform of $f$ and

[TABLE]

We call $q$ the symbol of the Feller process $(X_{t})_{t\geq 0}$ and of the pseudo-differential operator; $(b,Q,\nu)$ are the characteristics of the symbol $q$. For each fixed $x\in\mathds{R}^{d}$, $(b(x),Q(x),\nu(x,dy))$ is a Lévy triplet, i. e. $b(x)\in\mathds{R}^{d}$, $Q(x)\in\mathds{R}^{d\times d}$ is a symmetric positive semidefinite matrix and $\nu(x,dy)$ a $\sigma$-finite measure on $(\mathds{R}^{d}\backslash\{0\},\mathcal{B}(\mathds{R}^{d}\backslash\{0\}))$ satisfying $\int_{y\neq 0}\min\{|y|^{2},1\}\,\nu(x,dy)<\infty$. We use $q(x,D)$ to denote the pseudo-differential operator $L$ with continuous negative definite symbol $q$. A family of continuous negative definite functions $(q(x,\cdot))_{x\in\mathds{R}^{d}}$ is locally bounded if for any compact set $K\subseteq\mathds{R}^{d}$ there exists $c>0$ such that $|q(x,\xi)|\leq c(1+|\xi|^{2})$ for all $x\in K$, $\xi\in\mathds{R}^{d}$. By [14, Lemma 2.1, Remark 2.2], this is equivalent to

[TABLE]

If (3) holds for $K=\mathds{R}^{d}$, we say that $q$ has bounded coefficients. We will frequently use the following result.

Lemma 2.1 ()

Let $L$ be a pseudo-differential operator with continuous negative definite symbol $q$ and characteristics $(b,Q,\nu)$. Assume that $q(\cdot,0)=0$ and that $q$ is locally bounded.

1. (i).

$\lim_{|x|\to\infty}Lf(x)=0$* for all $f\in C_{c}^{\infty}(\mathds{R}^{d})$ if, and only if,*

[TABLE] 2. (ii).

If $\lim_{|x|\to\infty}\sup_{|\xi|\leq|x|^{-1}}|\operatorname{Re}q(x,\xi)|=0$, then (4) holds. 3. (iii).

$L(C_{c}^{\infty}(\mathds{R}^{d}))\subseteq C(\mathds{R}^{d})$* if, and only if, $x\mapsto q(x,\xi)$ is continuous for all $\xi\in\mathds{R}^{d}$.*

For a proof of Lemma 2.1(i),(ii) see [1, Lemma 3.26] or [8, Theorem 1.27]; 2.1(iii) goes back to Schilling [13, Theorem 4.4], see also [10, Theorem A.1].

If the symbol $q$ of a rich Feller process $(L_{t})_{t\geq 0}$ does not depend on $x$, i. e. $q(x,\xi)=q(\xi)$, then $(L_{t})_{t\geq 0}$ is a Lévy process. This is equivalent to saying that $(L_{t})_{t\geq 0}$ has stationary and independent increments and càdlàg sample paths. The symbol $q=q(\xi)$ is called characteristic exponent. Our standard reference for Lévy processes is the monograph [12] by Sato. Weak uniqueness holds for the Lévy-driven stochastic differential equation (SDE, for short)

[TABLE]

if any two weak solutions of the SDE have the same finite-dimensional distributions. We refer the reader to Situ [15] for further details.

Let $(A,\mathcal{D})$ be a linear operator with domain $\mathcal{D}\subseteq\mathcal{B}_{b}(E)$ and $\mu$ a probability measure on $(E,\mathcal{B}(E))$. A $d$-dimensional stochastic process $(X_{t})_{t\geq 0}$, defined on a probability space $(\Omega,\mathcal{A},\mathds{P}^{\mu})$, with càdlàg sample paths is a solution to the $(A,\mathcal{D})$-martingale problem with initial distribution $\mu$, if $X_{0}\sim\mu$ and

[TABLE]

is a $\mathds{P}^{\mu}$-martingale with respect to the canonical filtration of $(X_{t})_{t\geq 0}$ for any $f\in\mathcal{D}$. By considering the measure $\mathds{Q}^{\mu}$ induced by $(X_{t})_{t\geq 0}$ on $D([0,\infty),E)$ we may assume without loss of generality that $\Omega=D([0,\infty),E)$ is the Skorohod space and $X_{t}(\omega):=\omega(t)$ the canonical process. The $(A,\mathcal{D})$-martingale problem is well-posed if for any initial distribution $\mu$ there exists a unique (in the sense of finite-dimensional distributions) solution to the $(A,\mathcal{D})$-martingale problem with initial distribution $\mu$. For a comprehensive study of martingale problems see [2, Chapter 4].

3 Proof of the main results

In order to prove Theorem 1.1 we need the following statement which allows us to formulate the linear growth condition (G) in terms of the characteristics.

Lemma 3.1 ()

Let $(q(x,\cdot))_{x\in\mathds{R}^{d}}$ be a family of continuous negative definite functions with characteristics $(b,Q,\nu)$ such that $q(\cdot,0)=0$. Then

[TABLE]

if, and only if, there exists an absolute constant $c>0$ such that each of the following conditions is satisfied for $|x|\gg 1$.

1. (i).

$\left|b(x)+\int_{1\leq|y|<|x|/2}y\,\nu(x,dy)\right|\leq c(1+|x|)$. 2. (ii).

$|Q(x)|+\int_{|y|\leq|x|/2}|y|^{2}\,\nu(x,dy)\leq c(1+|x|^{2})$. 3. (iii).

$\nu(x,\{y\in\mathds{R}^{d};|y|\geq 1\vee|x|/2\})\leq c$.

If (G) holds and $q$ is locally bounded, cf. (3), then (i)–(iii) hold for all $x\in\mathds{R}^{d}$.

Proof.

First we prove that (i)–(iii) are sufficient for (G). Because of (i) and (ii) it suffices to show that

[TABLE]

satisfies the linear growth condition (G). Using the elementary estimates

[TABLE]

we find

[TABLE]

for all $|x|\geq 1$ which implies by (ii) and (iii) that

[TABLE]

It remains to prove that (G) implies (i)-(iii). For (ii) and (iii) we use a similar idea as in [13, proof of Theorem 4.4]. It is known that the function $g$ defined by

[TABLE]

has a finite second moment, i. e. $\int_{\mathds{R}^{d}}|\eta|^{2}g(\eta)\,d\eta<\infty$, and satisfies

[TABLE]

for all $z\in\mathds{R}^{d}$. As

[TABLE]

we obtain by applying Tonelli’s theorem

[TABLE]

Since

[TABLE]

for any continuous negative definite function $\psi$, cf. [1, Proposition 2.17d)], we get

[TABLE]

and this gives (iii) for $|x|\gg 1$. Next we prove (ii). First of all, we note that

[TABLE]

and therefore $|Q(x)|\leq c(1+|x|^{2})$ is a direct consequence of (G). On the other hand,

[TABLE]

implies that

[TABLE]

Using (5) and applying Tonelli’s theorem once more, we find

[TABLE]

Hence,

[TABLE]

and (ii) follows. Finally, as (ii) and (iii) imply that

[TABLE]

see the first part of the proof, a straightforward application of the triangle inequality gives

[TABLE]

which proves (i). ∎

Corollary 3.2 ()

Let $A$ be a pseudo-differential operator with continuous negative definite symbol $q$ such that $q(\cdot,0)=0$. If $A$ maps $C_{c}^{\infty}(\mathds{R}^{d})$ into $C_{\infty}(\mathds{R}^{d})$ and $q$ satisfies the linear growth condition (G), then there exists for any initial distribution $\mu$ a solution to the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem which is conservative.

Proof.

Since $A(C_{c}^{\infty}(\mathds{R}^{d}))\subseteq C_{\infty}(\mathds{R}^{d})$ and $A$ satisfies the positive maximum principle, it follows from [2, Theorem 4.5.4] that there exists an $\mathds{R}^{d}_{\Delta}$-valued solution to the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem with initial distribution $\mu:=\delta_{x}$. By considering the probability measure induced by $(X_{t})_{t\geq 0}$ on the Skorohod space $D([0,\infty),\mathds{R}^{d}_{\Delta})$, we may assume without loss of generality that $X_{t}(\omega):=\omega(t)$ is the canonical process on $\Omega:=D([0,\infty),\mathds{R}^{d}_{\Delta})$. Lemma 3.1 implies that

[TABLE]

and therefore [10, Corollary 3.2] shows that the solution with initial distribution $\delta_{x}$ does not explode in finite time with probability $1$. By construction, see [2, proof of Theorem 4.5.4], the mapping $x\mapsto\mathds{P}^{x}(B)$ is measurable for all $B\in\mathcal{F}_{\infty}^{X}:=\sigma(X_{t};t\geq 0)$. If we define

[TABLE]

then $\mathds{P}^{\mu}$ gives rise to a conservative solution to the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem with initial condition $\mu$. ∎

In Section 4 we will formulate Corollary 3.2 for solutions of stochastic differential equations, cf. Theorem 4.1. The next result is an important step to prove Theorem 1.1.

Lemma 3.3 ()

Let $L$ be a pseudo-differential operator with continuous negative definite symbol $p$ and characteristics $(b,Q,\nu)$ such that $p(\cdot,0)=0$ and $L(C_{c}^{\infty}(\mathds{R}^{d}))\subseteq C_{\infty}(\mathds{R}^{d})$. Assume that $\nu(x,\{y\in\mathds{R}^{d};|y|\geq|x|/2\})=0$ for $|x|\gg 1$ and

[TABLE]

1. (i).

For any initial distribution $\mu$ there exists a probability measure $\mathds{P}^{\mu}$ on $D([0,\infty),\mathds{R}^{d})$ such that the canonical process $(Y_{t})_{t\geq 0}$ solves the $(L,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem and

[TABLE] 2. (ii).

For any $t\geq 0$, $R>0$ and $\varepsilon>0$ there exist constants $\varrho>0$ and $\delta>0$ such that

[TABLE]

for any initial distribution $\mu$ such that $\mu(B(0,\varrho))\leq\delta$. 3. (iii).

For any $t\geq 0$, $\varepsilon>0$ and any compact set $K\subseteq\mathds{R}^{d}$ there exists $R>0$ such that

[TABLE]

Proof.

(i) is a direct consequence of Corollary 3.2; we have to prove (ii) and (iii). To keep notation simple we show the result only in dimension $d=1$. Since $L$ maps $C_{c}^{\infty}(\mathds{R})$ into $C_{\infty}(\mathds{R})$, the symbol $p$ is locally bounded, cf. [1, Proposition 2.27(d)], and therefore Lemma 3.1 shows that 3.1(i)–(iii) hold for all $x\in\mathds{R}$. Set $u(x):=1/(1+|x|^{2})$, $x\in\mathds{R}$, then

[TABLE]

Clearly, $|Lu(x)|\leq I_{1}+I_{2}$ where

[TABLE]

for all $|x|\gg 1$. By Lemma 3.1 and (9) there exists a constant $c_{1}>0$ such that $I_{1}\leq c_{1}u(x)$ for all $x\in\mathds{R}$. On the other hand, Taylor’s formula shows

[TABLE]

for some intermediate value $\zeta=\zeta(x,y)$ between $x$ and $x+y$. Since $|y|<|x|/2$, we have $|\zeta|\geq|x|/2$; hence, by (9),

[TABLE]

Applying Lemma 3.1, we find that there exists a constant $c_{2}>0$ such that

[TABLE]

Consequently, $|Lu(x)|\leq(c_{1}+c_{2})u(x)$ for all $|x|\gg 1$. As $Lu$ is bounded and $u$ is bounded away from [math] on compact sets, we can choose a constant $c_{3}>0$ such that

[TABLE]

Define $\tau_{R}:=\inf\{t\geq 0;|Y_{t}|<R\}$. Using a standard truncation and stopping technique it follows that

[TABLE]

Hence, by ($\star$ ‣ 3),

[TABLE]

An application of Gronwall’s inequality shows that there exists a constant $C>0$ such that

[TABLE]

By the Markov inequality, this implies that

[TABLE]

If $\mu$ is an initial distribution such that $\mu(B(0,\varrho))\leq\delta$, then $\mathds{E}^{\mu}u(Y_{0})\leq\delta+\varrho^{-2}$. Choosing $\varrho$ sufficiently large and $\delta>0$ sufficiently small, we get (7). The proof of (iii) is similar. If we set $v(x):=x^{2}+1$, then there exists by Lemma 3.1 a constant $c>0$ such that $|Lv(x)|\leq cv(x)$ for all $x\in\mathds{R}$. Applying Gronwall’s inequality another time, we find a constant $C>0$ such that

[TABLE]

where $\sigma_{R}:=\inf\{t\geq 0;|Y_{t}|\geq R\}$ denotes the exit time from the ball $B(0,R)$. Hence, by the Markov inequality,

[TABLE]

In particular we can choose for any compact set $K\subseteq\mathds{R}$ and any $\varepsilon>0$ some $R>0$ such that

[TABLE]

Now if $\mu$ is an initial distribution such that $\mu(K^{c})\leq\varepsilon/2$, then, by (6),

[TABLE]

For the proof of Theorem 1.1 we will use the following result which follows e. g. from [4, Theorem 4.1.16, Proof of Corollary 4.6.4].

Lemma 3.4 ()

Let $A$ be a pseudo-differential operator with negative definite symbol $q$ such that $A:C_{c}^{\infty}(\mathds{R}^{d})\to C_{b}(\mathds{R}^{d})$. If the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem is well-posed and the unique solution $(X_{t})_{t\geq 0}$ satisfies the compact containment condition

[TABLE]

for any compact set $K\subseteq\mathds{R}^{d}$, then $x\mapsto\mathds{E}^{x}f(X_{t})$ is continuous for all $f\in C_{b}(\mathds{R}^{d})$.

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1.

The well-posedness implies that the solution $(X_{t})_{t\geq 0}$ is a Markov process, see e. g. [2, Theorem 4.4.2], and by Corollary 3.2 the (unique) solution is conservative. In order to prove that $(X_{t})_{t\geq 0}$ is a Feller process, we have to show that the semigroup $T_{t}f(x):=\mathds{E}^{x}f(X_{t})$, $f\in C_{\infty}(\mathds{R}^{d})$, has the following properties, cf. [1, Lemma 1.4]:

1. (i).

continuity at $t=0$: $T_{t}f(x)\to f(x)$ as $t\to 0$ for any $x\in\mathds{R}^{d}$ and $f\in C_{\infty}(\mathds{R}^{d})$. 2. (ii).

Feller property:  $T_{t}(C_{\infty}(\mathds{R}^{d}))\subseteq C_{\infty}(\mathds{R}^{d})$ for all $t\geq 0$.

The first property is a direct consequence of the right-continuity of the sample paths and the dominated convergence theorem. Since we know that the martingale problem is well posed, it suffices to construct a solution to the martingale problem satisfying (ii). Write $\nu(x,dy)=\nu_{s}(x,dy)+\nu_{l}(x,dy)$ where

[TABLE]

are the small jumps and large jumps, respectively, and denote by $p$ the symbol with characteristics $(b,Q,\nu_{s})$. By Corollary 3.2 there exists for any initial distribution $\mu$ a conservative solution to the $(p(x,D),C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem, and the solution satisfies 3.3(ii) and 3.3(iii). Using the same reasoning as in [2, proof of Proposition 4.10.2] it is possible to show that we can use interlacing to construct a solution to the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem with initial distribution $\mu=\delta_{x}$:

[TABLE]

where

• •

$\tau_{k}:=\inf\{t\geq 0;N_{t}=k\}=\sum_{j=1}^{k}\sigma_{j}$ are the jump times of a Poisson process $(N_{t})_{t\geq 0}$ with intensity $\lambda:=\sup_{z\in\mathds{R}^{d}}\nu_{l}(z,\mathds{R}^{d}\backslash\{0\})$, i. e. $\sigma_{j}\sim\operatorname{Exp}(\lambda)$ are independent and identically distributed. Note that $\lambda<\infty$ by Lemma 3.1.

• •

$(Y^{(k,\mu_{k})}_{t})_{t\geq 0}:=(Y^{(k)}_{t})_{t\geq 0}$ is a solution to the $(p(x,D),C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem with initial distribution

[TABLE]

for $k\geq 1$ and $\mu_{0}(dy):=\delta_{x}(dy)$. Moreover, $Y^{(k)}$ and $(\sigma_{j})_{j\geq k+1}$ are independent for all $k\geq 0$.

• •

$\mathds{P}^{x}$ is a probability measure which depends on the initial distribution $\mu=\delta_{x}$ of $(X_{t})_{t\geq 0}$.

Note that if we define a linear operator $P$ by

[TABLE]

then (8) implies that

[TABLE]

Before we proceed with the proof, let us give a remark on the construction of $(X_{t})_{t\geq 0}$. The intensity of the Poisson process $(N_{t})_{t\geq 0}$, which announces the “large jumps”, is $\lambda=\sup_{z}\lambda(z)$ where $\lambda(z):=\nu_{l}(z,\mathds{R}^{d}\backslash\{0\})$ is the “state-space dependent intensity” of the large jumps. Roughly speaking the second term on the right-hand side of (10) is needed to thin out the large jumps; with probability $\lambda^{-1}\mathds{E}^{x}((\lambda-\lambda(Y_{\sigma_{k-1}-}^{(k-1)}))$ there is no large jump at time $\sigma_{k-1}$, and therefore the effective jump intensity at time $t=\sigma_{k-1}$ is $\lambda(Y_{\sigma_{k-1}-}^{(k-1)})$.

We will prove that $(X_{t})_{t\geq 0}$ has the Feller property. To this end, we first show that for any $t\geq 0$, $\varepsilon>0$, $k\geq 1$ and any compact set $K\subseteq\mathds{R}^{d}$ there exists $R>0$ such that

[TABLE]

we prove (13) by induction. Note that $\mu_{j}=\mu_{j}(x)$ depends on the initial distribution of $(X_{t})_{t\geq 0}$.

• •

$k=0$: This is a direct consequence of Lemma 3.3(ii) since $\mu_{0}(dy)=\delta_{x}(dy)$.

• •

$k\to k+1$: Because of Lemma 3.3(ii) and the induction hypothesis, it suffices to show that there exists a compact set $C\subseteq\mathds{R}^{d}$ such that $\mathds{P}^{x}(Y_{0}^{(k+1,\mu_{k+1})}\notin C)\leq\varepsilon/2$ for all $x\in K$. Choose $m\geq 0$ sufficiently large such that $\mathds{P}^{x}(\sigma_{k}\geq m)\leq\varepsilon^{\prime}:=\varepsilon/8$, and choose $R>0$ such that (13) holds with $\varepsilon:=\varepsilon^{\prime}$, $t:=m$. Then, by (12) and our choice of $R$,

[TABLE]

which implies for $r>R$, $x\in K$

[TABLE]

The second term on the right-hand side converges to [math] as $r\to\infty$, cf.  [13, Theorem 4.4] or [10, Theorem A.1], and therefore we can choose $r>0$ sufficiently large such that $\mathds{P}^{x}(|Y_{0}^{(k+1)}|\geq r)\leq 4\varepsilon^{\prime}=\varepsilon/2$ for all $x\in K$.

For fixed $\varepsilon>0$ choose $k\geq 1$ such that $\mathds{P}^{x}(N_{t}\geq k+1)\leq\varepsilon$. By definition of $(X_{t})_{t\geq 0}$ and (13), we get

[TABLE]

Thus, by Lemma 3.4, $x\mapsto T_{t}f(x)=\mathds{E}^{x}f(X_{t})$ is continuous for any $f\in C_{\infty}(\mathds{R}^{d})$. It remains to show that $T_{t}f$ vanishes at infinity; to this end we will show that for any $r>0$, $\varepsilon>0$ there exists a constant $M>0$ such that

[TABLE]

It follows from Lemma 3.1 and the very definition of $\lambda$ that $Pf$ defined in (11) is bounded and

[TABLE]

i. e. $Pf$ vanishes at infinity for any $f\in C_{\infty}(\mathds{R}^{d})$. We claim that for any $k\geq 0$, $\varepsilon>0$, $t\geq 0$ and $r>0$ there exists a constant $M>0$ such that

[TABLE]

We prove (15) by induction.

• •

$k=0$: This follows from Lemma 3.3(ii) since $\mu_{0}(dy)=\delta_{x}(dy)$.

• •

$k\to k+1$: For fixed $r>0$ choose $\delta>0$ and $\varrho>0$ as in 3.3(ii). By 3.3(ii) it suffices to show that there exists $M>0$ such that

[TABLE]

(Note that $\mu_{k+1}=\mu_{k+1}(x)$ depends on the initial distribution of $(X_{t})_{t\geq 0}$.) Pick a cut-off function $\chi\in C_{c}^{\infty}(\mathds{R}^{d})$ such that $\mathds{1}_{B(0,\varrho)}\leq\chi\leq\mathds{1}_{B(0,\varrho+1)}$, then by (10),

[TABLE]

If $\|P\chi\|_{\infty}=0$ this proves ($\star$ ‣ • ‣ 3). If $\|P\chi\|_{\infty}>0$, then we can choose $m\geq 1$ such that $\mathds{P}^{x}(\sigma_{1}\geq m)\leq\delta/(2\|P\chi\|_{\infty})$. Since $P\chi$ vanishes at infinity, we have $\sup_{|z|\geq R}|P\chi(z)|\leq\lambda\delta/4$ for $R>0$ sufficiently large. By the induction hypothesis, there exists $M>0$ such that (15) holds with $\varepsilon:=\lambda\delta/4$, $r:=R$ and $t:=m$. Then

[TABLE]

for all $s\leq m$ and $|x|\geq M$, and therefore

[TABLE]

For fixed $\varepsilon>0$ and $t\geq 0$ choose $k\geq 1$ such that $\mathds{P}^{x}(N_{t}\geq k+1)\leq\varepsilon$. Choose $M>0$ as in (15), then

[TABLE]

Consequently, we have shown that $(X_{t})_{t\geq 0}$ is a Feller process. Since $(X_{t})_{t\geq 0}$ solves the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem, we have

[TABLE]

where $\tau_{r}^{x}:=\inf\{t\geq 0;|X_{t}-x|\geq r\}$ denotes the exit time from the ball $B(x,r)$. Using that $A(C_{c}^{\infty}(\mathds{R}^{d}))\subseteq C_{\infty}(\mathds{R}^{d})$, it is not difficult to see that the generator of $(X_{t})_{t\geq 0}$ is, when restricted to $C_{c}^{\infty}(\mathds{R}^{d})$, a pseudo-differential operator with symbol $q$, see e. g. [7, Proof of Theorem 3.5, Step 2] for details. This means that $(X_{t})_{t\geq 0}$ is a rich Feller process with symbol $q$. ∎

Proof of Corollary 1.3.

By Corollary 3.2 there exists for any initial distribution $\mu$ a solution to the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem, and by assumption the martingale problem for the pseudo-differential operator $A_{k}$ with symbol $q_{k}$ is well-posed. Therefore [3, Theorem 5.3], see also [2, Theorem 4.6.2], shows that the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem is well-posed. Now the assertion follows from Theorem 1.1. ∎

4 Applications

In this section we apply our results to Lévy-driven stochastic differential equations (SDEs) and stable-like processes. Corollary 3.2 gives the following general existence result for weak solutions to Lévy-driven SDEs.

Theorem 4.1 ()

Let $(L_{t})_{t\geq 0}$ be a $k$-dimensional Lévy process with characteristic exponent $\psi$ and Lévy triplet $(b,Q,\nu)$. Let $\ell:\mathds{R}^{d}\to\mathds{R}^{d}$, $\sigma:\mathds{R}^{d}\to\mathds{R}^{d\times k}$ be continuous functions which grow at most linearly. If

[TABLE]

then the SDE

[TABLE]

has for any initial distribution $\mu$ a weak solution $(X_{t})_{t\geq 0}$ which is conservative.

Note that (16) is, in particular, satisfied if

[TABLE]

e. g. if $\sigma$ is at most of sublinear growth, cf. Lemma 2.1(ii).

Proof.

Denote by $A$ the pseudo-differential operator with symbol $q(x,\xi):=-i\ell(x)\cdot\xi+\psi(\sigma(x)^{T}\xi)$. Since $q$ is locally bounded and $x\mapsto q(x,\xi)$ is continuous for all $\xi\in\mathds{R}^{d}$ it follows from (17) that $A(C_{c}^{\infty}(\mathds{R}^{d}))\subseteq C_{\infty}(\mathds{R}^{d})$, cf. Lemma 2.1. Because $\ell$, $\sigma$ are at most of linear growth, $q$ satisfies the growth condition (G). Applying Corollary 3.2 we find that there exists a conservative solution $(X_{t})_{t\geq 0}$ to the $(A,C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem. By [6], $(X_{t})_{t\geq 0}$ is a weak solution to the SDE (17). ∎

For $\alpha\in(0,1]$ we denote by

[TABLE]

the space of (locally) Hölder continuous functions with Hölder exponent $\alpha$.

Theorem 4.2 ()

Let $(L_{t})_{t\geq 0}$ be a $k$-dimensional Lévy process with Lévy triplet $(b,Q,\nu)$ and characteristic exponent $\psi$. Suppose that there exist $\alpha,\beta\in(0,1]$ such that the Lévy-driven SDE

[TABLE]

has a unique weak solution for any initial distribution $\mu$ and any two bounded functions $f\in\mathcal{C}^{\alpha}(\mathds{R}^{d},\mathds{R}^{d})$ and $g\in\mathcal{C}^{\beta}(\mathds{R}^{d},\mathds{R}^{d\times k})$ such that

[TABLE]

for some constant $c>0$. Then the SDE

[TABLE]

has a unique weak solution for any $\ell\in C^{\alpha}_{\operatorname{loc}}(\mathds{R}^{d},\mathds{R}^{d})$, $\sigma\in C^{\beta}_{\operatorname{loc}}(\mathds{R}^{d},\mathds{R}^{d\times k})$ which are at most of linear growth and satisfy

[TABLE]

and

[TABLE]

The unique weak solution is a conservative rich Feller process with symbol

[TABLE]

Proof.

Let $\ell\in\mathcal{C}^{\alpha}_{\operatorname{loc}}(\mathds{R}^{d},\mathds{R}^{d})$ and $\sigma\in\mathcal{C}^{\beta}_{\operatorname{loc}}(\mathds{R}^{d},\mathds{R}^{d\times k})$ be two functions which grow at most linearly and satisfy (18), (19). Lemma 2.1 shows that the pseudo-differential operator $A$ with symbol $q$ satisfies $A(C_{c}^{\infty}(\mathds{R}^{d}))\subseteq C_{\infty}(\mathds{R}^{d})$. Moreover, since $\sigma$, $\ell$ are at most of linear growth, the growth condition (G) is clearly satisfied. Set

[TABLE]

By assumption, the SDE

[TABLE]

has a unique weak solution for any initial distribution $\mu$ for all $k\geq 1$. By [6] (see also [7, Lemma 3.3]) this implies that the $(A_{k},C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem for the pseudo-differential operator with symbol $q_{k}(x,\xi):=-i\ell_{k}(x)\cdot\xi+\psi(\sigma_{k}(x)^{T}\xi)$ is well-posed. Since $\sigma_{k}$ is bounded, we have

[TABLE]

and therefore Lemma 2.1 shows that $A_{k}$ maps $C_{c}^{\infty}(\mathds{R}^{d})$ into $C_{\infty}(\mathds{R}^{d})$. Now the assertion follows from Corollary 1.3. ∎

Applying Theorem 4.2 we obtain the following generalization of [9, Corollary 4.7], see also [11, Theorem 5.23].

Theorem 4.3 ()

Let $(L_{t})_{t\geq 0}$ be a one-dimensional Lévy process such that its characteristic exponent $\psi$ satisfies the following conditions:

1. (i).

$\psi$* has a holomorphic extension $\Psi$ to*

[TABLE]

for some $m\geq 0$ and $\vartheta\in(0,\pi/2)$.

2. (ii).

There exist $\alpha\in(0,2]$, $\beta\in(1,2)$ and constants $c_{1},c_{2}>0$ such that

[TABLE]

and

[TABLE] 3. (iii).

There exists a constant $c_{3}>0$ such that $|\Psi^{\prime}(z)|\leq c_{3}|z|^{\beta-1}$ for all $z\in U$, $|z|\gg 1$.

Let $\ell:\mathds{R}\to\mathds{R}$ and $\sigma:\mathds{R}\to(0,\infty)$ be two locally Hölder continuous functions which grow at most linearly. If

[TABLE]

then the SDE

[TABLE]

has a unique weak solution for any initial distribution $\mu$. The unique solution is a conservative rich Feller process with symbol $q(x,\xi):=-i\ell(x)\xi+\psi(\sigma(x)\xi)$.

Proof.

[9, Corollary 4.7] shows that the assumptions of Theorem 4.2 are satisfied, and this proves the assertion. ∎

Theorem 4.3 applies, for instance, to Lévy processes with the following characteristic exponents:

1. (i).

(isotropic stable) $\psi(\xi)=|\xi|^{\alpha}$, $\xi\in\mathds{R}$, $\alpha\in(1,2]$, 2. (ii).

(relativistic stable) $\psi(\xi)=(|\xi|^{2}+\varrho^{2})^{\alpha/2}-\varrho^{\alpha}$, $\xi\in\mathds{R}$, $\varrho>0$, $\alpha\in(1,2)$, 3. (iii).

(Lamperti stable) $\psi(\xi)=(|\xi|^{2}+\varrho)_{\alpha}-(\varrho)_{\alpha}$, $\xi\in\mathds{R}$, $\varrho>0$, $\alpha\in(1/2,1)$, where $(r)_{\alpha}:=\Gamma(r+\alpha)/\Gamma(r)$ denotes the Pochhammer symbol, 4. (iv).

(truncated Lévy process) $\psi(\xi)=(|\xi|^{2}+\varrho^{2})^{\alpha/2}\cos(\alpha\arctan(\varrho^{-1}|\xi|))-\varrho^{\alpha}$, $\xi\in\mathds{R}$, $\alpha\in(1,2)$, $\varrho>0$, 5. (v).

(normal tempered stable) $\psi(\xi)=(\kappa^{2}+(\xi-ib)^{2})^{\alpha/2}-(\kappa^{2}-b^{2})^{\alpha/2}$, $\xi\in\mathds{R}$, $\alpha\in(1,2)$, $b>0$, $|\kappa|>|b|$.

For further examples of Lévy processes satisfying the assumptions of Theorem 4.3 we refer to [9, 11].

We close this section with two further applications of Corollary 1.3. The first is an existence result for Feller processes with symbols of the form $p(x,\xi)=\varphi(x)q(x,\xi)$. Recall that $p(x,D)$ denotes the pseudo-differential operator with symbol $p$.

Theorem 4.4 ()

Let $A$ be a pseudo-differential operator with symbol $q$ such that $q(\cdot,0)=0$, $A(C_{c}^{\infty}(\mathds{R}^{d}))\subseteq C_{\infty}(\mathds{R}^{d})$ and

[TABLE]

Assume that for any continuous bounded function $\sigma:\mathds{R}^{d}\to(0,\infty)$ the $(\sigma(x)q(x,D),C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem for the pseudo-differential operator with symbol $\sigma(x)q(x,\xi)$ is well-posed. If $\varphi:\mathds{R}^{d}\to(0,\infty)$ is a continuous function such that

[TABLE]

and

[TABLE]

then there exists a conservative rich Feller process $(X_{t})_{t\geq 0}$ with symbol $p(x,\xi):=\varphi(x)q(x,\xi)$ and $(X_{t})_{t\geq 0}$ is the unique solution to the $(p(x,D),C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem.

Theorem 4.4 is more general than [10, Theorem 4.6]. Indeed: If there exists a rich Feller process $(X_{t})_{t\geq 0}$ with symbol $q$ and $C_{c}^{\infty}(\mathds{R}^{d})$ is a core for the infinitesimal generator of $(X_{t})_{t\geq 0}$, then, by [1, Theorem 4.2], there exists for any continuous bounded function $\sigma>0$ a rich Feller process with symbol $\sigma(x)q(x,\xi)$ and core $C_{c}^{\infty}(\mathds{R}^{d})$, and therefore the $(\sigma(x)q(x,D),C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem is well-posed, cf. [5, Theorem 4.10.3].

Proof of Theorem 4.4.

For given $\varphi$ define

[TABLE]

By assumption, the $(\varphi_{k}(x)q(x,D),C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem is well-posed. Moreover, it follows from the boundedness of $\varphi_{k}$ and the fact that $q(x,D)(C_{c}^{\infty}(\mathds{R}^{d}))\subseteq C_{\infty}(\mathds{R}^{d})$ that $\varphi_{k}(x)q(x,D)$ maps $C_{c}^{\infty}(\mathds{R}^{d})$ into $C_{\infty}(\mathds{R}^{d})$. On the other hand, (21) gives $p(x,D)(C_{c}^{\infty}(\mathds{R}^{d}))\subseteq C_{\infty}(\mathds{R}^{d})$, cf. Lemma 2.1. Applying Corollary 1.3 proves the assertion. ∎

4.5 Example

Let $\varphi:\mathds{R}^{d}\to(0,\infty)$ be a continuous fuction and $\alpha:\mathds{R}^{d}\to(0,2]$ a locally Hölder continuous function. If there exists a constant $c>0$ such that $\varphi(x)\leq c(1+|x|^{\alpha(x)})$ for all $x\in\mathds{R}^{d}$, then there exists a conservative rich Feller process $(X_{t})_{t\geq 0}$ with symbol

[TABLE]

and $(X_{t})_{t\geq 0}$ is the unique solution to the $(p(x,D),C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem.

Indeed: If we set

[TABLE]

then [8, Theorem 5.2] shows that there exists a rich Feller process with symbol $q_{j}(x,\xi):=|\xi|^{\alpha_{j}(x)}(x)$, and that $C_{c}^{\infty}(\mathds{R}^{d})$ is a core for the generator. By [1, Theorem 4.2], there exists for any continuous bounded function $\sigma>0$ a rich Feller process with symbol $\sigma(x)q_{j}(x,\xi)$ and core $C_{c}^{\infty}(\mathds{R}^{d})$. This implies that the $(\sigma(x)q_{j}(x,D),C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem is well posed, see e. g. [5, Theorem 4.10.3] or [8, Theorem 1.37]. Applying Theorem 4.4 we find that there exists a conservative rich Feller process with symbol $p_{j}(x,\xi):=\varphi(x)q_{j}(x,\xi)$, and that the $(p_{j}(x,D),C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem is well-posed. Now the assertion follows from Corollary 1.3.

Example 4.5 shows that Corollary 1.3 is useful to establish the existence of stable-like processes with unbounded coefficients. For relativistic stable-like processes we obtain the following general existence result.

Theorem 4.6 ()

Let $\alpha:\mathds{R}^{d}\to(0,2]$, $m:\mathds{R}^{d}\to(0,\infty)$ and $\kappa:\mathds{R}^{d}\to(0,\infty)$ be locally Hölder continuous functions. If

[TABLE]

and

[TABLE]

then there exists a conservative rich Feller process $(X_{t})_{t\geq 0}$ with symbol

[TABLE]

and $(X_{t})_{t\geq 0}$ is the unique solution to the $(q(x,D),C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem.

Note that $\kappa$ and $m$ do not need to be of linear growth; for instance if $\inf_{x}\alpha(x)>0$, then we can choose $m(x):=e^{|x|}$ and $\kappa(x):=(1+|x|^{k})$ for $k\geq 1$.

Proof of Theorem 4.6.

For a function $f:\mathds{R}^{d}\to\mathds{R}$ set

[TABLE]

and define

[TABLE]

Since $\kappa_{i}$, $\alpha_{i}$ and $m_{i}$ are bounded Hölder continuous functions which are bounded away from [math], it follows from [11], see also [8], that the $(q_{k}(x,D),C_{c}^{\infty}(\mathds{R}^{d}))$-martingale problem is well-posed. Consequently, the assertion follows from Corollary 1.3 if we can show that $q$ satisfies (G) and that the pseudo-differential operators $q(x,D)$ and $q_{i}(x,D)$, $i\geq 1$, map $C_{c}^{\infty}(\mathds{R}^{d})$ into $C_{\infty}(\mathds{R}^{d})$. An application of Taylor’s formula yields

[TABLE]

and by (22) this implies (G). It remains to prove the mapping properties of $q(x,D)$ and $q_{i}(x,D)$. Since $x\mapsto q_{i}(x,\xi)$ is continuous and

[TABLE]

it follows from Lemma 2.1 that $q_{i}(x,D)(C_{c}^{\infty}(\mathds{R}^{d}))\subseteq C_{\infty}(\mathds{R}^{d})$. To prove $q(x,D)(C_{c}^{\infty}(\mathds{R}^{d}))\subseteq C_{\infty}(\mathds{R}^{d})$ we note that $x\mapsto q(x,\xi)$ is continuous, and therefore it suffices to show by Lemma 2.1 that

[TABLE]

where $\nu(x,dy)$ is for each fixed $x\in\mathds{R}^{d}$ the Lévy measure of a relativistic stable Lévy process with parameters $(\kappa(x),m(x),\alpha(x))$. It is known that $\nu(x,dy)\leq c\kappa(x)e^{-|y|m(x)/2}\,dy$ on $B(0,1)^{c}$, and therefore

[TABLE]

For $|x|\gg 1$ and fixed $r>0$ we obtain from Taylor’s formula

[TABLE]

Acknowledgements

I would like to thank René Schilling for helpful comments and suggestions.