The Time-Dependent Symbol of a Non-homogeneous Itô Process and Corresponding Maximal Inequalities

Abstract

The probabilistic symbol is defined as the right-hand side derivative at time zero of the characteristic functions corresponding to the one-dimensional marginals of a time-homogeneous stochastic process. As described in various contributions to this topic, the symbol contains crucial information concerning the process. When leaving time-homogeneity behind, a modification of the symbol by inserting a time component is needed. In the present article, we show the existence of such a time-dependent symbol for non-homogeneous Itô processes. Moreover, for this class of processes, we derive maximal inequalities which we apply to generalize the Blumenthal–Getoor indices to the non-homogeneous case. These are utilized to derive several properties regarding the paths of the process, including the asymptotic behavior of the sample paths, the existence of exponential moments and the finiteness of p-variationa. In contrast to many situations where non-homogeneous Markov processes are involved, the space-time process cannot be utilized when considering maximal inequalities.

1 Introduction

The probabilistic symbol $p(x,\xi )$ of a Markov process X is the function $p:\mathbb {R}^d\times \mathbb {R}^d\rightarrow \mathbb {C}$ given by

$$\begin{aligned} p(x,\xi ):=- \lim _{t\downarrow 0}\mathbb {E}^x \frac{e^{i(X_{ \sigma \wedge t}-x)'\xi }-1}{t} \end{aligned}$$

(1)

if the limit exists and coincides for every $R>0$, where

$$\begin{aligned} \sigma :=\inf \{t\geqslant 0: \Vert X_t^x-x \Vert > R \}. \end{aligned}$$

This symbol proves to be a crucial concept for deriving a wide range of properties of the stochastic process, such as conservativeness (cf. [16], Theorem 5.5), asymptotic behavior (cf. [20], Theorems 3.11 and 3.12), strong $\gamma$-variation (cf. [19] Corollary 5.10), Hausdorff-dimension (cf. [17], Theorem 4) and Hölder conditions [18]. For a survey on recent results, we refer to [4, 14]. By now, all of these results are restricted to the time-homogeneous case.

Proving some of these results, the symbol is utilized to derive maximal inequalities. Inequalities of this kind have been proved for Lévy processes (cf. [12]), certain Feller processes (cf. [18]) and homogeneous diffusions with jumps (cf. [20]). One can find a throughout discussion of maximal inequalities for various classes of stochastic processes in [8]. Formulating these maximal inequalities, Blumenthal–Getoor indices are used (cf. [18, 20]) which allow for a governance of the process’s paths by the behavior of the symbol in the variable $\xi$, and, therefore, for the derivation of the properties stated above.

However, when leaving the time-homogeneity behind one does not expect the symbol, being the right-hand side derivative of the characteristic functions corresponding to the one-dimensional marginal at time zero, to yield any information regarding the entire process. To overcome this, a time component is added to the symbol or more precisely

$$\begin{aligned} p(\tau ,x,\xi ) := -\lim _{h \downarrow 0} \mathbb {E}^{\tau ,x} \left( \frac{e^{i(X_{ \sigma \wedge (\tau +h}))'\xi }-1}{h} \right) \text { for } x,\xi \in \mathbb {R}^d, \ \tau \geqslant 0. \end{aligned}$$

(2)

Moreover, the existence of such a time-dependent probabilistic symbol was shown for rich càdlàg Feller evolution processes, i.e., non-homogeneous càdlàg Markov processes such that

$$\begin{aligned} T_{\tau ,t}u(x):= \mathbb {E}^{\tau ,x}u(X_t) \end{aligned}$$

for $0\leqslant \tau \leqslant t$ and $u \in B_b(\mathbb {R}^d)$ forms a strongly continuous evolution system. In addition, the domain of the infinitesimal generator $A_\tau$

$$\begin{aligned} A_\tau f:= \lim _{h \downarrow 0} \frac{T_{\tau ,\tau +h}f-f}{h}, \quad \tau >0 \end{aligned}$$

given by

$$\begin{aligned} D(A_\tau ):= \left\{ f: \lim _{h \downarrow 0} \frac{T_{\tau ,\tau +h}f-f}{h} \text { exists } \right\} \end{aligned}$$

contains the test functions $C_c(\mathbb {R}^d)$. Theorem 4.5 of [13] shows that the generator $A_\tau |_{C_c^\infty }$ is a pseudo-differential operator with symbol $-q(\tau ,x,\xi )$, i.e.,

$$\begin{aligned} A_\tau f(x) = - \frac{1}{(2\pi )^{\frac{d}{2}}} \int _{\mathbb {R}^d} e^{ix'\xi }q(\tau ,x,\xi )\hat{f}(\xi ) \ \textrm{d}\xi \end{aligned}$$

for $\tau \geqslant 0, x,\xi \in \mathbb {R}^d$ and $\hat{f}$ is the Fourier transform of f. Moreover, it is shown that the symbol of the generator $q(\tau ,x,\xi )$ and the time-dependent probabilistic symbol defined in (2) $p(\tau ,x,\xi )$ coincide if the symbol is continuous in x.

In the present article, we prove the existence of the time-dependent symbol for non-homogeneous Itô processes (cf. Definition 2.2). We utilize this result to prove maximal inequalities, the existence of non-homogeneous generalizations of the Blumenthal–Getoor indices and an exemplary selection of properties of such processes. Before we do so, we fix some notations:

A family of $\sigma$-fields $(\mathcal {G}_t^\tau )_{0 \leqslant \tau \leqslant t}$ is called a two-parameter filtration if $\mathcal {G}_s^\tau \subset \mathcal {G}_t^\tau$ for all $0 \leqslant \tau \leqslant s \leqslant t$ and $\mathcal {G}_t^{\tau _2} \subset \mathcal {G}_t^{\tau _1}$ for $0\leqslant \tau _1 \leqslant \tau _2 \leqslant t$. The natural double filtration of X is denoted by $(\mathcal {F}^X)_t^\tau )_{0\leqslant \tau \leqslant t}$ and is defined as

$$\begin{aligned} \left( \mathcal {F}^X\right) _t^\tau := \sigma (X_s: \tau \leqslant s \leqslant t). \end{aligned}$$

Let $(\Omega ,\mathcal {M})$ be a measurable space equipped with the two-parameter filtration $(\mathcal {M}_t^\tau )_{0\leqslant \tau \leqslant t}$. We call a stochastic process X adapted to the two-parameter filtration if for all $0 \leqslant \tau \leqslant t$

$$\begin{aligned} \left( \mathcal {F}^X\right) _t^\tau \subset \mathcal {M}_t^\tau . \end{aligned}$$

We tacitly assume that every stochastic process $X:=(X_t)_{t \geqslant 0}$ is defined on a generic stochastic basis $(\Omega , \mathcal {A}, (\mathcal {A}_t)_{t \geqslant 0},\mathbb {P})$ takes values in $(\mathbb {R}^d, \mathcal {B}(\mathbb {R}^d))$ and is cádlág. Here, $\mathcal {B}(\mathbb {R}^d)$ is the $\sigma$-field of Lebesgue sets. Moreover, we call $\Delta X_t:= X_t - \lim _{s \uparrow t} X_s$ the jump of the process at time $t\geqslant 0$, and for a stopping time $\tau$ we call $X^\tau := X1_{\llbracket 0, \tau \rrbracket }+ X_\tau 1_{\llbracket \tau , \infty \llbracket }$ the stopped process. The stochastic interval $\llbracket \tau , \sigma \llbracket$ for two stopping times $\tau , \sigma$ is defined by $\{ (\omega , t) \in \Omega \times \mathbb {R}_+: \tau (\omega ) \leqslant t < \sigma (\omega )\}$. The stochastic intervals $\llbracket \tau , \sigma \rrbracket$, $\rrbracket \tau , \sigma \llbracket$, $\rrbracket \tau , \sigma \rrbracket$ are defined alike. Additionally, we define for $\omega \in \Omega$

$$\begin{aligned} \mu ^X(\omega ,\textrm{d}t,\textrm{d}x):= \sum _{s \geqslant 0} 1_{\{\Delta X_s(\omega ) \ne 0 \}} \delta _{(s,\Delta X_s(\omega ))}(\textrm{d}t,\textrm{d}x) \end{aligned}$$

to be the integer-valued random measure on $\mathbb {R}_+ \times \mathbb {R}^d$ associated with the jumps of the process X.

A (strong) Markov process $(\Omega , \mathcal {M}, ( \mathcal {M}_t^\tau )_{0\leqslant \tau \leqslant t}, (X_t)_{t \geqslant 0}, \mathbb {P}^{\tau ,x})_{\tau \in \mathbb {R}_+, x \in \mathbb {R}^d}$ satisfies

$$\begin{aligned} \mathbb {E}^{\tau ,x} \left[ f(X_t) \mid \mathcal {M}_s^\tau \right] = \mathbb {E}^{s,X_s}\left( f(X_t \right) ) ,\quad \mathbb {P}^{\tau ,x}\text {-a.s.} \end{aligned}$$

(3)

for all $\tau \leqslant s \leqslant t$ and all bounded Borel-measurable functions f. Moreover, every Markov process is normal, i.e., $\mathbb {P}^{\tau ,x}(X_\tau =x)=1$. For more information on Markov processes see [5, 24]. We associate an evolution system $(T_{\tau ,t})_{0\leqslant \tau \leqslant t}$ of operators on $B_b(\mathbb {R}^d)$ with every Markov process by setting

$$\begin{aligned} T_{\tau ,t} u(x):= \mathbb {E}^{\tau ,x} u(X_t). \end{aligned}$$

2 The Time-Dependent Probabilistic Symbol

In this section, $X:=(\Omega , \mathcal {M}, ( \mathcal {M}_t^\tau )_{0\leqslant \tau \leqslant t}, (X_t)_{t \geqslant 0}, \mathbb {P}^{\tau ,x})_{\tau \in \mathbb {R}_+, x \in \mathbb {R}^d}$ denotes a Markov process. Before we start with the main topic of this section, we properly define the space-time process of a Markov process. That is due to the fact that various different definitions are used in the literature. The following definitions follows [3].

Let $\hat{\Omega }:= \mathbb {R}_+ \times \Omega$ and the $\sigma$-field $\hat{\mathcal {M}}:= \{ B \subset \hat{\Omega }: B_s \in \mathcal {M}\ \forall s \in \mathbb {R}_+\}$ where $B_s$ denotes the s-slice of B for $s \geqslant 0$. We define a process $\hat{X}$ with values in ${ \hat{E}:=}\mathbb {R}_+ \times \mathbb {R}^d$ by

$$\begin{aligned} \hat{X}_t(\hat{\omega })= \hat{X}_t((c, \omega )):= (t+c, X_{t+c}(\omega )). \end{aligned}$$

Moreover, we set

$$\begin{aligned} \hat{\theta }_\tau : \hat{\Omega } \rightarrow \hat{\Omega }; (s,\omega ) \mapsto (s+\tau , \omega ) \end{aligned}$$

and

$$\begin{aligned} \mathbb {P}^{(\tau ,x)}(B):= \mathbb {P}^{\tau ,x} \left( \pi _0^{-1} \left( \hat{\theta }_\tau ^{-1} (\{\tau \} \times B_\tau ) \right) \right) \end{aligned}$$

(4)

where $\pi _0:\Omega \rightarrow \hat{\Omega }; \omega \mapsto (0, \omega )$, and $B \in \hat{\mathcal {M}}$.

We call the homogeneous Markov process

$$\begin{aligned} \hat{X}:=(\hat{\Omega }, {\hat{\mathcal {M}}}, {(\mathcal {F}^{\hat{X}})_{t\geqslant 0}}, (\hat{X}_t)_{t\geqslant 0}, (\hat{\theta }_t)_{t \geqslant 0},\hat{\mathbb {P}}^{(\tau ,x)})_{(\tau ,x)\in \mathbb {R}_+ \times \mathbb {R}^d} \end{aligned}$$

the space-time process associated with X. The transition probability function of X is given by

$$\begin{aligned} \hat{P}(t,(\tau ,x), B):= P(\tau ,x; t+\tau , B_{t+ \tau }), \end{aligned}$$

and for any $\hat{\mathcal {M}}$-measurable random variable Y it holds true that

$$\begin{aligned} Y=Y \circ \hat{\theta }_\tau \circ \pi _0, \quad \mathbb {P}^{(\tau ,x)}\text {-a.s.}, \end{aligned}$$

i.e., for $\mathbb {P}^{(\tau ,x)}$-almost all $(c,\omega ) \in \hat{\Omega }$ we have

$$\begin{aligned} Y(c,\omega )= Y(\tau ,\omega ). \end{aligned}$$

(5)

That is due to

$$\begin{aligned}&\mathbb {P}^{(\tau ,x)}(\{ (c,\omega ) \in \hat{\Omega }: Y(c,\omega )= Y(\tau ,\omega )\}) \\&\quad = \ \mathbb {P}^{\tau ,x}(\pi _0^{-1}(\hat{\theta }_\tau ^{-1}(\{\tau \} \times \{ \omega \in \Omega : Y(\tau ,\omega )= Y(\tau ,\omega )\})\\&\quad = \ \mathbb {P}^{\tau ,x}(\Omega )\\&\quad = \ 1. \end{aligned}$$

This will be used frequently throughout the following calculations.

Definition 2.1

We call a Markov process X non-homogeneous Markov semimartingale if for every $\mathbb {P}^{\tau ,x}$, $\tau \geqslant 0, x \in \mathbb {R}^d$ the process $(X_t)_{t \geqslant 0}$ is a semimartingale on $[\tau , \infty )$.

Definition 2.2

We call a non-homogeneous Markov semimartingale non-homogeneous Itô process if its characteristics $(B,C,\nu )$ are of the form

$$\begin{aligned} B_t^{(i)}&= \int _\tau ^t b^{(i)}(s,X_s) \ \textrm{d}s \quad \mathbb {P}^{\tau ,x}\text {-a.s.} \nonumber \\ C_t^{(ij)}&= \int _\tau ^t c^{(ij)}(s,X_s) \ \textrm{d}s \quad \mathbb {P}^{\tau ,x}\text {-a.s.}\nonumber \\ \nu ( ; \textrm{d}t,\textrm{d}x)&= \textrm{d}t N_t(X_t,\textrm{d}x) \quad \mathbb {P}^{\tau ,x}\text {-a.s.} \end{aligned}$$

(6)

for $t \geqslant \tau \geqslant 0, x \in { \mathbb {R}^d}$ and $i,j \in \{1,...,d\}$.

Definition 2.3

Let X be a Markov process and let

$$\begin{aligned} \sigma := \sigma _R^{\tau ,x}:= \inf \{ h \geqslant \tau : \Vert X^{\tau ,x}_{h} -x \Vert> R\}=\tau +\inf \{ h \geqslant 0: \Vert X^{\tau ,x}_{\tau +h} -x \Vert > R\} \end{aligned}$$

(7)

be the first exit time from the ball of radius $R>0$ after $\tau \geqslant 0$, and $\Vert \cdot \Vert$ the maximum norm. The function $p:\mathbb {R}_+\times \mathbb {R}^d \times \mathbb {R}^d \rightarrow \mathbb {C}$ defined by

$$\begin{aligned} p(\tau ,x,\xi ) := -\lim _{h \downarrow 0} \mathbb {E}^{\tau ,x} \left( \frac{e^{i(X^\sigma _{\tau +h}-x)'\xi }-1}{h} \right) \end{aligned}$$

(8)

is called the time-dependent probabilistic symbol of the process, if the limit exists for every $\tau \geqslant 0$ and $x,\xi \in \mathbb {R}^d$ independently of the choice of R.

Example 2.4

• 1. Let $(X_t)_{t \geqslant 0}$ be an additive process on $(\Omega , \mathcal {F}, (\mathcal {F}_t)_{t\geqslant 0},\mathbb {P})$ in the sense of Definition 1.6 of [15], i.e., X has independent increments, is stochastically continuous and càdlàg and starts in 0 at time $t=0$. We define a family of probability measures on $(\Omega , \mathcal {F})$ by

  $$\begin{aligned} \mathbb {P}^{\tau ,x}(X_t \in B):= \mathbb {P}( X_t -X_\tau \in B-x) \end{aligned}$$

  for $B \in { \mathcal {B}(\mathbb {R}^d)}$ and $\tau \in \mathbb {R}_+, x \in { \mathbb {R}^d}$. For this family, it holds true that $\mathbb {P}^{\tau ,x}(X_\tau =x)=1$, and $(\Omega , \mathcal {F}, (\mathcal {F}_t)_{t\geqslant 0},(X_t)_{t \geqslant 0} ,\mathbb {P}^{\tau ,x})_{\tau \geqslant 0, x \in { \mathbb {R}^d}}$ is a Markov process. Let, in addition, X be a semimartingale for all $\mathbb {P}^{\tau ,x}$ which is quasi-left-continuous, and possesses the characteristics $(B,C,\nu )$. Theorem II.4.15 of [7] provides the existence of a version of $(B,C,\nu )$ that is deterministic. Hence, in the following, we assume $(B,C,\nu )$ to be deterministic. By Corollary II.4.18 of [7], X has no fixed times of discontinuity, i.e.,

  $$\begin{aligned} \{t\geqslant 0: \nu (\{t\} \times { \mathbb {R}^d})>0\} = \emptyset . \end{aligned}$$

  Therefore, when calculating the time-dependent probabilistic symbol of X, we derive with Theorem II.4.15 of [7] that

  $$\begin{aligned}&p(\tau ,x,\xi )\\&\quad = -\lim _{h \downarrow 0} \mathbb {E}^{\tau ,x} \left( \frac{e^{i(X_{\tau +h}-x)'\xi }-1}{h} \right) \\&\quad = -\lim _{h \downarrow 0} \mathbb {E}\left( \frac{e^{i(X_{\tau +h}-X_\tau )'\xi }-1}{h} \right) \\&\quad = -\lim _{h \downarrow 0} \frac{e^{ i \xi (B_{\tau +h} -B_\tau ) - \frac{1}{2}\xi ' ( C_{\tau +h}-C_\tau )\xi + \int _{\mathbb {R}^d \setminus \{0\}}\left( e^{i\xi 'y}-1-i\xi 'y\chi (y)\right) \ \nu ((\tau ,\tau +h],\textrm{d}y) }-1}{h}. \end{aligned}$$

  This limit exists if and only if

  $$\begin{aligned} -\lim _{h \downarrow 0} \frac{ i \xi ' (B_{\tau +h} -B_\tau ) - \frac{1}{2}\xi ' ( C_{\tau +h}-C_\tau )\xi + \int _{\mathbb {R}^d \setminus \{0\}} \left( e^{i\xi 'y}-1-i\xi 'y\chi (y) \right) \ \nu ((\tau ,\tau +h],\textrm{d}y)}{h} \end{aligned}$$

  exists. If $B^{(i)}$ and $C^{(ij)}$ are right-differentiable for all $i,j \in \{1,...,d\}$ and if the function

  $$\begin{aligned} \tau \mapsto \int _{\mathbb {R}^d \setminus \{0\}} \left( e^{i\xi 'y}-1-i\xi 'y\chi (y)\right) \ \nu ((0,\tau ],\textrm{d}y) \end{aligned}$$

  is right-differentiable the time-dependent symbol exists and is of the form

  $$\begin{aligned} p(\tau ,x,\xi )=&\, -i \xi \partial _+ B_{\tau } + \frac{1}{2}\xi ' \partial _+ C_{\tau } \xi - \partial _+\\&\int _{0}^\tau \int _{\mathbb {R}^d \setminus \{0\}}\left( e^{i\xi 'y}-1-i\xi 'y\chi (y)\right) \ \nu (\textrm{d}s,\textrm{d}y), \end{aligned}$$

  where $\partial _+$ denotes the right-derivative.

• 2. Let $(X_t)_{t \geqslant 0}$ be a one-dimensional Brownian motion with variance function $\sigma ^2(t)$ on $(\Omega , \mathcal {F}, (\mathcal {F}_t)_{t\geqslant 0},\mathbb {P})$. The process X is additive and a continuous semimartingale and

  $$\begin{aligned} \mathbb {E}\left( e^{i(X_{t+\tau }-X_\tau )'\xi } \right) = \exp \left( \frac{1}{2}(\sigma ^2(t+\tau )-\sigma ^2(\tau ))\xi ^2 \right) . \end{aligned}$$

  By the previous example, the (non-homogeneous) probabilistic symbol exists if and only if the variance function is right-differentiable with right-derivative $\partial _+ \sigma ^2$. In this case, we have

  $$\begin{aligned} p(\tau ,x,\xi ) = \frac{1}{2}\xi ^2 \partial _+ \sigma ^2(\tau ). \end{aligned}$$

We have seen in (2) that under some mild conditions for a rich càdlàg Feller evolution process the symbol of the generator $q(\tau ,x,\xi )$ and the time-dependent probabilistic symbol $p(\tau ,x,\xi )$ coincide. Additionally, Corollary 3.5 of [3] states that the symbol of the generator of the homogeneous space-time process $\hat{X}$ corresponding to X is given by

$$\begin{aligned} q(\hat{x},\hat{\xi }) = -i\xi _0 + q(\tau ,x,\xi ) \end{aligned}$$

with $\tau \geqslant 0, x \in \mathbb {R}^d, \xi \in \mathbb {R}^d$, $\hat{x}=(\tau ,x),\hat{\xi }=(\xi _0,\xi ) \in \mathbb {R}^{d+1}$. Therefore, we expect the space-time process to be useful when calculating the symbol of a non-homogeneous Itô process, provided the characteristics of the space-time process are the ones of a homogeneous diffusion with jumps.

Hence, the following lemma states the characteristics of the space-time process associated to a non-homogeneous Markov semimartingale. Note that the proof is omitted since it is quite straightforward but needs tedious calculations.

Lemma 2.5

Let X be a non-homogeneous Markov semimartingale with characteristics $(B,C,\nu )$. In this case, the space-time process $\hat{X}$ associated with X is semimartingale for all $\mathbb {P}^{(\tau ,x)}, (\tau ,x)\in \mathbb {R}_+ \times \mathbb {R}^d$ and its characteristics $(\hat{B},\hat{C},\hat{\nu })$ are given by

$$\begin{aligned} \hat{B}_t(c, \omega )&= (t,B_{\tau +t}(\omega )), \quad \mathbb {P}^{(\tau ,x)}\text {-a.s.}\end{aligned}$$

(9)

$$\begin{aligned} \hat{C}_t(c, \omega )&= \begin{pmatrix} 0&\begin{matrix} \cdots &{} \hspace{-1mm} 0 \end{matrix} \\ \begin{matrix} \vdots \\ 0 \end{matrix} &{} \begin{matrix}\\ {C_{\tau +t}(\omega )} \end{matrix} \end{pmatrix}, \quad \mathbb {P}^{(\tau ,x)}\text {-a.s.} \end{aligned}$$

(10)

$$\begin{aligned} \hat{\nu }((c,\omega ); \textrm{d}s, \textrm{d}u \times \textrm{d}y)&= \nu (\omega ; \textrm{d}s+\tau , \textrm{d}y) \delta _0(\textrm{d}u), \quad \mathbb {P}^{(\tau ,x)}\text {-a.s.} \end{aligned}$$

(11)

for $\tau \leqslant t$ and $(c,\omega ) \in \hat{\Omega }$.

Before we state the main theorem of this section, the following lemma provides that non-homogeneous Itô process are indeed a generalization of rich Feller evolution processes.

Lemma 2.6

Let $(X_t)_{t\geqslant 0}$ be a rich Feller evolution process on $C_\infty (\mathbb {R}^d)$ with generator $A_s$ and time-dependent symbol $p: \mathbb {R}_+ \times \mathbb {R}^d \times \mathbb {R}^d \rightarrow \mathbb {C}$ such that $p(\cdot , x,\xi )$ is continuous for all $x, \xi \in \mathbb {R}^d$. Then, X is a non-homogeneous Itô process.

Proof

Let $(X_t)_{t\geqslant 0}$ be a rich Feller evolution system on $C_\infty (\mathbb {R}^d)$. Analogously to the proof of Theorem 3.1 of [23], but under usage of the non-homogeneous version of Dynkin’s formula as mentioned in [13], we show that $(X_t)_{t\geqslant 0}$ is a semimartingale with characteristics $(B,C,\nu )$. Since it is well-known that X is a Markov process, it suffices to show that the characteristics are of the form mentioned in Definition 2.2:

By Theorem 3.2 and Lemma 3.7 of [3] the space-time process $\hat{X}$ associated to X is a rich Feller process on $C_\infty (\mathbb {R}_+ \times \mathbb {R}^d)$. Hence, Theorem 3.10 of [23] provides that the characteristics $(\hat{B},\hat{C},\hat{\nu })$ of $\hat{X}$ are of the form

$$\begin{aligned} \hat{B}_t^{(i)}&= \int _0^t b^{(i)}(\hat{X}_s) \ \textrm{d}s, \quad \mathbb {P}^{(\tau ,x)}\text {-a.s.} \\ \hat{C}_t^{(ij)}&= \int _0^t c^{(ij)}(\hat{X}_s) \ \textrm{d}s, \quad \mathbb {P}^{(\tau ,x)}\text {-a.s.}\\ \hat{\nu }(\; ; \textrm{d}t,\textrm{d}\hat{x})&= \textrm{d}t N(\hat{X}_t,\textrm{d}\hat{x}), \quad \mathbb {P}^{(\tau ,x)}\text {-a.s.} \end{aligned}$$

and, therefore, we conclude with equations (9) to (11) that

$$\begin{aligned} B_t^{(i)}&= \int _0^{t-\tau } b^{(i)}(\hat{X}_s) \ \textrm{d}s= \int _0^{t-\tau } b^{(i)}(s+\tau ,X_{s+\tau }) \ \textrm{d}s= \int _\tau ^{t} b^{(i)}(s,X_{s}) \ \textrm{d}s, \\ C_t^{(ij)}&= \int _0^{t-\tau } c^{(ij)}(\hat{X}_s) \ \textrm{d}s= \int _0^{t-\tau } c^{(ij)}(s+\tau ,X_{s+\tau }) \ \textrm{d}s = \int _\tau ^{t} c^{(ij)}(s,X_{s}) \ \textrm{d}s, \\ \nu ( \; ; \textrm{d}t,\textrm{d}x)&= \textrm{d}t N(X_{t},\textrm{d}x) \end{aligned}$$

for $t \geqslant \tau$, where all equalities are meant $\mathbb {P}^{(\tau ,x)}\text {-a.s.}$. Transition from $\mathbb {P}^{(\tau ,x)}$ to $\mathbb {P}^{\tau ,x}$ yields the statement. $\square$

Theorem 2.7

Let X be a non-homogeneous Itô process and let $\ell = (\ell ^{(j)})_{1 \leqslant j \leqslant d}$ and $Q= (q^{(ik)})_{1 \leqslant j,k\leqslant d}$ be continuous, N be such that the function

$$\begin{aligned} (s,x) \mapsto \int _{y \ne 0} (1 \wedge y^2) \ N_s(x,\textrm{d}y) \end{aligned}$$

is continuous. In this case, the time-dependent symbol exists and equals

$$\begin{aligned} p(\tau ,x,\xi )= & {} -i \ell (\tau ,x)'\xi + \frac{1}{2} \xi ' Q(\tau ,x) \xi \\{} & {} - \int _{y \ne 0} \left( e^{iy' \xi } -1 - iy' \xi \cdot \chi (y) \right) \ N_\tau (x,\textrm{d}y). \end{aligned}$$

Proof

Let X be a non-homogeneous Itô process, and let $\hat{X}$ be the associated space-time process. The characteristics of $\hat{X}$ are given by

$$\begin{aligned} \hat{B}_t^{(1)}&= t = \int _0^t 1 \ \textrm{d}s\\ \hat{B}_t^{(i)}&= \int _\tau ^{t+\tau } \ell ^{(i-1)}(s,X_s) \ \textrm{d}s =\int _0^{t} \ell ^{(i-1)}(s+\tau ,X_{s+\tau }) \ \textrm{d}s =\int _0^{t} \ell ^{(i-1)}(\hat{X}_{s}) \ \textrm{d}s ,\\ \hat{C}_t^{(1j)}&= \int _0^{t} 0 \ \textrm{d}s \\ \hat{C}_t^{(ij)}&= \int _0^{t} c^{(i-1,j-1)}(\hat{X}_{s}) \ \textrm{d}s \end{aligned}$$

for $i,j \in \{2,...,d+1\}$. All of the equations above are meant $\hat{\mathbb {P}}^{(\tau ,x)} \text {-a.s.}$

For $T \in \mathcal {B}(\mathbb {R}_+)$ and $B \in \mathcal {B}(\hat{E})$ we have

$$\begin{aligned} \hat{\nu }( \; ; T,B)&= \nu (\; ; T+\tau , B_0) = \int _{T+\tau } N_{s}(X_{s},B_0)\ \textrm{d}s\\&=\int _{T} N_{s+\tau }(X_{s+\tau },B_0)\ \textrm{d}s =\int _{T} \hat{N}((s+\tau ,X_{s+\tau }),B)\ \textrm{d}s \\&=\int _{T} \hat{N}(\hat{X}_s,B)\ \textrm{d}s, \quad \mathbb {P}^{(\tau ,x)}\text {-a.s.} \end{aligned}$$

where $\hat{N}((c,\omega ),B)=N_c(\omega ,B_0)$ is a transition kernel from $\hat{\Omega }\times \hat{E}$ to $\mathbb {R}^d$. Consequently, $\hat{X}$ is a Itô process and we denote by $\hat{p}(\hat{x},\hat{\xi })$ its (homogeneous) probabilistic symbol.

For the stopping time $\hat{\sigma }:= \inf \{ h \geqslant 0: \Vert \hat{X}_h -(\tau ,x) \Vert > R \}$ it holds true that

$$\begin{aligned} \hat{\sigma }(\hat{\theta }_\tau (\pi _0(\omega )))&:= \inf \{ h \geqslant 0: \Vert \hat{X}_h(\hat{\theta }_\tau (\pi _0(\omega ))) -(\tau ,x) \Vert> R \}\\&= \inf \{ h \geqslant 0: \Vert \hat{X}_h(\tau ,\omega ) -(\tau ,x) \Vert> R \}\\&= \inf \{ h \geqslant 0: \Vert (\tau +h,X_{\tau +h}(\omega )) -(\tau ,x) \Vert> R \}\\&= \inf \{ h \geqslant 0: \Vert (h,X_{\tau +h}(\omega )-x) \Vert> R \}\\&= R \wedge \inf \{ h \geqslant 0: \Vert X_{\tau +h}(\omega )-x \Vert > R \}\\&= R \wedge ( \sigma -\tau ). \end{aligned}$$

We compute for $\hat{\xi }=(0,\xi )$, $\hat{\ell }=(1,\ell ), \hat{c}=(\hat{c}^{(i,j)}{ )_{i,j \in \{1,...,d+1\}}}$ with $\hat{c}^{(1,j)}=\hat{c}^{(i,1)}=0$ and $\hat{c}^{(i,j)}=c^{((i-1),(j-1))}$ for $i,j \in \{2,...,d+1\}$:

$$\begin{aligned} p(\tau ,x,\xi )&= -\lim _{h \downarrow 0} \mathbb {E}^{\tau ,x} \left( \frac{e^{i (X^\sigma _{\tau +h} -x)'\xi }-1}{h} \right) \\ {}&= -\lim _{h \downarrow 0} \mathbb {E}^{\tau ,x} \left( \frac{e^{i (X_{\tau +(h\wedge (\sigma -\tau ))} -x)'\xi }-1}{h} \right) \\ {}&= -\lim _{\underset{h<R}{h \downarrow 0}} \mathbb {E}^{\tau ,x} \left( \frac{e^{i (X_{\tau +(h\wedge (\sigma -\tau )\wedge R)} -x)'\xi }-1}{h} \right) \\ {}&= -\lim _{\underset{h<R}{h \downarrow 0}} \mathbb {E}^{\tau ,x} \left( \frac{e^{i (\hat{X}_{h\wedge (\hat{\sigma }\circ \hat{\theta }_\tau \circ \pi _0)} \circ \hat{\theta }_\tau \circ \pi _0 -(\tau ,x))'\hat{\xi }}-1}{h} \right) \\ {}&= -\lim _{\underset{h <R}{h \downarrow 0}} \hat{\mathbb {E}}^{(\tau ,x)} \left( \frac{e^{i (\hat{X}^{\hat{\sigma }}_h -(\tau ,x))'\hat{\xi }}-1}{h} \right) \\ {}&= \hat{p}((\tau ,x), \hat{\xi })\\ {}&= -i \ell (\tau ,x)'\hat{\xi }+ \frac{1}{2} \hat{\xi }' c(\tau ,x) \hat{\xi }\\&\qquad - \int _{\hat{y}\ne 0} \left( e^{i\hat{y}'\hat{\xi }}-1-i \hat{y}' \hat{\xi }\chi (\hat{y}) \right) \hat{N}((\tau ,x),\textrm{d}y)\delta _0(\textrm{d}u)\\ {}&= -i \ell (\tau ,x)'\xi + \frac{1}{2} \xi ' c(\tau ,x) \xi - \int _{y\ne 0} \left( e^{iy'\xi }-1-i y' \xi \chi (y) \right) N_\tau (x,\textrm{d}y). \end{aligned}$$

$\square$

We have observed, for instance in Example 2.4 that a process does not necessary need to be a non-homogeneous Itô process to possess a time-dependent symbol. However, the processes considered up to this point are quasi-left continuous. We will see in the subsequent example that, when leaving quasi-left continuity behind, the time-dependent symbol does not contain the same information on the process as before. This is not unexpected, as we have encountered similar situations in the homogeneous case.

Example 2.8

Let us consider the following (deterministic!) process:

$$\begin{aligned} X^{\tau ,x}_t(\omega ):= x+1_{[1,\infty )}(t-\tau ), \quad t \geqslant \tau , \end{aligned}$$

where $\tau \geqslant 0$ and $x \in \mathbb {R}.$ Here, we write $X^{\tau ,x}$ for the process $(X, \mathbb {P}^{\tau ,x})$ (cf. [21, 22]). The following figure depicts the path of $X^{0,0}$.

The process X is a non-homogeneous Markov process, and considering the cut-off function $\chi (x)=1_{\{|x| \leqslant 0.5\}}$ we receive a non-homogeneous Markov semimartingale with characteristics $B \equiv 0, C \equiv 0$ and $\nu (\textrm{d}t,\textrm{d}x)= \delta _1(\textrm{d}t)\delta _1(\textrm{d}x)$. In this case, the time-dependent symbol exists and is given by

$$\begin{aligned} p(\tau ,x,\xi ) = -\lim _{h \downarrow 0} \mathbb {E}^{\tau ,x} \left( \frac{e^{i (X^\sigma _{\tau +h} -x)'\xi }-1}{h} \right) \equiv 0. \end{aligned}$$

Nevertheless, the symbol does not provide any information regarding the jumps of the process.

3 Maximal Inequalities and Time-Dependent Blumenthal–Getoor Indices

As we have pointed out before, for homogeneous processes like multivariate $\alpha$-stable processes [1], more general Lévy processes [2], Feller processes satisfying some mild conditions [18] and homogeneous diffusion with jumps [20], there exists a set of indices, called Blumenthal–Getoor indices, which utilize the symbol to derive maximal inequalities. For a historical overview we refer to [14]. Equally, we now want to consider maximal inequalities for non-homogeneous processes with the help of the time-dependent symbol. In this framework, the non-homogeneous growth and sector condition of the symbol play an important role:

$$\begin{aligned}&\Vert q(s, x, \xi ) \Vert \leqslant c(1+\Vert \xi \Vert ^2), \end{aligned}$$

(IG)

$$\begin{aligned}&| \mathrm{Im\,}(q(s,x,\xi )) | \leqslant c_0 \ \mathrm{Re\,}(p(s,x,\xi )) \end{aligned}$$

(IS)

for every $s\geqslant 0, x,\xi \in \mathbb {R}^d$ and $c,c_0 >0$.

Specifically, we want to use the maximal inequalities to examine the paths of the process, including the asymptotic behavior of the sample paths, the p-variation of the paths and the existence of the exponential moments of the process. The following indices generalize the Blumenthal–Getoor indices as defined in Definitions 4.2 and 4.5 of [18] or Definition 3.8 of [20] to the non-homogeneous case.

Definition 3.1

The quantities

$$\begin{aligned} \beta _0&:=\sup \left\{ \lambda \geqslant 0 : \limsup _{R\rightarrow \infty } R^\lambda H(R) =0 \right\} \\ \underline{\beta _0}&:=\sup \left\{ \lambda \geqslant 0 : \liminf _{R\rightarrow \infty } R^\lambda H(R) =0 \right\} \\ \overline{\delta _0}&:=\sup \left\{ \lambda \geqslant 0 : \limsup _{R\rightarrow \infty } R^\lambda h(R) =0 \right\} \\ \delta _0&:=\sup \left\{ \lambda \geqslant 0 : \liminf _{R\rightarrow \infty } R^\lambda h(R) =0 \right\} \end{aligned}$$

are called time-dependent indices of X in the starting point, where

$$\begin{aligned} H(R)&:= \sup _{s \geqslant 0} \sup _{y \in \mathbb {R}^d} \sup _{\left\| \varepsilon \right\| \leqslant 1} \left| p\left( s,y,\frac{\varepsilon }{R}\right) \right| , \text { and} \end{aligned}$$

(12)

$$\begin{aligned} h(R)&:= \inf _{s \geqslant 0} \inf _{y \in \mathbb {R}^d} \; \sup _{\left\| \varepsilon \right\| \leqslant 1} \mathrm{Re\,}\left( p\left( s,y,\frac{\varepsilon }{4\kappa R} \right) \right) \end{aligned}$$

(13)

with $\kappa =(4 \arctan (1/2 c_0))^{-1}$ where $c_0$ comes from the sector condition (IS).

Definition 3.2

Let $\tau \in \mathbb {R}_+, x\in \mathbb {R}^d$ and $R>0$. The quantities

$$\begin{aligned} \beta _\infty ^{\tau ,x}&:=\inf \left\{ \lambda> 0 : \limsup _{R\rightarrow 0} R^\lambda H(\tau ,x,R) =0 \right\} \\ \underline{\beta _\infty ^{\tau ,x}}&:=\inf \left\{ \lambda> 0 : \liminf _{R\rightarrow 0} R^\lambda H(\tau ,x,R) =0 \right\} \\ \overline{\delta _\infty ^{\tau ,x}}&:=\inf \left\{ \lambda> 0 : \limsup _{R\rightarrow 0} R^\lambda h(\tau ,x,R) =0 \right\} \\ \delta _\infty ^{\tau ,x}&:=\inf \left\{ \lambda > 0 : \liminf _{R\rightarrow 0} R^\lambda h(\tau ,x,R) =0 \right\} \end{aligned}$$

are the time-dependent indices of X at infinity, where

$$\begin{aligned} H(\tau ,x,R)&:= \sup _{\left\| y-x \right\| \leqslant 2R} \sup _{\left\| \varepsilon \right\| \leqslant 1} \left| p\left( \tau ,y,\frac{\varepsilon }{R}\right) \right| , \text { and} \end{aligned}$$

(14)

$$\begin{aligned} h(\tau , x,R)&:= \inf _{\left\| y-x \right\| \leqslant 2R} \; \sup _{\left\| \varepsilon \right\| \leqslant 1} \mathrm{Re\,}\left( p\left( \tau ,y,\frac{\varepsilon }{4\kappa R} \right) \right) \end{aligned}$$

(15)

with $\kappa =(4 \arctan (1/2 c_0))^{-1}$ where $c_0$ comes from the sector condition (IS).

Example 3.3

• 1. Let $(X_t)_{t \geqslant 0}$ be a one-dimensional Brownian motion with right-differentiable variance function $\sigma ^2(t)$ such that $\partial _+ \sigma ^2(t)$ is bounded (cf. Definition I.4.9 of [7]). In this case, the time-dependent symbol is given by

  $$\begin{aligned} p(\tau ,x,\xi ) = \frac{1}{2}\xi ^2 \partial _+ \sigma ^2(\tau ). \end{aligned}$$

  Additionally, we have

  $$\begin{aligned} H(R)&= \frac{1}{2}\frac{1}{R^2} \sup _{s \geqslant 0} \left| \partial _+ \sigma ^2(s) \right| ,\\ H(s,x,R)&= \frac{1}{2}\frac{1}{R^2}\left| \partial _+ \sigma ^2(s) \right| ,\\ h(R)&= \frac{1}{2}\frac{1}{4^2 \kappa ^2 R^2} \inf _{s \geqslant 0} (\partial _+ \sigma ^2(s)), \\ h(\tau ,x,R)&= \frac{1}{2}\frac{1}{4^2 \kappa ^2 R^2} (\partial _+ \sigma ^2(s)), \end{aligned}$$

  and, therefore,

  $$\begin{aligned} \beta _0 =\underline{\beta _0}= \beta _\infty ^{\tau ,x}= \underline{\beta _\infty ^{\tau ,x}}= 2 =\delta _0=\overline{\delta _0} = \delta ^{\tau ,x}_\infty = \overline{\delta ^{\tau ,x}_\infty }. \end{aligned}$$

• 2. Let $(X_t)_{t \geqslant 0}$ the unique solution of the SDE

  $$\begin{aligned} dX_t = \mu (X_t,t)\ \textrm{d}t + \sigma (X_t,t) \ dB_t, \quad t \geqslant 0, \end{aligned}$$

  where $\mu ,\sigma \in C^{1,1}_b(\mathbb {R}\times \mathbb {R}_+)$ non-decreasing, $\sigma >0$ and $(B_t)_{t \geqslant 0}$ is a standard Brownian motion. By Lemma 4.6 of [13], the time-dependent symbol of X is given by

  $$\begin{aligned} p(\tau ,x,\xi )=-i\mu (x,\tau )\xi + \frac{1}{2}\sigma (x,\tau )^2\xi ^2. \end{aligned}$$

  Hence,

  $$\begin{aligned} H(R)&= \frac{1}{R}\sup _{s \geqslant 0} \sup _{x \in \mathbb {R}} \left| \frac{1}{2R} \sigma (x,s)^2+i\mu (x,s) \right| \\ h(R)&=\frac{1}{R^2}\inf _{s \geqslant 0} \inf _{x \in \mathbb {R}} \frac{1}{2}\sigma (x,s)^2\\ H(s,x,R)&= \frac{1}{R} \left| \frac{1}{2R}\sigma (x+2R,s)^2+i \mu (x+2R,s) \right| \\ h(s,x,R)&= \frac{1}{R^2} \frac{1}{2}\sigma (x+2R,s)^2, \end{aligned}$$

  and, therefore,

  $$\begin{aligned} \beta _0&= \underline{\beta _0}=1,\\ \overline{\delta _0}&=\delta _0= 2,\\ \beta ^{\tau ,x}_\infty&= \underline{\beta ^{\tau ,x}_\infty }= \left( 2-\sup \left\{ \lambda> 0 : \lim _{R\rightarrow 0} \frac{\sigma (x+2R,\tau )^2}{R^{\lambda }} =0 \right\} \right) \wedge \\&\quad \hspace{1.1cm} \left( 1-\sup \left\{ \lambda> 0 : \lim _{R\rightarrow 0} \frac{\mu (x+2R,\tau ) =0}{R^{\lambda }} \right\} \right) , \\ \overline{\delta ^{\tau ,x}_\infty }&= \delta ^{\tau ,x}_\infty = 2- \sup \left\{ \lambda > 0 : \lim _{R\rightarrow 0} R^{-\lambda }\sigma (x+2R,\tau )^2 =0 \right\} . \end{aligned}$$

The proofs of the previous section crucially rely on the space-time process to transfer properties of homogeneous Markov processes to the non-homogeneous framework. However, when deriving properties like the asymptotic behavior of sample paths or maximal inequalities we do not expect the space-time process to be of much use. This is due to the fact that if we utilize the space-time process, a deterministic drift with slope 1 is added. This obscures the path-behavior of the original process. Additionally, the indices of the space-time process do, in general, not allow for a calculation of the time-dependent indices of the underlying process. Considering the symbol of the space-time process

$$\begin{aligned} q(\hat{x},\hat{\xi }) = -i\xi _0 + q(\tau ,x,\xi ) \end{aligned}$$

with $\tau \geqslant 0, x \in \mathbb {R}^d, \xi \in \mathbb {R}^d$, $\hat{x}=(\tau ,x),\hat{\xi }=(\xi _0,\xi ) \in \mathbb {R}^{d+1}$ and the definition of the indices, one easily sees that when calculating the (homogeneous) indices of the space-time process a linear term (in R) is added to the functions H resp. h along with a supremum in time for the indices at infinity of the space-time process.

In this section, we assume all characteristics to be encountered to be with respect to the truncation function $id\cdot \chi$, where $\chi \in C_c^\infty (\mathbb {R}^d)$ is a symmetric cut-off function with

$$\begin{aligned} 1_{B_{ R}(0)} \leqslant \chi \leqslant 1_{B_{2R}(0)} \end{aligned}$$

where $R >0$.

Theorem 3.4

Let X be a non-homogeneous Itô process with characteristics as in Theorem 2.7. In this case, we have

$$\begin{aligned} \mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s- x \Vert \geqslant R \right) \leqslant c_d \cdot t \cdot \sup _{\tau < s \leqslant \tau +t} H(s,x,R) \end{aligned}$$

(16)

for $t\geqslant 0$, $R>0$ and a constant $c_d>0$ which only depends on the dimension d. If, in addition, (IS) holds true, we have

$$\begin{aligned} \mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s- x \Vert< R \right) \leqslant c_k \cdot \frac{1}{t} \cdot \frac{1}{\inf _{\tau < s \leqslant \tau +t} \ h(s,x,R)} \end{aligned}$$

(17)

where $c_k>0$ only depends on $c_0$ of the sector condition (IS).

Remark 3.5

Throughout the following proof, we often make use of Lemma 5.2 of [20] although it is a statement for time-homogeneous symbols. A closer look shows that an analogous statement holds for the time-dependent probabilistic symbol and the proof also works analogously.

Proof

The proof of (16) closely follows the proof of Proposition 3.10 of [20]. We omit the proof of (17) since it is generalized from the proof of Lemma 6.3 of [18] analogously to the following generalization. However, let us mention that one has to use Dynkin’s formula for non-homogeneous processes as stated in [13] when Corollary 3.6 is utilized in [18].

In order to prove (16), let X be a non-homogeneous Itô process such that the differential characteristics $(\ell ,Q,\nu )$ of X are locally bounded and continuous. At first, we show that for S, R and $\sigma := \inf \{ t \geqslant \tau : \Vert X_t -x \Vert > S \}$ as above we have

$$\begin{aligned} \mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant t} \Vert X_s^\sigma -x \Vert \geqslant 2R \right) \leqslant c_d \cdot (t-\tau ) \cdot \sup _{\tau \leqslant s \leqslant t} \sup _{\left\| y-x \right\| \leqslant S} \sup _{\left\| \varepsilon \right\| \leqslant 1} \left| p\left( s,y,\frac{\varepsilon }{2R} \right) \right| \end{aligned}$$

(18)

where $c_d=4d+16\widetilde{c}_d$ and $t \geqslant \tau$. We introduce the stopping time

$$\begin{aligned} \tau _R:= \inf \{t\geqslant \tau : \left\| \Delta X_t^\sigma \right\| >R \}, \end{aligned}$$

as the first time the jumps of $X^\sigma$ exceed R, and estimate the following

$$\begin{aligned}&\mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant t} \Vert X_s^\sigma -x \Vert \geqslant 2R \right) \nonumber \\&\qquad \leqslant \mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant t} \Vert X_s^\sigma -x \Vert \geqslant 2R, \tau _R > t \right) + \mathbb {P}^{\tau ,x} \Big (\tau _R \leqslant t \Big ). \end{aligned}$$

(19)

We deal with the terms on the right-hand side one after another, starting with the first one.

Again we separate the first term (19) in order to get control over the big jumps. For $t \geqslant \tau$ let

$$\begin{aligned} \check{X}_t:= X_t- \sum _{\tau \leqslant s \leqslant t} \Delta X_s(1-\chi (\Delta X_s)). \end{aligned}$$

The process $\check{X}$ is a special semimartingale on $[\tau , \infty )$ with characteristics

$$\begin{aligned} \check{B}_t^{(i)}&= \int _\tau ^t b^i(s,X_s) \ \textrm{d}s, \quad \mathbb {P}^{\tau ,x}\text {-a.s.} \\ \check{C}_t^{(ij)}&= \int _\tau ^t c^{ij}(s,X_s) \ \textrm{d}s, \quad \mathbb {P}^{\tau ,x}\text {-a.s.}\\ \check{\nu }( ; \textrm{d}t,{ \textrm{d}y})&= \chi (y) 1_{\left[ [0, \sigma ] \right] } (t) \ N(X_t,\textrm{d}y) \ \textrm{d}t, \quad \mathbb {P}^{\tau ,x}\text {-a.s.} \end{aligned}$$

Now let $u=(u_1,...,u_d)':\mathbb {R}^d \rightarrow \mathbb {R}^d$ be such that $u_j \in C_b^2(\mathbb {R}^d)$ is 1-Lipschitz continuous, $u_j$ depends only on $x^{(j)}$ and is zero in zero for $j=1,...,d$. We define the auxiliary process for $t \geqslant \tau$:

$$\begin{aligned} \check{M}_t:=u(\check{X}_t^\sigma -x) - \int _\tau ^{t\wedge \sigma } \sum _{j=1}^d F^{(j)}_s \ \textrm{d}s \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} F_s^{(j)}&= \partial _j u(\check{X}_{s-}-x) \ell ^{(j)}(s,X_{s-}) -\frac{1}{2} \partial _j \partial _j u(\check{X}_{s-}-x) Q^{(jj)} (s,X_{s-}) \\&\quad -\int _{z\ne 0} \Big ( u_j(\check{X}_{s-}-x+z)-u_j(\check{X}_{s-}-x) - \chi (z) z^{(j)} \partial _j u(\check{X}_{s-}-x)\Big ) \chi (z) \\&\quad N_s(X_{s-}, \textrm{d}z). \end{aligned} \end{aligned}$$

(20)

$\check{M}$ is a $\mathbb {P}^{\tau ,x}$-local martingale on $[\tau ,\infty )$ by [7] Theorem II.2.42. Applying Lemma 3.7 of [23] we have under (IG)

$$\begin{aligned} \left| F_s^{(j)} \right| \leqslant const \cdot \sum _{0\leqslant \left| \alpha \right| \leqslant 2} \left\| \partial ^\alpha u \right\| _\infty \end{aligned}$$

since $u_j\in C_b^2(\mathbb {R}^d)$. Let us mention, that although Lemma 3.7 of [23] considers homogeneous diffusion with jumps only the proof is alike for non-homogeneous Itô processes. In particular, since $\check{M}$ is uniformly bounded it is an $L^2$-martingale on $[\tau ,t]$. We define

$$\begin{aligned} D:=\left\{ \omega \in \Omega :\int _\tau ^{t\wedge \sigma (\omega )} \left\| F_s(\omega ) \right\| \ \textrm{d}s \leqslant R \right\} \end{aligned}$$

and obtain for $t \geqslant \tau$:

$$\begin{aligned}&\mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant t} \Vert X_s^\sigma -x \Vert \geqslant 2R, \tau _R> t \right) \nonumber \\&\quad \leqslant \ \mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant t} \Vert X_s^\sigma -x \Vert \geqslant 2R, \tau _R > t, D \right) + \mathbb {P}^{\tau ,x} (D^c). \end{aligned}$$

(21)

For the left summand of (21), we estimate for $t \geqslant \tau$:

$$\begin{aligned}&\mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant t} \Vert u(X_s^\sigma - x) \Vert \geqslant 2R, \tau _R> t, D \right) \\&\quad = \ \mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant t} \Vert u(\check{X}_s^\sigma - x) \Vert \geqslant 2R, \tau _R> t, D \right) \\&\quad \leqslant \ \mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant t} \Vert u(X_s^\sigma - x) \Vert -\int _\tau ^{t \wedge \sigma } F_s \ \textrm{d}s \geqslant R, \tau _R > t, D \right) \\&\quad \leqslant \ \mathbb {P}^{\tau ,x}\left( \sup _{\tau \leqslant s \leqslant t} \Vert \check{M}_{s \wedge \sigma } \Vert \geqslant R \right) \\&\quad = \ \mathbb {P}^{\tau ,x} \left( \sup _{0 \leqslant s \leqslant t-\tau } \Vert \check{M}_{(s+\tau ) \wedge \sigma } \Vert \geqslant R \right) \\&\quad \leqslant \ \frac{1}{R^2} \mathbb {E}^{\tau ,x} \left( \left\| \check{M}_t^\sigma \right\| ^2 \right) \\&\quad \leqslant \ \frac{1}{R^2} \sum _{j=1}^d \mathbb {E}^{\tau ,x} \left( \left[ \check{M}^{(j)}, \check{M}^{(j)} \right] ^\sigma _t \right) \\&\quad \leqslant \ \frac{1}{R^2} \sum _{j=1}^d \mathbb {E}^{\tau ,x} \left( [\check{X}_\cdot ^{(j)}, \check{X}_\cdot ^{(j)}]_t^\sigma \right) , \end{aligned}$$

where we used Doob’s inequality for the martingale $\check{M}^\sigma$ and the Lipschitz property of u in combination with Corollary II.3 of [11]. Since

$$\begin{aligned}&\mathbb {E}^{\tau ,x} \left( [\check{X}_\cdot ^{(j)}, \check{X}_\cdot ^{(j)}]_t^\sigma \right) \\&\quad = \ \mathbb {E}^{\tau ,x} \left( \left<\check{X}_\cdot ^{(j),c}, \check{X}_\cdot ^{(j),c}\right>_t^\sigma \right) + \mathbb {E}^{\tau ,x} \left( \int _\tau ^{t\wedge \sigma } \int _{z\ne 0} (z^{(j)})^2 \chi (z)^2 \ N_s(X_s,\textrm{d}z) \ \textrm{d}s\right) . \end{aligned}$$

we obtain

$$\begin{aligned}&\mathbb {P}^{\tau ,x} \Big ( \sup _{\tau \leqslant s \leqslant t} \Vert u(X_s^\sigma - x) \Vert \geqslant 2R, \tau _R > t, D \Big ) \\&\quad \leqslant \frac{1}{R^2} \sum _{j=1}^d \mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } Q^{(jj)}(s,X_s) \ \textrm{d}s + \mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \int _{z\ne 0} \frac{\left\| z \right\| ^2}{R^2} \chi (z)^2 \ N_s(X_s,z) \ \textrm{d}s \\&\quad \leqslant 4 \sum _{j=1}^d \mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \left( \frac{e_j}{2R} ' Q(s,X_s) \frac{e_j}{2R}\right) \textrm{d}s \\&\qquad + 4^2 \mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \int _{z\ne 0} \left( \left\| \frac{z}{2R} \right\| ^2 \wedge 1 \right) N_s(X_s,\textrm{d}z) \textrm{d}s \\&\quad \leqslant 4 {(t\wedge \sigma -\tau )}\sum _{j=1}^d \mathbb {E}^{\tau ,x} \sup _{\tau< s<t\wedge \sigma } \left( \frac{e_j}{2R} ' Q(s,X_s) \frac{e_j}{2R}\right) \\&\qquad + 4^2 {(t\wedge \sigma -\tau )} \mathbb {E}^{\tau ,x} \sup _{\tau< s<t\wedge \sigma } \int _{z\ne 0} \left( \left\| \frac{z}{2R} \right\| ^2 \wedge 1 \right) N_s(X_s,\textrm{d}z) \\&\quad \leqslant 4(t-\tau ) \sum _{j=1}^d \sup _{\tau< s<t\wedge \sigma } \sup _{\left\| y-x \right\| \leqslant S} \mathrm{Re\,}p\left( s,y, \frac{e_j}{2R} \right) \\&\qquad + 4^2(t-\tau ) \sup _{\tau< s<t\wedge \sigma } \sup _{\left\| y-x \right\| \leqslant S} \int _{z\ne 0} \left( \left\| \frac{z}{2R} \right\| ^2 \wedge 1 \right) N_s(y,\textrm{d}z) \\ {}&\quad \leqslant 4(t-\tau )d \sup _{\tau< s<t\wedge \sigma } \sup _{\left\| y-x \right\| \leqslant S} \sup _{\left\| \varepsilon \right\| \leqslant 1} \left| p\left( s,y,\frac{\varepsilon }{2R} \right) \right| \\ {}&\qquad + 4^2 t \sup _{\tau<s<t\wedge \sigma } \sup _{\left\| y-x \right\| \leqslant S} \widetilde{c}_d \sup _{\left\| \varepsilon \right\| \leqslant 1} \left| p\left( s,y,\frac{\varepsilon }{2R}\right) \right| \end{aligned}$$

where we have used Lemma 5.2 of [20] on the second term. By choosing a sequence $(u_n)_{n\in \mathbb {N}}$ of functions of the type described above which tends to the identity in a monotonous way we obtain

$$\begin{aligned}&\mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant t} \Vert X_s^\sigma - x \Vert \geqslant 2R, \tau _R > t, D \right) \nonumber \\&\quad \leqslant \, (4d+ 4^2 \widetilde{c}_d ) (t-\tau ) \sup _{\tau<s<t\wedge \sigma }\sup _{\left\| y-x \right\| \leqslant S} \sup _{\left\| \varepsilon \right\| \leqslant 1} \left| p\left( s,y,\frac{\varepsilon }{2R} \right) \right| . \end{aligned}$$

(22)

Now let us consider the term $\mathbb {P}^{\tau ,x}{( D^c)}$ of (21). The Markov inequality provides

$$\begin{aligned} \mathbb {P}^{\tau ,x}(D^c)=\mathbb {P}^{\tau ,x} \left( \int _\tau ^{t\wedge \sigma } \left\| F_s \right\| \ \textrm{d}s > R \right) \leqslant \frac{1}{R} \mathbb {E}^{\tau ,x} \Big (\int _\tau ^{t\wedge \sigma } \sup _{j=1,...,d} \left| F_s^{(j)} \right| \ \textrm{d}s \Big )=:(*) \end{aligned}$$

Again we chose a sequence $(u_n)_{n\in \mathbb {N}}$ of functions as we described in (20), but this time it is important that the first and second derivatives are uniformly bounded. Since the $u_n$ converge to the identity, the first partial derivatives tend to 1 and the second partial derivatives to 0. In the limit ($n\rightarrow \infty$), we obtain

$$\begin{aligned} (*)&\leqslant \frac{1}{R} \mathbb {E}^{\tau ,x} \int _\tau ^{t \wedge \sigma } \sup _{j=1,...,d} \nonumber \\&\quad \left| \ell ^{(j)}(s,X_s) + \int _{z\ne 0} (-z^{(j)} \chi (z) + (\chi (z))^2 z^{(j)} ) \ N_s(X_s,\textrm{d}z) \right| \ \textrm{d}s \nonumber \\ {}&\leqslant 2\sum _{j=1}^d \mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \left| \frac{\ell ^{(j)}(s,X_s)}{2R} + \int _{z\ne 0} \sin \left( \frac{z'e_j}{2R} \right) - \frac{z^{(j)}\chi (z)}{2R} \ N_s(X_s,\textrm{d}z) \right| \ \textrm{d}s \end{aligned}$$

(23)

$$\begin{aligned}&\quad +2 \mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \sup _{j=1,...,d} \left| \int _{z\ne 0} \frac{(\chi (z))^2 z^{(j)}}{2R} - \sin \left( \frac{z'e_j}{2R} \right) \ N(X_s,\textrm{d}z) \right| \ \textrm{d}s. \end{aligned}$$

(24)

For term (23), we get

$$\begin{aligned}&2 \sum _{j=1}^d \mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \left| \frac{\ell (s,X_s)'e_j}{2R} + \int _{z\ne 0} \sin \left( \frac{z'e_j}{2R} \right) - \frac{z'e_j \chi (z)}{2R} \ N_s(X_s,\textrm{d}z) \right| \ \textrm{d}s \nonumber \\ {}&\quad \leqslant 2(t-\tau )\sum _{j=1}^d \sup _{\tau< s\leqslant t\wedge \sigma } \mathbb {E}^{\tau ,x}\nonumber \\&\quad \left| \frac{\ell (s,X_s)'e_j}{2R} + \int _{z\ne 0} \sin \left( \frac{z'e_j}{2R} \right) - \frac{z'e_j \chi (z)}{2R} \ N_s(X_s,\textrm{d}z) \right| \nonumber \\ {}&\quad \leqslant 2(t-\tau )d \sup _{\tau < s\leqslant t}\sup _{\left\| y-x \right\| \leqslant S} \sup _{\left\| \varepsilon \right\| \leqslant 1} \left| \mathrm{Im\,}p\left( s,y,\frac{\varepsilon }{2R} \right) \right| \end{aligned}$$

(25)

and for term (24)

$$\begin{aligned}&2 \mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \sup _{j=1,...,d} \left| \int _{z\ne 0} \frac{(\chi (z))^2 z'e_j}{2R}- \sin \left( \frac{z'e_j}{2R} \right) \right| \ N_s(X_s,\textrm{d}z) \ \textrm{d}s \nonumber \\&\quad \leqslant 2 \mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \sup _{j=1,...,d} \int _{z\ne 0} \ \left| \chi (z)^2 \left( \frac{z'e_j}{2R}- \sin \left( \frac{z'e_j}{2R} \right) \right) \right| \ N_s(X_s,\textrm{d}z) \ \textrm{d}s \nonumber \\ {}&\qquad + 2\mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \sup _{j=1,...,d}\int _{z\ne 0} \ \left| \left( (1-\chi (z)^2)\sin \left( \frac{z'e_j}{2R} \right) \right) \right| \ N_s(X_s,\textrm{d}z) \ \textrm{d}s \nonumber \\ {}&\quad \leqslant 2 \mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \int _{z\ne 0} \ \sup _{j=1,...,d} \left| \chi (z)^2 \left( \frac{z'e_j}{2R}- \sin \left( \frac{z'e_j}{2R} \right) \right) \right| \ N_s(X_s,\textrm{d}z) \ \textrm{d}s \nonumber \\ {}&\qquad + 2\mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \int _{z\ne 0} \ \sup _{j=1,...,d} \left| \left( (1-\chi (z)^2)\sin \left( \frac{z'e_j}{2R} \right) \right) \right| \ N_s(X_s,\textrm{d}z) \ \textrm{d}s \nonumber \\ {}&\quad \leqslant 2 \mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \int _{B_{2R}(0)\setminus \{ 0 \}} \ \sup _{j=1,...,d} \left| \left( \frac{z'e_j}{2R}- \sin \left( \frac{z'e_j}{2R} \right) \right) \right| \ N_s(X_s,\textrm{d}z) \ \textrm{d}s \nonumber \\ {}&\qquad + 2\mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \int _{B_R(0)^c} \ \sup _{j=1,...,d} \left| \sin \left( \frac{z'e_j}{2R} \right) \right| \ N_s(X_s,\textrm{d}z) \ \textrm{d}s \nonumber \\ {}&\quad \leqslant 2 \mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \int _{B_{2R}(0)\setminus \{0\}} \ \sup _{j=1,...,d} \left| \left( 1- \cos \left( \frac{z'e_j}{2R} \right) \right) \right| \ N_s(X_s,\textrm{d}z) \ \textrm{d}s \nonumber \\ {}&\qquad + 2\mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \int _{B_{2R}(0) \setminus B_R(0)} \ \left| \sin \left( \frac{\sup _{j=1,...,d} z'e_j}{2R} \right) \right| \ N_s(X_s,\textrm{d}z) \ \textrm{d}s\nonumber \\&\qquad +\int _{B_{2R}(0)^c} 1 \ N_s(X_s,\textrm{d}z) \textrm{d}s \nonumber \\&\quad \leqslant 10 \sum _{j=1}^d\mathbb {E}^{\tau ,x} \int _\tau ^{t\wedge \sigma } \int _{z \ne 0} \ \left( 1- \cos \left( \frac{z'e_j}{2R} \right) \right) \ N_s(X_s,\textrm{d}z) \ \textrm{d}s \nonumber \\&\qquad +\int _{B_{2R}(0)^c} 1 \ N_s(X_s,\textrm{d}z) \textrm{d}s \nonumber \\ {}&\quad \leqslant 10(t-\tau )d \sup _{\tau< s\leqslant t} \sup _{\left\| y-x \right\| \leqslant S} \sup _{\left\| \varepsilon \right\| \leqslant 1} \mathrm{Re\,}p\left( s,y,\frac{\varepsilon }{2R} \right) \nonumber \\ {}&\qquad + 2^2 (t-\tau )d \sup _{\tau < s\leqslant t} \sup _{\left\| y-x \right\| \leqslant S} \widetilde{c}_d \sup _{\left\| \varepsilon \right\| \leqslant 1} \left| p\left( s,y,\frac{\varepsilon }{2R}\right) \right| \end{aligned}$$

(26)

where we have used again Lemma 5.2 of [20] on the second term, and the inequalities

$$\begin{aligned} |\sin (\alpha ) -\alpha | \leqslant 1-\cos (\alpha ), \ |\alpha |\leqslant 1 \quad \text { and }\quad |\sin (\beta )|\leqslant 4(1-\cos (\beta )), \ \frac{1}{2}< |\beta | \leqslant 1. \end{aligned}$$

It remains to deal with the second term of (19). Let $\delta >0$ be fixed (at first) and $m:\mathbb {R}\rightarrow ]1,1+\delta [$ a strictly monotone increasing auxiliary function. Since $m\geqslant 1$ and since we have at least one jump of size $>R$ on $\{\tau _R\leqslant t\}$ we obtain

$$\begin{aligned} \mathbb {P}^{\tau ,x}(\tau _R\leqslant t)&\leqslant \mathbb {P}^{\tau ,x} \left( \int _\tau ^t \int _{\left\| z \right\| \geqslant R} m(\left\| z \right\| ) \ \mu ^{X^\sigma }(\cdot ;ds,\textrm{d}z) \geqslant m(R) \right) \\ {}&\leqslant \frac{1}{m(R)} \mathbb {E}^{\tau ,x} \left( \int _0^t \int _{\left\| z \right\| \geqslant R} m(\left\| z \right\| ) 1_{\left[ [0, \sigma ] \right] }(s) \ \mu ^X(\cdot ;ds,\textrm{d}z) \right) \\ {}&= \frac{1}{m(R)} \mathbb {E}^{\tau ,x} \left( \int _\tau ^t \int _{z\ne 0} m(\left\| z \right\| ) 1_{[ [0 , \sigma [ [}(s) 1_{B_R(0)^c} (z) \ N_s(X_s,\textrm{d}z) \ \textrm{d}s \right) \\ {}&\leqslant (1+\delta ) (t-\tau ) \sup _{\tau< s\leqslant t\wedge \sigma } \mathbb {E}^{\tau ,x} (N_s(X_s, B_R(0)^c)) \\ {}&\leqslant (1+\delta ) (t-\tau ) \sup _{\tau< s\leqslant t} \sup _{\left\| y-x \right\| \leqslant S} N_s(y,B_R(0)^c) \\&\leqslant (1+\delta ) 4(t-\tau ) \sup _{\tau < s\leqslant t} \sup _{\left\| y-x \right\| \leqslant S} \int _{z\ne 0} \left( \left\| \frac{z}{2R} \right\| ^2 \wedge 1 \right) \ N_s(y,\textrm{d}z) \end{aligned}$$

because $m(\left\| z \right\| ) 1_{[ [0 , \sigma [ [}(s) 1_{B_R(0)^c} (z)$ is in class $F_p^1$ of Ikeda and Watanabe (see [6], Section II.3). Since $\delta$ can be chosen arbitrarily small we obtain by Lemma 5.2 of [20]

$$\begin{aligned} \mathbb {P}^{\tau ,x}(\tau _R\leqslant t) \leqslant 4(t-\tau ) \sup _{\tau < s\leqslant t} \sup _{\left\| y-x \right\| \leqslant S} \widetilde{c}_d \sup _{\left\| \varepsilon \right\| \leqslant 1} \left| p\left( s,y,\frac{\varepsilon }{2R}\right) \right| . \end{aligned}$$

(27)

Plugging together (22), (25), (26) and (27) we obtain (18). For the particular case $\sigma ={\sigma _{3\widetilde{R}}^{\tau ,x}}$, we have

$$\begin{aligned} \left\{ \sup _{\tau \leqslant s \leqslant t} \Vert u(X_s^\sigma - x) \Vert \geqslant 2\widetilde{R} \right\} = \left\{ \sup _{\tau \leqslant s \leqslant t} \Vert u(X_s- x) \Vert \geqslant 2\widetilde{R} \right\} , \end{aligned}$$

and therefore, for every $\widetilde{R}>0$

$$\begin{aligned}&\mathbb {P}^{\tau ,x}\left( \sup _{\tau \leqslant s \leqslant t} \Vert u(X_s^\sigma - x) \Vert \geqslant 2\widetilde{R}\right) \nonumber \\&\qquad \leqslant c_d \cdot (t-\tau ) \cdot \sup _{\tau < s\leqslant t} \sup _{\left\| y-x \right\| \leqslant 3\widetilde{R}} \sup _{\left\| \varepsilon \right\| \leqslant 1} \left| p\left( s,y,\frac{\varepsilon }{2\widetilde{R}} \right) \right| . \end{aligned}$$

(28)

Setting $\widetilde{R}:=(1/2)R$ we obtain (16). $\square$

The maximal inequalities (16) and (17) allow for statements of the asymptotic behavior of the sample paths: First, we state a result concerning the behavior near the starting point $x \in \mathbb {R}^d$ at time $\tau \geqslant 0$ with respect to the measure $\mathbb {P}^{\tau ,x}$. The second statement, treats the same behavior at infinity. The proof of both statements is inspired by Theorem 4.3 and Theorem 4.6 of [18] but takes the time-component $\tau$ into account. Let us mention, that for a function $f:\mathbb {R}_+ \rightarrow \mathbb {R}_+$ we denote by $f(t^+):= \limsup _{s \downarrow t} f(s)$ and $f(t_+):= \liminf _{s \downarrow t} f(s)$.

Theorem 3.6

Let X be a non-homogeneous Itô process such that the differential characteristics of X are locally bounded and continuous. Then, we have

$$\begin{aligned} \lim _{t\rightarrow 0} \ t^{-1/\lambda }\sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s-x \Vert&=0 \text { for all }{ \lambda > \beta _\infty ^{\tau ^+,x} } \end{aligned}$$

(29)

$$\begin{aligned} \liminf _{t\rightarrow 0 } \ t^{-1/\lambda }\sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s-x \Vert&=0 \text { for all } { \beta _\infty ^{\tau ^+, x} \geqslant \lambda > \underline{\beta _\infty ^{\tau ^+,x}}}. \end{aligned}$$

(30)

If the symbol $p(\tau ,x,\xi )$ of the process X satisfies (IS), then we have in addition

$$\begin{aligned} \limsup _{t\rightarrow 0 } \ t^{-1/\lambda }\sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s-x \Vert =\infty&\text { for all } {\overline{\delta _\infty ^{\tau _+,x}} > \lambda \geqslant \delta _\infty ^{\tau _+,x}} \end{aligned}$$

(31)

$$\begin{aligned} \lim _{t\rightarrow 0 } \ t^{-1/\lambda }\sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s-x \Vert =\infty&\text { for all } { \delta _\infty ^{\tau _+,x} > \lambda }. \end{aligned}$$

(32)

All these limits are meant $\mathbb {P}^{\tau ,x}$-a.s with respect to every $\tau \in \mathbb {R}_+$ and $x\in \mathbb {R}^d$.

Proof

Here, we only prove (29) and (31) and omit the proofs of (30) and (32) since (29) and (32) and (30) and (31) are very similar, respectively.

Let $\varepsilon >0$, $\tau \in \mathbb {R}_+$ and $x\in \mathbb {R}^d$. We start with proving (29):

Let $\lambda > \sup _{\tau < s \leqslant \tau + \varepsilon } \beta _\infty ^{s,x}$ and choose $\sup _{\tau< s \leqslant \tau + \varepsilon } \beta _\infty ^{s,x}< \alpha _2< \alpha _1 <\lambda$. For $t<T_0^\varepsilon$ with $T_0^\varepsilon$ sufficiently small, for (16) one obtains:

$$\begin{aligned} \mathbb {P}^{\tau ,x}\left( \sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s-x \Vert \geqslant t^{1/\alpha _1}\right)&\leqslant c_d \cdot t \cdot \sup _{\tau < s\leqslant \tau +\varepsilon } H(s,x,t^{1/\alpha _1}) \\&\leqslant c_d' \cdot t (t^{1/\alpha _1})^{-\alpha _2}\\&=c_d' t^{1-(\alpha _2/\alpha _1)}. \end{aligned}$$

Now let $t_k:=(1/2)^k$ for $k\in \mathbb {N}$. Since

$$\begin{aligned} \sum _{k=k_0 ^\varepsilon }^\infty \mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant \tau +t_k} \Vert X_s-x \Vert \geqslant (t_k)^{1/\alpha _1} \right) \leqslant c_d' \sum _{k=k_0^\varepsilon }^\infty 2^{-k(1-(\alpha _2/\alpha _1))} <+\infty \end{aligned}$$

where $k_0^\varepsilon$ depends on $T_0^\varepsilon$, the Borel-Cantelli Lemma derives

$$\begin{aligned} \mathbb {P}^{\tau ,x} \left( \limsup _{k\rightarrow \infty } \sup _{\tau \leqslant s \leqslant \tau +t_k} \Vert X_s-x \Vert \geqslant (t_k)^{1/\alpha _1}\right) =0, \end{aligned}$$

and, hence, $\sup _{\tau \leqslant s \leqslant \tau +t_k} \Vert X_s-x \Vert < (t_k)^{1/\alpha _1}$ for all $k\geqslant k_1^\varepsilon (\omega )$ on a set of probability one. For fixed $\omega$ in this set and $t \in [t_{k+1}, t_k]$ and $k\geqslant k^\varepsilon _1(\omega ) \geqslant k^\varepsilon _0$, we have

$$\begin{aligned} \sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s(\omega )-x \Vert \leqslant \sup _{\tau \leqslant s \leqslant \tau +t_k} \Vert X_s(\omega )-x \Vert \leqslant t_k^{1/\alpha _1} \leqslant 2^{1/\alpha _1} t^{1/\alpha _1} \end{aligned}$$

and since $\lambda >\alpha _1$

$$\begin{aligned} t^{-1/\lambda } \sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s-x \Vert \leqslant 2^{1/\alpha _1} t^{(1/\alpha _1)-(1/\lambda )} \end{aligned}$$

which converges $\mathbb {P}^{\tau ,x}$-a.s to zero as $t\downarrow 0$. Since $\lambda >\sup _{\tau < s \leqslant \tau + \varepsilon } \beta _\infty ^{s,x}$ and $\varepsilon >0$ arbitrary, $\varepsilon \downarrow 0$ provides the statement.

In order to prove (31), we derive the following:

Let $\inf _{\tau < s \leqslant \tau +\varepsilon } \overline{\delta _\infty ^{s,x}}> \lambda ' > \lambda$. Moreover, let $(t_k)_{k \in \mathbb {N}}$ be a sequence such that

$$\begin{aligned} \lim _{k \rightarrow \infty } (t_k)^{\lambda '} h(s,x,t_k)= \infty , \quad \forall s \in ]\tau , \tau +\varepsilon ]. \end{aligned}$$

Hence, the maximal inequality (17) provides for k sufficiently large

$$\begin{aligned} \mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant \tau +(t_k)^{\lambda '}} \Vert X_s- x \Vert< t_k \right)&\leqslant c_d \cdot \frac{1}{(t_k)^{\lambda '}} \cdot \frac{1}{\inf _{\tau< s \leqslant \tau +(t_k)^{\lambda '}} h(s,x,t_k)} \\&\leqslant c_d \cdot \frac{1}{(t_k)^{\lambda '}} \cdot \frac{1}{\inf _{\tau < s \leqslant \tau +\varepsilon } h(s,x,t_k)} \\&\underset{k \rightarrow \infty }{\longrightarrow }0. \end{aligned}$$

Fatou’s lemma implies

$$\begin{aligned} 0&= \liminf _{k \rightarrow \infty } \mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant \tau +(t_k)^{\lambda '}} \Vert X_s- x \Vert < t_k \right) \\&= 1- \limsup _{k \rightarrow \infty } \mathbb {P}^{\tau ,x} \left( \sup _{\tau \leqslant s \leqslant \tau +(t_k)^{\lambda '}} \Vert X_s- x \Vert \geqslant t_k \right) \\&\geqslant 1- \mathbb {P}^{\tau ,x} \left( \limsup _{k \rightarrow \infty } \left\{ \sup _{\tau \leqslant s \leqslant \tau +(t_k)^{\lambda '}} \Vert X_s- x \Vert \; \geqslant t_k \right\} \right) . \end{aligned}$$

Hence,

$$\begin{aligned} \mathbb {P}^{\tau ,x} \left( \left( \frac{1}{t_k}\right) \sup _{\tau \leqslant s \leqslant \tau +(t_k)^{\lambda '}} \Vert X_s- x \Vert \geqslant 1 \text {, infinitely often}\right) =1, \end{aligned}$$

and, therefore,

$$\begin{aligned} \limsup _{k \rightarrow \infty } \left( \frac{1}{t_k}\right) ^{-\frac{1}{\lambda '}}\sup _{\tau \leqslant s \leqslant \tau +t_k} \Vert X_s- x \Vert \geqslant 1. \end{aligned}$$

Since $\lambda < \lambda '$ we observe, that

$$\begin{aligned} \limsup _{t \rightarrow 0} \left( \frac{1}{t}\right) ^{-\frac{1}{\lambda }}\sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s- x \Vert = \infty . \end{aligned}$$

Since $\varepsilon >0$ arbitrary, the statement follows. $\square$

The proof of the following theorem parallels Lemma 3.6, and hence, we omit details of the proof.

Theorem 3.7

Let X be a non-homogeneous Itô process such that the differential characteristics of X are locally bounded and continuous. Then, we have

$$\begin{aligned} \lim _{t\rightarrow \infty } \ t^{-1/\lambda }\sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s-x \Vert&=0 \text { for all } \lambda < \beta _0 \end{aligned}$$

(33)

$$\begin{aligned} \liminf _{t\rightarrow \infty } \ t^{-1/\lambda }\sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s-x \Vert&=0 \text { for all } \underline{\beta _0} > \lambda \geqslant \beta _0 \end{aligned}$$

(34)

If the symbol $p(\tau ,x,\xi )$ of the process X satisfies (IS) then we have in addition

$$\begin{aligned} \limsup _{t\rightarrow \infty } \ t^{-1/\lambda }\sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s-x \Vert =\infty&\text { for all } \overline{\delta _0} < \lambda \leqslant \delta _0 \end{aligned}$$

(35)

$$\begin{aligned} \lim _{t\rightarrow \infty } \ t^{-1/\lambda }\sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s-x \Vert =\infty&\text { for all } \delta _0 < \lambda . \end{aligned}$$

(36)

All these limits are meant $\mathbb {P}^{\tau ,x}$-a.s with respect to every $\tau \in \mathbb {R}_+$ and $x\in \mathbb {R}^d$.

The time homogeneous version of the following result is Lemma 5.11 of [4].

Lemma 3.8

Let X be a non-homogeneous Itô process such that the differential characteristics of X are continuous, and let (IG) hold true. Then

$$\begin{aligned} \mathbb {E}^{\tau ,x}\left( e^{X_t'\xi } \right) < \infty \end{aligned}$$

for all $t \geqslant \tau \geqslant 0$ and $x,\xi \in \mathbb {R}^d$.

Proof

At first, let us reconsider the stopping time $\sigma$ as defined in (7):

$$\begin{aligned} \sigma _R^{\tau ,x}:= \sigma := \inf \{ h \geqslant \tau : \Vert X^{\tau ,x}_{h} -x \Vert > R\} \end{aligned}$$

and let $p(\tau ,x,\xi )$ be the symbol of X given by

$$\begin{aligned} p(\tau ,x,\xi )= & {} -i \ell (\tau ,x)'\xi + \frac{1}{2} \xi ' Q(\tau ,x) \xi \\{} & {} - \int _{y \ne 0} \left( e^{iy' \xi } -1 - iy' \xi \cdot \chi (y) \right) \ N_\tau (x,\textrm{d}y). \end{aligned}$$

In order to apply Gronwall’s Lemma, we estimate the following:

$$\begin{aligned} \mathbb {E}^{\tau ,x} \left( e^{(X_t^{\sigma }-x){ '}\xi } \right)&= \mathbb {E}^{\tau ,x} \left[ \int _{\tau }^{t\wedge \sigma } e^{(X^\sigma _{s}-x){ '}\xi } \left( \xi { '} \ell (s,X_s)- \frac{1}{2}\xi { '}\ Q(s,X_s) \xi \right. \right. \\&\quad \quad + \left. \left. \int _{ \{y\ne 0\}} (e^{\xi { '} y}-1 - \xi { '} y\chi (y)) \ N_s(X_s,\textrm{d}y) \right) \ \textrm{d}s\right] \\&\leqslant b(\xi ) \int _\tau ^t \mathbb {E}^{\tau ,x} \left( e^{(X_s^{\sigma }-x){ '}\xi } \right) \ \textrm{d}s, \end{aligned}$$

(IG) provides (see Lemma 3.3 of [23]) the finiteness of the constant

$$\begin{aligned} b(\xi ):= & {} \sup _{s\geqslant 0, x \in \mathbb {R}^d} \left| \xi { '} \ell (s,x)- \frac{1}{2}\xi { '}\ Q(s,x) \xi \right. \nonumber \\{} & {} \left. + \int _{ \{y\ne 0\}} (e^{\xi { '} y}-1 - \xi { '} y\chi (y)) \ N_s(x,\textrm{d}y) \right| . \end{aligned}$$

Application of Gronwall’s Lemma provides

$$\begin{aligned} \mathbb {E}^{\tau ,x} \left( e^{(X_s^{\sigma }-x){ '}\xi } \right)&\leqslant 1+ b(\xi ) \int _\tau ^t e^{b(\xi )(t-s)} \ \textrm{d}s = e^{b(\xi )(t-\tau )}, \end{aligned}$$

and with Fatou’s Lemma follows

$$\begin{aligned} \mathbb {E}^{\tau ,x} \left( e^{(X_s-x){ '}\xi } \right) \leqslant \liminf _{R \rightarrow \infty } \mathbb {E}^{\tau ,x} \left( e^{(X_s^{\sigma }-x){ '}\xi } \right) \leqslant e^{b(\xi )(t-\tau )}, \end{aligned}$$

where the first inequality follows by Proposition 3.4. $\square$

Finally, we generalize Theorem 2.10 of [10] which is a nice and applicable criterion for the finiteness of p-variation.

Theorem 3.9

Let X be a non-homogeneous Itô process such that the differential characteristics of X are continuous. For every $t \geqslant \tau \geqslant 0$ and $p > \sup _{\tau \geqslant 0} \beta _\infty ^{\tau ,x}$ the following holds true

$$\begin{aligned} \sup _{\pi _n} \sum _{j=1}^n \Vert X_{t_j}-X_{t_{j-1}} \Vert ^p < + \infty , \quad \mathbb {P}^{\tau ,x}\text {-a.s.} \end{aligned}$$

(37)

where the supremum is taken over all finite partitions $\pi _n= (t_i)_{i=1,...,n}$ with $\tau =t_0< t_1< \cdots <t_n=t$. I.e., the p-variation of the paths of X is $\mathbb {P}^{\tau ,x}$-a.s finite for $p > \sup _{\tau \geqslant 0} \ \beta _\infty ^{\tau ,x}$.

Proof

Let $t, r >0$ and $\lambda >p$. Then, Theorem 3.4 provides the following estimation

$$\begin{aligned} \alpha (t,r)&:= \sup \left\{ \mathbb {P}^{\tau ,x}(\Vert X_s-x \Vert \geqslant r) : \tau \geqslant 0, s \in [\tau , \tau +t], x \in \mathbb {R}^d \right\} \\&\leqslant \sup \left\{ \mathbb {P}^{\tau ,x}\left( \sup _{\tau \leqslant s \leqslant \tau +t} \Vert X_s-x \Vert \geqslant r\right) : \tau \geqslant 0, x \in \mathbb {R}^d \right\} \\&\leqslant c_d \cdot t \cdot \sup _{\tau \geqslant 0} \ \sup _{\tau < s \leqslant \tau +t} H(s,x,r) \\&\leqslant c_d \cdot t \cdot \sup _{\tau \geqslant 0} H(\tau ,x,r) \\&\leqslant c_d \cdot t \cdot K \cdot r^{-\lambda } \end{aligned}$$

for r small enough and $K>0$. Hence, Theorem 1.3 of [9] provides the statement. $\square$