The Time-Dependent Symbol of a Non-Homogeneous It\^o Process and corresponding Maximal Inequalities

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 It\^o 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 patsh, the existence of exponential moments and the finiteness of p-variation. In contrast to many situations where non-homogeneous Markov processes are involved, the space-time process cannot be utilized when considering maximal inequalities.


Introduction
The probabilistic symbol p(x, ξ) of a Markov process X is the function p : R d × R d → C given by p(x, ξ) if the limit exists and coincides for every R > 0, where σ := inf{t 0 : X x t − x > R}.
This symbol proves to be a crucial concept for deriving a wide range of properties of the stochastic process, such as conservativeness (cf.[19], Theorem 5.5), asymptotic behavior (cf.[25],Theorems 3.11 and 3.12), strong γ-variation (cf.[24] Corollary 5.10), Hausdorff-dimension (cf.[20], Theorem 4) and Hölder conditions [21].For a survey on recent results we refer to [6] and [16].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.[15]), certain Feller processes (cf.[21]) and homogeneous diffusions with jumps (cf.[25]).One can find a throughout discussion of maximal inequalities for various classes of stochastic processes in [11].Formulating these maximal inequalities, Blumenthal-Getoor indices are used (cf.[21], [25]) which allow for a governance of the process's paths by the behavior of the symbol in the variable ξ, 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, [17] proposed adding a time component to the symbol or more precisely p(τ, x, ξ) 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 T τ,t u(x) := E τ,x u(X t ) for 0 τ t and u ∈ B b (R d ) forms a strongly continuous evolution system.In addition, the domain of the infinitesimal generator A τ given by D(A τ ) := f : lim h↓0 T τ,τ +h f − f h exists contains the test functions C c (R d ).Theorem 4.5 of [17] shows that the generator for τ 0, x, ξ ∈ R d and f is the Fourier transform of f .Moreover, it is shown that the symbol of the generator q(τ, x, ξ) and the time-dependent probabilistic symbol defined in (2) p(τ, x, ξ) 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 σ-fields (G τ t ) 0 τ t is called a two-parameter filtration if G τ s ⊂ G τ t for all 0 τ s t and G τ1 t ⊂ G τ2 t for 0 τ 1 τ 2 t.The natural double filtration of X is denoted by (F X ) τ t ) 0 τ t and is defined as Let (Ω, M) be a measurable space equipped with the two-parameter filtration (M τ t ) 0 τ t .We call a stochastic process X adapted to the two-parameter filtration if for all 0 τ t We tacitly assume that every stochastic process X := (X t ) t 0 is defined on a generic stochastic basis (Ω, A, (A t ) t 0 , P) takes values in (R d , B(R d )) and is cádlág.Here, B(R d ) is the σ-field of Lebesgue sets.Moreover, we call ∆X t := X t − lim s↑t X s the jump of the process at time t 0, and for a stopping time τ we call X τ := X1 0,τ + X τ 1 τ,∞ the stopped process.The stochastic interval τ, σ for two stopping times τ, σ is defined by for all τ s t and all bounded Borel-measurable functions f .Moreover, every Markov process is normal, i.e., P τ,x (X τ = x) = 1.For a more information on Markov processes see [8] and [31].We associate an evolution system (T τ,t ) 0 τ t of operators on B b (R d ) with every Markov process by setting T τ,t u(x) := E τ,x u(X t ).

The Time-Dependent Probabilistic Symbol
In this section X := (Ω, M, (M τ t ) 0 τ t , (X t ) t 0 , P τ,x ) τ ∈R+,x∈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 [5].
Definition 2.1.We call a Markov process X non-homogeneuous Markov semimartingale if for every P τ,x , τ 0, x ∈ R d the process (X t ) t 0 is a semimartingale on [τ, ∞).
Definition 2.3.Let X be a Markov process and let be the first exit time from the ball of radius R > 0 after τ 0, and • the maximum norm.The function p : R is called the time-dependent probabilistic symbol of the process, if the limit exists for every τ 0 and x, ξ ∈ R d independently of the choice of R.
Example 2.4.(a.)Let (X t ) t 0 be an additive process on (Ω, F , (F t ) t 0 , P) in the sense of Definition 1.6 of [18], 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 (Ω, F ) by For this family it holds true that P τ,x (X τ = x) = 1, and (Ω, F , (F t ) t 0 , (X t ) t 0 , P τ,x ) τ 0,x∈E is a Markov process.Let, in addition, X be a semimartingale for all P τ,x which is quasi-left-continuous, and possesses the characteristics (B, C, ν).Theorem II.4.15 of [10] provides the existence of a version of (B, C, ν) that is deterministic.Hence, in the following we assume (B, C, ν) to be deterministic.By Corollary II.4.18 of [10] X has no fixes times of discontinuity, i.e., Therefore, when calculating the time-dependent probabilistic symbol of X we derive with Theorem II.4.15 of [10] that This limit exists if and only if i) and C (ij) are right-differentiable for all i, j ∈ {1, ..., d} and if the function is right-differentiable the time-dependent symbol exists and is of the form (b.) Let (X t ) t 0 be a one-dimensional Brownian motion with variance function σ 2 (t) on (Ω, F , (F t ) t 0 , P).The process X is additive and a continuous semimartingale and By the previous example the (non-homogeneous) probabilistic symbol exists if and only if the variance function is right-differentiable with right-derivative ∂ + σ 2 .In this case we have We have seen in ( 2) that under some mild conditions for a rich càdlàg Feller evolution process the symbol of the generator q(τ, x, ξ) and the time-dependent probabilistic symbol p(τ, x, ξ) coincide.Additionally, Corollary 3.5 of [5] states that the symbol of the generator of the homogeneous spacetime process X corresponding to X is given by . 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 nonhomogeneous 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, ν).In this case the space-time process X associated with X is semimartingale for all P (τ,x) , (τ, x) ∈ R + ×R d and its characteristics ( B, Ĉ, ν) are given by Bt (c, ω) = (t, B τ +t (ω)), P (τ,x) -a.s. ( ν((c, ω); ds, du × dy) = ν(ω; ds + τ, dy)δ 0 (du), P (τ,x) -a.s.(11) for τ t and (c, ω) ∈ Ω.
Before we state the main theorem of this section, the following lemma provides that nonhomogeneous Itô process are indeed a generalization of rich Feller evolution processes: Lemma 2.6.Let (X t ) t 0 be a rich Feller evolution process on C ∞ (R d ) with generator A s and time dependent symbol p : Proof.Let (X t ) t 0 be a rich Feller evolution system on C ∞ (R d ).Analogously to the proof of Theorem 3.1 of [30], but under usage of the non-homogeneous version of Dynkin's formula as mentioned in [17], we show that (X t ) t 0 is a semimartingale with characteristics (B, C, ν).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 [5] the space-time process X associated to X is a rich Feller process on C ∞ (R + × R d ).Hence, Theorem 3.10 of [30] provides that the characteristics ( B, Ĉ, ν) of X are of the form ν( ; dt, dx) = dtN ( Xt , dx), P (τ,x) -a.s.
and, therefore, we conclude with equations ( 9) to (11) that for t τ .Transition from P (τ,x) to P τ,x yields the statement.
We have observed, for instance in Example 2.4 that a process does not necessary need to be a nonhomogeneous 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!)example: By adding all other starting points in a Markovian manner (c.f.[26]), and considering the truncation function h(x) = x1 {|x| 0.5} we receive a non-homogeneous Markov semimartingale with characteristics B ≡ 0, C ≡ 0 and ν(dt, dx) = δ 1 (dt)δ 1 (dx).In this case, the time-dependent symbol exists and is given by Nevertheless, the symbol does not provide any information regarding the process.

Maximal Inequalitys and Time-Dependent Blumenthal-Getoor Indices
As we have pointed out before, for homogeneous processes like multivariate α-stable processes [2], more general Lévy processes [3], Feller processes satisfying some mild conditions [21] and homogeneous diffusion with jumps [25], 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 [16].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: for every s 0, x, ξ ∈ 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 [21] or Definition 3.8 of [25] to the non-homogeneous case.
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.
In this section we assume all characteristics to be encountered to be with respect to the truncation where R > 0.
Theorem 3.3.Let X be a non-homogeneous Itô process with characteristics as in Theorem 2.7.In this case we have for t 0, R > 0 and a constant c d > 0 which only depends on the dimension d.If, in addition, (IS) holds true, we have where c k > 0 only depends on c 0 of the sector condition (IS).
Remark 3.4.Throughout the following proof we often make use of Lemma 5.2 of [25] 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 [25].We omit the proof of (17) since it is generalized from the proof of Lemma 6.3 of [21] analogously to the following generalization.However, let us mention that one has to use Dynkin's formula for non-homogeneous processes as stated in [17] when Corollary 3.6 is utilized in [21].
In order to prove (16) let X be a non-homogeneous Itô process such that the differential characteristics (ℓ, Q, ν) of X are locally bounded and continuous.At first we show that for S, R and σ := inf{t τ : X t − x > S} as above we have where c d = 4d + 16 c d and t τ .We introduce the stopping time as the first time the jumps of X σ exceed R, and estimate the following 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 τ let Xt : The process X is a special semimartingale on [τ, ∞) with characteristics ) 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 τ : where M is a P τ,x -local martingale on [τ, ∞) by [10] Theorem II.2.42.Applying Lemma 3.7 of [30] we have under (IG) Let us mention, that although Lemma 3.7 of [30] considers homogeneous diffusion with jumps only the proof is alike for non-homogeneous Itô processes.In particular, since M is uniformly bounded it is a L 2 -martingale on [τ, t].We define and obtain for t τ : For the left summand of (21) we estimate for again for t τ : where we used Doob's inequality for the martingale M σ and the Lipschitz property of u in combination with Corollary II.3 of [14].Since we obtain where we have used Lemma 5.2 of [25] on the second term.By choosing a seqence (u n ) n∈N of functions of the type described above which tends to the identity in a monotonous way we obtain Now let us consider the term P τ,x of (21).The Markov inequality provides Again we chose a sequence (u n ) n∈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 → ∞) we obtain + 2 For term (23) we get and for term (24) where we have used again Lemma 5.2 of [25] on the second term.It remains to deal with the second term of (19).Let δ > 0 be fixed (at first) and m : R →]1, 1 + δ[ a strictly monotone increasing auxiliary function.Since m 1 and since we have at least one jump of size > R on {τ R t} we obtain In order to apply Gronwall's Lemma we estimate the following: where the first inequality follows by Proposition 3.3.
Finally we generalize Theorem 2.10 of [13] which is a nice and applicable criterion for the finiteness of p-variation.Theorem 3.8.Let X be a non-homogeneous Itô process such that the differential characteristics of X are locally bounded and continuous.For every t τ 0 and p > sup τ 0 β τ,x ∞ the following holds true sup πn n j=1 X tj − X tj−1 p < +∞, P τ,x -a.s. ( where the supremum is taken over all finite partitions π n = (t i ) i=1,...,n with τ = t 0 < t 1 < ... < t n = t.I.e., the p-variation of the paths of X are P τ,x -a.s finite for p > sup τ 0 β τ,x ∞ .Proof.Let t, r > 0 and λ > p. Then Theorem 28 provides the following estimation α(t, r) := sup P τ,x ( X s − x r) : τ 0, s ∈ for r small enough and K > 0. Hence, Theorem 1.3 of [12] provides the statement.