On Martingale Problems and Feller Processes

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.


Introduction
Let (Lt)t≥0 be a k-dimensional Lévy process with characteristic exponent ψ ∶ R d → C and σ ∶ R d → R d×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) and the pseudo-differential operator A with symbol q(x, ξ) ∶= ψ(σ(x) T ξ), i. e.
wheref 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 µ is equivalent to the well-posedness of the (A, C ∞ c (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 ν satisfies ν({y ∈ R k ; σ(x) ⋅ y + x ≤ r}) x →∞ → 0 for all r > 0 which is equivalent to saying that A maps C ∞ c (R d ) into C∞(R d ), the space of continuous functions vanishing at infinity.
In this paper, we are interested in the following more general question: Consider a pseudodifferential operator A with continuous negative definite symbol q, (1 − e iy⋅ξ + iy ⋅ ξ1 (0,1) ( y )) ν(x, dy), x, ξ ∈ R d , such that the (A, C ∞ c (R d ))-martingale problem is well-posed, i. e. for any initial distribution µ there exists a unique solution to the (A, C ∞ c (R d ))-martingale problem. Under which assumptions does the well-posedness of the (A, C ∞ c (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 (R d ), the pseudo-differential operator A, it is clear that A has to satisfy Af ∈ C∞(R d ) for all f ∈ C ∞ c (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.
1.1 Theorem Let A be a pseudo-differential operator with continuous negative definite symbol q such that q(⋅, 0) = 0 and A maps C ∞ c (R d ) into C∞(R d ). If the (A, C ∞ c (R d ))-martingale problem is well-posed and lim x →∞ then the solution (Xt)t≥0 to the martingale problem is a conservative rich Feller process with symbol q.

Remark (i). If the martingale problem is well-posed and
The growth condition (G) is needed to prove the Feller property; that is, to show that Ttf vanishes at infinity for any f ∈ C∞(R d ) and t ≥ 0.
(ii). There is a partial converse to Theorem 1.1: If (Xt)t≥0 is a Feller process and C ∞ c (R d ) is a core for the generator A of (Xt)t≥0, then the (A, C ∞ c (R d ))-martingale problem is well-posed, see e. g. [5,Theorem 4.10.3] or [11, Theorem 1.37] for a proof.
can be equivalently formulated in terms of the symbol q and its characteristics, cf. Lemma 2.1.
(iv). For the particular case that A is the pseudo-differential operator associated with the SDE (1), i. e. q(x, ξ) = ψ(σ(x) T ξ), we recover [7, Theorem 1.1]. Note that the growth condition (G) is automatically satisfied for any function σ 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 Let
A be a pseudo-differential operator with symbol q such that q(⋅, 0) = 0, Assume that there exists a sequence (q k ) k∈N of symbols such that q k (x, ξ) = q(x, ξ) for all x < k, ξ ∈ R d , and the pseudo-differential operator )-martingale problem is well posed for all k ≥ 1, then there exists conservative rich Feller process (Xt)t≥0 with symbol q, and (Xt)t≥0 is the unique solution to the (A, C ∞ c (R d ))-martingale problem.

Preliminaries
We consider R d endowed with the Borel σ-algebra B(R d ) and write B(x, r) for the open ball centered at x ∈ R d with radius r > 0; R d ∆ is the one-point compactification of R d . If a certain statement holds for x ∈ R d with x sufficiently large, we write "for x ≫ 1". For a metric space (E, d) we denote by C(E) the space of continuous functions f ∶ E → R; C∞(E) (resp. C b (E)) is the space of continuous functions which vanish at infinity (resp. are bounded).
if f is right-continuous and has left-hand limits in E. We will always consider E = R d or E = R d ∆ . An E-valued Markov process (Ω, A, P x , x ∈ E, Xt, t ≥ 0) with càdlàg (right-continuous with left-hand limits) sample paths is called a Feller process if the associated semigroup (Tt)t≥0 defined by has the Feller property, i. e. Ttf ∈ C∞(E) for all f ∈ C∞(E), and (Tt)t≥0 is strongly continuous → 0 for any f ∈ C∞(E). Following [13] we call a Markov process (Xt)t≥0 with càdlàg sample paths a C b -Feller process if If the smooth functions with compact support C ∞ c (R d ) are contained in the domain of the generator (L, D(L)) of a Feller process (Xt)t≥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 R d -valued rich Feller process is, when restricted to C ∞ c (R d ), a pseudo-differential operator with negative definite symbol: (1 − e iy⋅ξ + iy ⋅ ξ1 (0,1) ( y )) ν(x, dy).
We call q the symbol of the Feller process (Xt)t≥0 and of the pseudo-differential operator; (b, Q, ν) are the characteristics of the symbol q. For each fixed x ∈ R d , (b(x), Q(x), ν(x, dy)) is a Lévy triplet, i. e. b(x) ∈ R d , Q(x) ∈ R d×d is a symmetric positive semidefinite matrix and ν(x, dy) a σ-finite measure on (R d {0}, B(R d {0})) satisfying ∫ y≠0 min{ y 2 , 1} ν(x, dy) < ∞. 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, ⋅)) x∈R d is locally bounded if for any compact set K ⊆ R d there exists c > 0 such that q(x, ξ) ≤ c(1 + ξ 2 ) for all x ∈ K, ξ ∈ R d . By [ If (3) holds for K = R d , we say that q has bounded coefficients. We will frequently use the following result.

Lemma
Let L be a pseudo-differential operator with continuous negative definite symbol q and characteristics (b, Q, ν). Assume that q(⋅, 0) = 0 and that q is locally bounded.
(ii). If lim x →∞ sup ξ ≤ x −1 Re q(x, ξ) = 0, then (4) holds. If the symbol q of a rich Feller process (Lt)t≥0 does not depend on x, i. e. q(x, ξ) = q(ξ), then (Lt)t≥0 is a Lévy process. This is equivalent to saying that (Lt)t≥0 has stationary and independent increments and càdlàg sample paths. The symbol q = q(ξ) 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) if any two weak solutions of the SDE have the same finite-dimensional distributions. We refer the reader to Situ [15] for further details.
is a P µ -martingale with respect to the canonical filtration of (Xt)t≥0 for any f ∈ D.

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
Let (q(x, ⋅)) x∈R d be a family of continuous negative definite functions with char- if, and only if, there exists an absolute constant c > 0 such that each of the following conditions is satisfied for x ≫ 1.
If (G) holds and q is locally bounded, cf.
Proof. First we prove that (i)-(iii) are sufficient for (G). Because of (i) and (ii) it suffices to show that satisfies the linear growth condition (G). Using the elementary estimates for all x ≥ 1 which implies by (ii) and (iii) that lim sup 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 has a finite second moment, i. e. ∫ R d η 2 g(η) dη < ∞, and satisfies we obtain by applying Tonelli's theorem for any continuous negative definite function ψ, cf. [1, Proposition 2.17d)], we get and this gives (iii) for x ≫ 1. Next we prove (ii). First of all, we note that and therefore Q(x) ≤ c(1 + x 2 ) is a direct consequence of (G). On the other hand, Using (5) and applying Tonelli's theorem once more, we find Hence, and (ii) follows. Finally, as (ii) and (iii) imply that lim sup see the first part of the proof, a straightforward application of the triangle inequality gives lim sup which proves (i).

Corollary
Let A be a pseudo-differential operator with continuous negative definite symbol and q satisfies the linear growth condition (G), then there exists for any initial distribution µ a solution to the (A, C ∞ c (R d ))martingale problem which is conservative.
and therefore [10,Corollary 3.2] shows that the solution with initial distribution δx does not explode in finite time with probability 1. By construction, see [2, proof of Theorem 4.5.4], the mapping then P µ gives rise to a conservative solution to the (A, C ∞ c (R d ))-martingale problem with initial condition µ.
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
Let L be a pseudo-differential operator with continuous negative definite symbol p and characteristics (b, Q, ν) such that p(⋅, 0) = 0 and L( (i). For any initial distribution µ there exists a probability measure P µ on D([0, ∞), R d ) such that the canonical process (Yt)t≥0 solves the (L, C ∞ c (R d ))-martingale problem and (ii). For any t ≥ 0, R > 0 and ε > 0 there exist constants > 0 and δ > 0 such that for any initial distribution µ such that µ(B(0, )) ≤ δ.
(iii). For any t ≥ 0, ε > 0 and any compact set K ⊆ R d there exists R > 0 such that 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 (R) into C∞(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 ∈ R. Set u(x) ∶= 1 (1 + x 2 ), x ∈ R, then Clearly, for all x ≫ 1. By Lemma 3.1 and (9) there exists a constant c1 > 0 such that I1 ≤ c1u(x) for all x ∈ R. On the other hand, Taylor's formula shows for some intermediate value ζ = ζ(x, y) between x and x+y. Since y < x 2, we have ζ ≥ x 2; hence, by (9), Applying Lemma 3.1, we find that there exists a constant c2 > 0 such that Consequently, Lu(x) ≤ (c1 + c2)u(x) for all x ≫ 1. As Lu is bounded and u is bounded away from 0 on compact sets, we can choose a constant c3 > 0 such that Define τ R ∶= inf{t ≥ 0; Yt < R}. Using a standard truncation and stopping technique it follows that

Lu(Ys) ds .
Hence, by (⋆), An application of Gronwall's inequality shows that there exists a constant C > 0 such that By the Markov inequality, this implies that If µ is an initial distribution such that µ(B(0, )) ≤ δ, then E µ u(Y0) ≤ δ + −2 . Choosing sufficiently large and δ > 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) ≤ cv(x) for all x ∈ R. Applying Gronwall's inequality another time, we find a constant C > 0 such that where σ R ∶= inf{t ≥ 0; Yt ≥ R} denotes the exit time from the ball B(0, R). Hence, by the Markov inequality, In particular we can choose for any compact set K ⊆ R and any ε > 0 some R > 0 such that Now if µ is an initial distribution such that µ(K c ) ≤ ε 2, then, by (6), 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 Let
A be a pseudo-differential operator with negative definite symbol q such that )-martingale problem is well-posed and the unique solution (Xt)t≥0 satisfies the compact containment condition Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1. The well-posedness implies that the solution (Xt)t≥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 (Xt)t≥0 is a Feller process, we have to show that the semigroup (i). continuity at t = 0: Ttf (x) → f (x) as t → 0 for any x ∈ R d and f ∈ C∞(R d ).
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 ν(x, dy) = νs(x, dy) + ν l (x, dy) where are the small jumps and large jumps, respectively, and denote by p the symbol with characteristics (b, Q, νs). By Corollary 3.2 there exists for any initial distribution µ a conservative solution to the (p(x, D), C ∞ c (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 (R d ))-martingale problem with initial distribution µ = δx: σj are the jump times of a Poisson process (Nt)t≥0 with intensity λ ∶= sup z∈R d ν l (z, R d {0}), i. e. σj ∼ Exp(λ) are independent and identically distributed. Note that λ < ∞ by Lemma 3.1.
• P x is a probability measure which depends on the initial distribution µ = δx of (Xt)t≥0.
Note that if we define a linear operator P by then (8) implies that Before we proceed with the proof, let us give a remark on the construction of (Xt)t≥0. The intensity of the Poisson process (Nt)t≥0, which announces the "large jumps", is 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 there is no large jump at time σ k−1 , and therefore the effective jump intensity at time We will prove that (Xt)t≥0 has the Feller property. To this end, we first show that for any t ≥ 0, ε > 0, k ≥ 1 and any compact set K ⊆ R d there exists R > 0 such that for all x ∈ K, j = 0, . . . , k; we prove (13) by induction. Note that µj = µj(x) depends on the initial distribution of (Xt)t≥0.
• k → k + 1: Because of Lemma 3.3(ii) and the induction hypothesis, it suffices to show that there exists a compact set Choose m ≥ 0 sufficiently large such that P x (σ k ≥ m) ≤ ε ′ ∶= ε 8, and choose R > 0 such that (13) holds with ε ∶= ε ′ , t ∶= m. Then, by (12) and our choice of R, The second term on the right-hand side converges to 0 as r → ∞, cf. [13,Theorem 4.4] or [10, Theorem A.1], and therefore we can choose r > 0 sufficiently large such that For fixed ε > 0 choose k ≥ 1 such that P x (Nt ≥ k + 1) ≤ ε. By definition of (Xt)t≥0 and (13), we get Thus, by Lemma 3.4, x ↦ Ttf (x) = E x f (Xt) is continuous for any f ∈ C∞(R d ). It remains to show that Ttf vanishes at infinity; to this end we will show that for any r > 0, ε > 0 there exists a constant M > 0 such that It follows from Lemma 3.1 and the very definition of λ that P f defined in (11) is bounded and i. e. P f vanishes at infinity for any f ∈ C∞(R d ). We claim that for any k ≥ 0, ε > 0, t ≥ 0 and r > 0 there exists a constant M > 0 such that We prove (15) by induction.
• k → k + 1: For fixed r > 0 choose δ > 0 and > 0 as in 3.3(ii). By 3.3(ii) it suffices to show that there exists M > 0 such that (Note that µ k+1 = µ k+1 (x) depends on the initial distribution of (Xt)t≥0.) Pick a cut-off , then by (10), If P χ ∞ = 0 this proves (⋆). If P χ ∞ > 0, then we can choose m ≥ 1 such that P x (σ1 ≥ m) ≤ δ (2 P χ ∞). Since P χ vanishes at infinity, we have sup z ≥R P χ(z) ≤ λδ 4 for R > 0 sufficiently large. By the induction hypothesis, there exists M > 0 such that (15) holds with ε ∶= λδ 4, r ∶= R and t ∶= m. Then for all s ≤ m and x ≥ M , and therefore For fixed ε > 0 and t ≥ 0 choose k ≥ 1 such that P x (Nt ≥ k + 1) ≤ ε. Choose M > 0 as in (15), then Consequently, we have shown that (Xt)t≥0 is a Feller process. Since (Xt)t≥0 solves the where τ x r ∶= inf{t ≥ 0; Xt − x ≥ r} denotes the exit time from the ball B(x, r). Using that , it is not difficult to see that the generator of (Xt)t≥0 is, when restricted to C ∞ c (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 (Xt)t≥0 is a rich Feller process with symbol q.
Proof of Corollary 1.3. By Corollary 3.2 there exists for any initial distribution µ a solution to the (A, C ∞ c (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 (R d ))-martingale problem is well-posed. Now the assertion follows from Theorem 1.1.

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
Let (Lt)t≥0 be a k-dimensional Lévy process with characteristic exponent ψ and Lévy triplet (b, Q, ν). Let ∶ R d → R d , σ ∶ R d → R d×k be continuous functions which grow at most linearly. If has for any initial distribution µ a weak solution (Xt)t≥0 which is conservative.
e. g. if σ is at most of sublinear growth, cf. Lemma 2.1(ii).
Since q is locally bounded and x ↦ q(x, ξ) is continuous for all ξ ∈ R d it follows from (17) that cf. Lemma 2.1. Because , σ are at most of linear growth, q satisfies the growth condition (G). Applying Corollary 3.2 we find that there exists a conservative solution (Xt)t≥0 to the (A, C ∞ c (R d ))-martingale problem. By [6], (Xt)t≥0 is a weak solution to the SDE (17).

Theorem
Let (Lt)t≥0 be a k-dimensional Lévy process with Lévy triplet (b, Q, ν) and characteristic exponent ψ. Suppose that there exist α, β ∈ (0, 1] such that the Lévy-driven SDE has a unique weak solution for any initial distribution µ and any two bounded has a unique weak solution for any ∈ C α loc (R d , R d ), σ ∈ C β loc (R d , R d×k ) which are at most of linear growth and satisfy and The unique weak solution is a conservative rich Feller process with symbol Proof. Let ∈ C α loc (R d , R d ) and σ ∈ C β loc (R d , R d×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 (R d )) ⊆ C∞(R d ). Moreover, since σ, are at most of linear growth, the growth condition (G) is clearly satisfied. Set By assumption, the SDE has a unique weak solution for any initial distribution µ for all k ≥ 1. By [6] (see also [7,Lemma 3.3]) this implies that the (A k , C ∞ c (R d ))-martingale problem for the pseudo-differential operator with symbol q k (x, ξ) → 0 for all r > 0, and therefore Lemma 2.1 shows that . Now the assertion follows from Corollary 1.3.
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, ξ) = ϕ(x)q(x, ξ). Recall that p(x, D) denotes the pseudo-differential operator with symbol p.