$L^{p}$-Wasserstein distance for stochastic differential equations driven by Lévy processes

Wang, Jian

doi:10.3150/15-bej705

Cited by 37 publications

(51 citation statements)

References 26 publications

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“…Wang [27] obtained the exponential decay in the L 2 -Wasserstein distance under B(K 1 r, K 2 r, l 0 ), i.e., (1.3) holds with Φ 1 (r) = K 1 r for some K 1 > 0; moreover, he establishes similar results for the L p -Wasserstein distance for all p ≥ 1 provided that the diffusion semigroup is ultracontractive. Some developments in the jump case can be found in [31,17] under B(K 1 r, K 2 r, l 0 ). In particular, the second author [31] obtained exponential convergence rate in the L p -Wasserstein distance for any p ≥ 1 when the Lévy noise in (1.1) has an α-stable component.…”

Section: )mentioning

confidence: 98%

“…Therefore, some new ideas are required for the construction of the coupling. One key ingredient of the proof in the paper relies, similarly to [10,31,17], on using Wasserstein distances of type W ψ defined by (1.5) with appropriately chosen concave test functions ψ, which, in some sense, are comparable with W 1 for the estimate (1.7), or are intermediate between W 1 and the total variation for (1.9). It is worth pointing out that our choice of the concave test function ψ satisfying ψ(r) ≍ r (see Theorem 4.2 below) is quite simple.…”

Section: Exponential Convergence In Wasserstein-type Distancesmentioning

confidence: 99%

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Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

Luo

Wang

2019

Stochastic Processes and their Applications

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We establish the exponential convergence with respect to the L 1 -Wasserstein distance and the total variation for the semigroup corresponding to the stochastic differential equationwhere (Z t ) t≥0 is a pure jump Lévy process whose Lévy measure ν fulfillsfor some constant κ 0 > 0, and the drift term b satisfies that for anywith some positive constants K 2 , l 0 and positive measurable function Φ 1 . The method is based on the refined basic coupling for Lévy jump processes. As a byproduct, we obtain sufficient conditions for the strong ergodicity of the process (X t ) t≥0 .where Φ 1 and Φ 2 are two nonnegative measurable functions, and l 0 ≥ 0 is a constant. For example, when Φ 2 (r) = K 2 r for some positive constantThis holds if the drift term b is dissipative outside some compact set. In particular, when Φ 1 (r) = K 1 r for some constant K 1 ≥ 0, it follows from (1.3) that for anywhich, along with (1.3), yields that the SDE (1.1) has a non-explosive and pathwise unique strong solution, see [1, Chapter 6, Theorem 6.2.3] (in the standard Lipschitz case) or [11, Theorem 2] and [25, Chapter 3, Theorem 115] (in the one-sided Lipschitz case). Note that, sincewe are sometimes concerned with only measurable drift term b, non-Lipschitz condition like B(K 1 , K 2 r, l 0 ) will also be adopted in our results below. The reader can refer to [8,20,22,26,32] and references therein for recent studies on the existence and uniqueness of strong solution to (1.1) with non-regular drift term. In particular, assuming that Z is the truncated symmetric α-stable process on R d with α ∈ (0, 2), and b is bounded and β-Hölder continuous with β > 1 − α/2, it was proved in [8, Corollary 1.4(i)] that the SDE (1.1) has a unique strong solution for each x ∈ R d . Furthermore, in the one-dimensional case, if α > 1, then the SDE (1.1) also enjoys a unique strong solution for each x ∈ R, even if the drift b is only bounded and measurable (see [26, Remark 1, p. 82]). Denote by ν the Lévy measure of the pure jump Lévy process Z. We assume that there is a constant κ 0 > 0 such that(1.4)Condition (1.4) was first used in [23] to study the coupling property of Lévy processes. It is satisfied by a large class of Lévy measures. For instance, iffor some z 0 ∈ R d and some ε > 0 such that ρ 0 (z) is positive and continuous on B(z 0 , ε), then such Lévy measure ν fulfills (1.4), see [24, Proposition 1.5] for details. Actually, as shown in Proposition 6.5, the condition (1.4) implies that there is a nonnegative measurable function ρ on R d such that ν(dz) ≥ ρ(z) dz andLet (P t ) t≥0 be the transition semigroup associated with the process (X t ) t≥0 . In this paper we are interested in the asymptotics of the Wasserstein-type distances (including the L 1 -Wasserstein distance and the total variation) between probability distributions δ x P t = P t (x, ·) and δ y P t = P t (y, ·) for any x, y ∈ R d , when the drift term b is dissipative outside some compact set, i.e. b satisfies B(Φ 1 (r), K 2 r, l 0 ) for some positive measurable function Φ 1 , and some const...

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Section: )mentioning

confidence: 98%

Section: Exponential Convergence In Wasserstein-type Distancesmentioning

confidence: 99%

Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

Luo

Wang

2019

Stochastic Processes and their Applications

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“…For a large class of Lévy processes whose associated Lévy measure has a rotationally invariant absolutely continuous component, Majka obtained in [15] the exponential convergence rates with respect to both the L 1 -Wasserstein distance and the total variation. Recently, the results of [15,28] are extended and improved in [14], where the associated Lévy measure of Lévy process is only assumed to have an absolutely continuous component. It is noticed that all the works above are restricted to the additive noise case.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling

Liang

Wang

2020

Stochastic Processes and their Applications

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We consider SDEs driven by multiplicative pure jump Lévy noises, where Lévy processes are not necessarily comparable to α-stable-like processes. By assuming that the SDE has a unique solution, we obtain gradient estimates of the associated semigroup when the drift term is locally Hölder continuous, and we establish the ergodicity of the process both in the L 1 -Wasserstein distance and the total variation, when the coefficients are dissipative for large distances. The proof is based on a new explicit Markov coupling for SDEs driven by multiplicative pure jump Lévy noises, which is derived for the first time in this paper.

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“…We also emphasize that the above argument still works if Z is of the form Z = Z ′ + Z ′′ where Z ′ , Z ′′ are independent Lévy processes and Z ′′ is rotationally symmetric, see e.g. [31]. Let ν be the Lévy measure of the Lévy process Z = (Z t ) t 0 .…”

Section: 1mentioning

confidence: 99%

A unified approach to coupling SDEs driven by Lévy noise and some applications

Liang¹,

Schilling²,

Wang³

2020

Bernoulli

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We present a general method to construct couplings of stochastic differential equations driven by Lévy noise in terms of coupling operators. This approach covers both coupling by reflection and refined basic coupling which are often discussed in the literature. As an application, we establish regularity results for the transition semigroups of the solutions to stochastic differential equations driven by additive Lévy noise.2010 Mathematics Subject Classification. 60J35; 60G51; 60H10; 60J25; 60J75.

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$L^{p}$-Wasserstein distance for stochastic differential equations driven by Lévy processes

Cited by 37 publications

References 26 publications

Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling

A unified approach to coupling SDEs driven by Lévy noise and some applications

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