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

Liang, Mingjie; Schilling, René L.; Wang, Jian

doi:10.3150/19-bej1148

Cited by 9 publications

(8 citation statements)

References 41 publications

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“…then there exists a unique weak solution (X t ) t≥0 to (35), and it gives rise to a Feller process with infinitesimal generator (A, D(A)). The associated Favard space F 1 of order 1 satisfies…”

Section: Applicationsmentioning

confidence: 99%

“…Intuitively one would expect that (38) holds for α ∈ (0, 2). If we knew that the semigroup (P t ) t≥0 of the solution to (35) satisfies…”

Section: Applicationsmentioning

confidence: 99%

“…(iii) Using coupling methods, Liang et. al [35] recently studied the regularity of semigroups associated with solutions to SDEs with additive noise…”

Section: Applicationsmentioning

confidence: 99%

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Schauder Estimates for Poisson Equations Associated with Non-local Feller Generators

Kühn

2020

J Theor Probab

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We show how Hölder estimates for Feller semigroups can be used to obtain regularity results for solutions to the Poisson equation Af = g associated with the (extended) infinitesimal generator of a Feller process. The regularity of f is described in terms of Hölder-Zygmund spaces of variable order and, moreover, we establish Schauder estimates. Since Hölder estimates for Feller semigroups have been intensively studied in the last years, our results apply to a wide class of Feller processes, e. g. random time changes of Lévy processes and solutions to Lévydriven stochastic differential equations. Most prominently, we establish Schauder estimates for the Poisson equation associated with the fractional Laplacian of variable order. As a by-product, we obtain new regularity estimates for semigroups associated with stable-like processes.2010 Mathematics Subject Classification. Primary: 60J25. Secondary: 45K05,35B65,60J35, 60J75.

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Section: Applicationsmentioning

confidence: 99%

“…Intuitively one would expect that (38) holds for α ∈ (0, 2). If we knew that the semigroup (P t ) t≥0 of the solution to (35) satisfies…”

Section: Applicationsmentioning

confidence: 99%

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Schauder Estimates for Poisson Equations Associated with Non-local Feller Generators

Kühn

2020

J Theor Probab

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“…Observe that in both situations non-degeneracy of Φ 2 (x)Φ ′ 2 (x) has been assumed. In the case when Φ 3 (x) ≡ Φ 3 ∈ R d×q the problem has been considered in [2,50,53], and for non-constant (and non-degenerated) Φ 3 (x) in [51].…”

Section: On Structural Properties Of Ltpsmentioning

confidence: 99%

“…Also, when the underlying process is given as a solution to an SDE of the form eq. (1.7), we refer to [50,51,53] and the references therein.…”

Section: On Structural Properties Of Ltpsmentioning

confidence: 99%

Periodic Homogenization of a Lévy-Type Process with Small Jumps

Sandrić¹,

Valentić²,

Wang³

2019

Preprint

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In this article, we consider the problem of periodic homogenization of a Feller process generated by a pseudo-differential operator, the so-called Lévytype process. Under the assumptions that the generator has rapidly periodically oscillating coefficients, and that it admits "small jumps" only (that is, the jump kernel has finite second moment), we prove that the appropriately centered and scaled process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the generator. The presented results generalize the classical and well-known results related to periodic homogenization of a diffusion process.

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Periodic homogenization of a Lévy-type process with small jumps

2020

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In this article, we consider the problem of periodic homogenization of a Feller process generated by a pseudo-differential operator, the so-called Lévy-type process. Under the assumptions that the generator has rapidly periodically oscillating coefficients, and that it admits “small jumps” only (that is, the jump kernel has finite second moment), we prove that the appropriately centered and scaled process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the generator. The presented results generalize the classical and well-known results related to periodic homogenization of a diffusion process.

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

Cited by 9 publications

References 41 publications

Schauder Estimates for Poisson Equations Associated with Non-local Feller Generators

Schauder Estimates for Poisson Equations Associated with Non-local Feller Generators

Periodic Homogenization of a Lévy-Type Process with Small Jumps

Periodic homogenization of a Lévy-type process with small jumps

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