Stochastic differential equations with Sobolev drifts and driven by α-stable processes

Xicheng Zhang

doi:10.1214/12-AIHP476

Stochastic differential equations with Sobolev drifts and driven by α-stable processes

Xicheng Zhang^*

^*Corresponding author for this work

Wuhan University

Research output: Contribution to journal › Article › peer-review

39 Citations (Scopus)

Abstract

In this article we prove the pathwise uniqueness for stochastic differential equations in ℝ^d with time-dependent Sobolev drifts, and driven by symmetric α-stable processes provided that α ∈ (1, 2) and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when α ∈ (2d/d+1, 2). Our proof is based on some estimates of Krylov's type for purely discontinuous semimartingales.

Original language	English
Pages (from-to)	1057-1079
Number of pages	23
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	49
Issue number	4
DOIs	https://doi.org/10.1214/12-AIHP476
Publication status	Published - Nov 2013
Externally published	Yes

Keywords

Fractional Sobolev space
Krylov's estimate
Pathwise uniqueness
Symmetric α-stable process

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10.1214/12-AIHP476

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abstract = "In this article we prove the pathwise uniqueness for stochastic differential equations in ℝd with time-dependent Sobolev drifts, and driven by symmetric α-stable processes provided that α ∈ (1, 2) and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when α ∈ (2d/d+1, 2). Our proof is based on some estimates of Krylov's type for purely discontinuous semimartingales.",

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PY - 2013/11

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N2 - In this article we prove the pathwise uniqueness for stochastic differential equations in ℝd with time-dependent Sobolev drifts, and driven by symmetric α-stable processes provided that α ∈ (1, 2) and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when α ∈ (2d/d+1, 2). Our proof is based on some estimates of Krylov's type for purely discontinuous semimartingales.

AB - In this article we prove the pathwise uniqueness for stochastic differential equations in ℝd with time-dependent Sobolev drifts, and driven by symmetric α-stable processes provided that α ∈ (1, 2) and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when α ∈ (2d/d+1, 2). Our proof is based on some estimates of Krylov's type for purely discontinuous semimartingales.

KW - Fractional Sobolev space

KW - Krylov's estimate

KW - Pathwise uniqueness

KW - Symmetric α-stable process

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JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

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ER -

Stochastic differential equations with Sobolev drifts and driven by α-stable processes

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Keywords

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