Strong existence and uniqueness for stable stochastic differential equations with distributional drift

Siva Athreya; Oleg Butkovsky; Leonid Mytnik

doi:10.1214/19-AOP1358

Strong existence and uniqueness for stable stochastic differential equations with distributional drift

Siva Athreya, Oleg Butkovsky, Leonid Mytnik

Data and Decision Sciences

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

Abstract

We consider the stochastic differential equation dX_t b(X_t +dL_t, where the drift b is a generalized function and L is a symmetric one dimensional α-stable Levy processes, α ∈ (1, 2). We define the notion of solution to this equation and establish strong existence and uniqueness whenever b belongs to the Besov-Holder space Cβ for β >1/2-α/2.

Original language	English
Pages (from-to)	178-210
Number of pages	33
Journal	Annals of Probability
Volume	48
Issue number	1
DOIs	https://doi.org/10.1214/19-AOP1358
State	Published - Jan 2020

Keywords

Regularization by noise
Stable processes
Stochastic differential equations
Strong solution
Zvonkin transformation

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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abstract = "We consider the stochastic differential equation dXt b(Xt +dLt, where the drift b is a generalized function and L is a symmetric one dimensional α-stable Levy processes, α ∈ (1, 2). We define the notion of solution to this equation and establish strong existence and uniqueness whenever b belongs to the Besov-Holder space Cβ for β >1/2-α/2.",

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N2 - We consider the stochastic differential equation dXt b(Xt +dLt, where the drift b is a generalized function and L is a symmetric one dimensional α-stable Levy processes, α ∈ (1, 2). We define the notion of solution to this equation and establish strong existence and uniqueness whenever b belongs to the Besov-Holder space Cβ for β >1/2-α/2.

AB - We consider the stochastic differential equation dXt b(Xt +dLt, where the drift b is a generalized function and L is a symmetric one dimensional α-stable Levy processes, α ∈ (1, 2). We define the notion of solution to this equation and establish strong existence and uniqueness whenever b belongs to the Besov-Holder space Cβ for β >1/2-α/2.

KW - Regularization by noise

KW - Stable processes

KW - Stochastic differential equations

KW - Strong solution

KW - Zvonkin transformation

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Strong existence and uniqueness for stable stochastic differential equations with distributional drift

Abstract

Keywords

ASJC Scopus subject areas

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