On multi-dimensional SDEs with locally integrable coefficients

V. P. Kurenok; A. N. Lepeyev

doi:10.1216/RMJ-2008-38-1-139

On multi-dimensional SDEs with locally integrable coefficients

V. P. Kurenok
, A. N. Lepeyev

Department of Electrical & Systems Engineering

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

6 Scopus citations

Abstract

We consider the multi-dimensional stochastic equation X_t = x₀ + ∫₀^t B(s, X_s dW_s + ∫₀^t A(s, X_s) d_s where x ₀ is an arbitrary initial value, W is a d-dimensional Wiener process and B : [0, + ∞) × R^d → R^d2, A : [0, + ∞) × R^d -rarr; R^d are measurable diffusion and drift coefficients, respectively. Our main result states sufficient conditions for the existence of (possibly, exploding) weak solutions. These conditions are some local integrability conditions of coefficients B and A. From one side, they extend the conditions from [3] where the corresponding SDEs without drift were considered. On the other hand, our results generalize the existence theorems for one-dimensional SDEs with drift studied in [4]. We also discuss the time-independent case.

Original language	English
Pages (from-to)	139-174
Number of pages	36
Journal	Rocky Mountain Journal of Mathematics
Volume	38
Issue number	1
DOIs	https://doi.org/10.1216/RMJ-2008-38-1-139
State	Published - 2008

Keywords

Krylov's estimates
Locally integrable coefficients
Multi-dimensional stochastic differential equations
Weak convergence
Wiener process

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10.1216/RMJ-2008-38-1-139

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@article{9f7928dd15354fed95a17c0a90150639,

title = "On multi-dimensional SDEs with locally integrable coefficients",

abstract = "We consider the multi-dimensional stochastic equation Xt = x0 + ∫0t B(s, Xs dWs + ∫0t A(s, Xs) ds where x 0 is an arbitrary initial value, W is a d-dimensional Wiener process and B : [0, + ∞) × Rd → Rd2, A : [0, + ∞) × Rd -rarr; Rd are measurable diffusion and drift coefficients, respectively. Our main result states sufficient conditions for the existence of (possibly, exploding) weak solutions. These conditions are some local integrability conditions of coefficients B and A. From one side, they extend the conditions from [3] where the corresponding SDEs without drift were considered. On the other hand, our results generalize the existence theorems for one-dimensional SDEs with drift studied in [4]. We also discuss the time-independent case.",

keywords = "Krylov's estimates, Locally integrable coefficients, Multi-dimensional stochastic differential equations, Weak convergence, Wiener process",

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year = "2008",

doi = "10.1216/RMJ-2008-38-1-139",

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volume = "38",

pages = "139--174",

journal = "Rocky Mountain Journal of Mathematics",

issn = "0035-7596",

number = "1",

}

TY - JOUR

T1 - On multi-dimensional SDEs with locally integrable coefficients

AU - Kurenok, V. P.

AU - Lepeyev, A. N.

PY - 2008

Y1 - 2008

N2 - We consider the multi-dimensional stochastic equation Xt = x0 + ∫0t B(s, Xs dWs + ∫0t A(s, Xs) ds where x 0 is an arbitrary initial value, W is a d-dimensional Wiener process and B : [0, + ∞) × Rd → Rd2, A : [0, + ∞) × Rd -rarr; Rd are measurable diffusion and drift coefficients, respectively. Our main result states sufficient conditions for the existence of (possibly, exploding) weak solutions. These conditions are some local integrability conditions of coefficients B and A. From one side, they extend the conditions from [3] where the corresponding SDEs without drift were considered. On the other hand, our results generalize the existence theorems for one-dimensional SDEs with drift studied in [4]. We also discuss the time-independent case.

AB - We consider the multi-dimensional stochastic equation Xt = x0 + ∫0t B(s, Xs dWs + ∫0t A(s, Xs) ds where x 0 is an arbitrary initial value, W is a d-dimensional Wiener process and B : [0, + ∞) × Rd → Rd2, A : [0, + ∞) × Rd -rarr; Rd are measurable diffusion and drift coefficients, respectively. Our main result states sufficient conditions for the existence of (possibly, exploding) weak solutions. These conditions are some local integrability conditions of coefficients B and A. From one side, they extend the conditions from [3] where the corresponding SDEs without drift were considered. On the other hand, our results generalize the existence theorems for one-dimensional SDEs with drift studied in [4]. We also discuss the time-independent case.

KW - Krylov's estimates

KW - Locally integrable coefficients

KW - Multi-dimensional stochastic differential equations

KW - Weak convergence

KW - Wiener process

UR - https://www.scopus.com/pages/publications/42449104371

U2 - 10.1216/RMJ-2008-38-1-139

DO - 10.1216/RMJ-2008-38-1-139

M3 - Article

AN - SCOPUS:42449104371

SN - 0035-7596

VL - 38

SP - 139

EP - 174

JO - Rocky Mountain Journal of Mathematics

JF - Rocky Mountain Journal of Mathematics

IS - 1

ER -

On multi-dimensional SDEs with locally integrable coefficients

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Keywords

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