Pathwise uniqueness of multi-dimensional stochastic differential equations with Hölder diffusion coefficients

Abstract

We extend Yamada-Watababe’s criterion [J. Math. Kyoto Univ., 1971, 11: 553–563] on the pathwise uniqueness of one-dimensional stochastic differential equations to a special class of multi-dimensional stochastic differential equations.

Keywords

Stochastic differential equation (SDE) / strong solution / pathwise uniqueness / Hölder continuity

References

[1] Dawson D. A., Fleischmann K., Xiong J. Strong uniqueness for cyclically symbiotic branching diffusions. Statist Probab Lett, 2005, 73, 251-257

[2] Fang S. Canonical Brownian motion on the diffeomorphism group of the circle. J Funct Anal, 2002, 196, 162-179

[3] Fang S., Imkeller P., Zhang T. Global flows for stochastic differential equations without global Lipschitz conditions. Ann of Probab, 2007, 35, 180-205

[4] Fang S., Luo D. Flow of homeomorphisms and stochastic transport equations. Stoch Anal Appl, 2007, 25, 1079-1108

[5] Fang S., Luo D., Thalmaier A. Stochastic differential equations with coefficients in Sobolev spaces. J Funct Anal, 2010, 259, 1129-1168

[6] Fang S., Zhang T. A study of a class of differential equations with non-Lipschitzian coefficients. Probab Theory Relat Fields, 2005, 132, 356-390

[7] Fang S., Zhang T. Isotropic stochastic flow of homeomorphisms on Sd for the critical Sobolev exponent. J Math Pures et Appl, 2006, 85, 580-597

[8] He H. Strong uniqueness for a class of singular SDEs for catalytic branching diffusions. Statist Probab Lett, 2009, 79, 182-187

[9] Ikeda N., Watanabe S. Stochastic Differential Equations and Diffusion Processes, 1989, 2nd ed., Amsterdam: North-Holland.

[10] Krylov N. V., Röckner M. Strong solutions of stochastic equations with singular time dependent drift. Prob Theory Relat Fields, 2005, 131, 154-196

[11] Kunita H. Stochastic Flows and Stochastic Differential Equations, 1990, Cambridge: Cambridge University Press.

[12] Luo D. Regularity of solutions to differential equations with non-Lipschitz coefficients. Bull Sci Math, 2008, 132, 257-271

[13] Malliavin P. The canonical diffusion above the diffeomorphism group of the circle. C R Acad Sci, 1999, 329, 325-329

[14] Suresh Kumar K. A class of degenerate stochastic differential equations with non-Lipschitz coefficients. http://arxiv.org/abs/0904.2629

[15] Yamada T., Watanabe S. On the uniqueness of stochastic differential equations. J Math Kyoto Univ, 1971, 11, 553-563

[16] Zhang X. Homeomorphic flows for multi-dimensional SDEs with non-Lipschitz coefficients. Stochastic Process Appl, 2005, 115, 435-448

[17] Zhang X. Strong solutions of SDEs with singular drift and Sobolev diffusion coefficients. Stochastic Process Appl, 2005, 115, 1805-1818

[18] Zhang X. Stochastic flows of SDEs with irregular coefficients and stochastic transport equations. Bull Sci Math, 2010, 134, 340-378