Articlegreen2012

PATHWISE UNIQUENESS FOR SINGULAR SDEs DRIVEN BY STABLE PROCESSES

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Stochastic processes and financial applications Nonlinear Partial Differential Equations Differential Equations and Boundary Problems

Abstract

We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric $\alpha$-stable L\'evy processes with values in $\R^d$ having a bounded and $\beta$-Holder continuous drift term. We assume $\displaystyle{ \beta > 1 - {\alpha}/{2} }$ and $\alpha \in [ 1, 2)$. The proof requires analytic regularity results for the associated integro-differential operators of Kolmogorov type. We also study differentiability of solutions with respect to initial conditions and the homeomorphism property.

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2005Stochastic Integration and Differential Equations 2004Lévy Processes and Stochastic Calculus 1974A TRANSFORMATION OF THE PHASE SPACE OF A DIFFUSION PROCESS THAT REMOVES THE DRIFT 1981ON STRONG SOLUTIONS AND EXPLICIT FORMULAS FOR SOLUTIONS OF STOCHASTIC INTEGRAL EQUATIONS 1996On the existence of smooth densities for jump processes 2005Strong solutions of SDES with singular drift and Sobolev diffusion coefficients 2007Uniqueness of Solutions of Stochastic Differential Equations

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