Stochastic flows for Lévy processes with Hölder drifts

By Zhen-Qing Chen, Renming Song, Xicheng Zhang. Revista matemática iberoamericana, 2015

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In this paper, we study the following stochastic differential equation (SDE) in R d Rd: dX t =dZ t +b(t,X t )dt,X 0 =x, whereZ is a Levy process. We show that for a large class of Levy processes Z and Holder continuous drifts b , the SDE above has a unique strong solution for every starting point x∈Rd. Moreover, these strong solutions form a C1 -stochastic flow. As a consequence, we show that, when Z is an α -stable-type Levy process with α∈(0,2) and b is a bounded β -Holder continuous function with β∈(1−α/2,1) , the SDE above has a unique strong solution. When α∈(0,1) , this in particular partially solves an open problem from Priola. Moreover, we obtain a Bismut type derivative formula for ∇E x f(X t ) ∇Exf(Xt) when Z is a subordinate Brownian motion. To study the SDE above, we first study the following nonlocal parabolic equation with Holder continuous b and f: ∂ t u+Lu+b⋅∇u+f=0,u(1,⋅)=0

Subfields: Finance, Management Science and Operations Research, Mathematical Physics

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