Stochastic flows for Lévy processes with Hölder drifts

Abstract

In this paper we study the following stochastic differential equation (SDE) in ${\mathbb{R}}^d$: $$\mathrm{d}{X}_t=\mathrm{d}{Z}_t+b(t,X_t)\mathrm{d}t,X_0=x,$$ where $Z$ is a Lévy process. We show that for a large class of Lévy processes $Z$ and Hölder continuous drift $b$, the SDE above has a unique strong solution for every starting point $x\in {\mathbb{R}}^d$. Moreover, these strong solutions form a $C^1$-stochastic flow. As a consequence, we show that, when $Z$ is an $\alpha$-stable-type Lévy process with $\alpha \in (0,2)$ and $b$ is bounded and $\beta$-Hölder continuous with $\beta \in (1-\alpha /2,1)$, the SDE above has a unique strong solution. When $\alpha \in (0,1)$, this in particular solves an open problem from Priola \cite{Pr1}. Moreover, we obtain a Bismut type derivative formula for $\nabla {\mathbb{E}}_{x}f(X_t)$ when $Z$ is a subordinate Brownian motion. To study the SDE above, we first study the following nonlocal parabolic equation with Hölder continuous $b$ and $f$: $${\partial }_{t}u+\mathcal{L}u+b\cdot \nabla u+f=0,u(1,\cdot )=0,$$ where $\mathcal{L}$ is the generator of the Lévy process $Z$.

22pages

References in corpus (1)

• Derivative formula and gradient estimate for SDEs driven by $\alpha$-stable processes

Cited by in corpus (1)

• Numerical algorithms for mean exit time and escape probability of stochastic systems with asymmetric Lévy motion