Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts

Abstract

In this paper, we study the following time-dependent stochastic differential equation (SDE) in ${\mathbf{R}}^d$: $$d{X}_t={\sigma }_t(X_{t-})d{Z}_t+b_t(X_t)dt,X_0=x\in {\mathbf{R}}^d,$$ where $Z$ is a $d$-dimensioanl nondegenerate $\alpha$-stable-like process with $\alpha \in (0,2)$ (including cylindrical case), and uniform in $t\ge 0$, $x\mapsto {\sigma }_t(x):{\mathbf{R}}^d\to {\mathbf{R}}^d\otimes {\mathbf{R}}^d$ is Lipchitz and uniformly elliptic and $x\mapsto b_t(x)$ is $\beta$-order Hölder continuous with $\beta \in (1-\alpha /2,1)$. Under these assumptions, we show the above SDE has a unique strong solution for every starting point $x\in {\mathbf{R}}^d$. When ${\sigma }_t(x)={\mathbf{I}}_{d\times d}$, the $d\times d$ identity matrix, our result in particular gives an affirmative answer to the open problem of Priola (2015).

References in corpus (3)

• Pathwise uniqueness and continuous dependence for SDEs with nonregular drift
• Stochastic functional differential equations driven by Lévy processes and quasi-linear partial integro-differential equations
• Uniqueness of stable-like processes

Cited by in corpus (11)

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• Existence of densities for stochastic differential equations driven by Lévy processes with anisotropic jumps
• Dirichlet problem for supercritical nonlocal operators
• Nonlocal elliptic equation in Hölder space and the martingale problem
• Schauder's estimate for nonlocal kinetic equations and its applications
• Schauder's estimates for nonlocal equations with singular Lévy measures
• Regularity properties of jump diffusions with irregular coefficients
• Hölder regularity and gradient estimates Hölder regularity and gradient estimates for SDEs driven by cylindrical $\alpha$-stable processes
• On the Cauchy problem for nondegenerate parabolic integro-differential equations in the scale of generalized Hölder spaces
• Linear Response Theory for Nonlinear Stochastic Differential Equations with $\alpha$-stable Lévy Noises
• Weak uniqueness for SDEs driven by supercritical stable processes with Holder drifts