Strong Existence and Uniqueness for Singular SDEs Driven by Stable Processes

Abstract

We consider the one-dimensional stochastic differential equation \begin{equation*} X_t = x_0 + L_t + \int_0^t μ(X_s)ds, \quad t \geq 0, \end{equation*} where $\mu$ is a finite measure of Kato class $K_{\eta }$ with $\eta \in (0,\alpha -1]$ and $(L_t{)}_{t\ge 0}$ is a symmetric $\alpha$-stable process with $\alpha \in (1,2)$. We derive weak and strong well posedness for this equation when $\eta \le \alpha -1$ and $\eta <\alpha -1$, respectively, and show that the condition $\eta \le \alpha -1$ is sharp for weak existence. We furthermore reformulate the equation in terms of the local time of the solution $(X_t{)}_{t\ge 0}$ and prove its well posedness. To this end, we also derive a Tanaka-type formula for a symmetric, $\alpha$-stable processes with $\alpha \in (1,2)$ that is perturbed by an adapted, right-continuous process of finite variation.

References in corpus (7)

• Fractal free energy landscapes in structural glasses
• Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains
• Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation
• Levy stable noise induced transitions: stochastic resonance, resonant activation and dynamic hysteresis
• Fluctuation-driven directed transport in the presence of Levy flights
• Multidimensional SDE with distributional drift and Lévy noise
• One-dimensional Stochastic Differential Equations with Generalized and Singular Drift