General path integrals and stable SDEs

TL;DR

This paper investigates the existence and uniqueness of weak solutions to stochastic differential equations driven by symmetric alpha-stable Lévy processes, extending classical results to jump processes using Markov process theory.

Contribution

It provides a complete characterization of weak solution existence and uniqueness for SDEs with jumps for alpha in (0,1), using a novel approach based on Markov process theory.

Findings

Derived integral tests for finiteness of path integrals.

Extended results of Zanzotto on SDEs with jumps.

Characterized solutions for alpha-stable processes with alpha in (0,1).

Abstract

The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and has been largely understood for several decades. For stochastic differential equations with jumps the picture is still incomplete, and even some of the most basic questions are only partially understood. In the present article we study existence and uniqueness of weak solutions to \[ {\rm d}Z_t=\sigma(Z_{t-}){\rm d} X_t \]driven by a (symmetric) $\alpha$-stable L\'evy process, in the spirit of the classical Engelbert-Schmidt time-change approach. Extending and completing results of Zanzotto we derive a complete characterisation for existence und uniqueness of weak solutions for $\alpha\in(0,1)$. Our approach is not based on classical stochastic calculus arguments but on the general theory of Markov processes. We proof integral tests for finiteness of path integrals under minimal assumptions.