Stochastic equations with discontinuous drift

Abstract:

We study stochastic differential equations, $dx = adt + \sigma d\beta$ where $\beta$ denotes a Brownian motion. By relaxing the definition of solutions we are able to prove existence theorems assuming only that $a$ is measurable, $\sigma$ is continuous and that both grow linearly at infinity. Nondegeneracy is not assumed. The relaxed definition of solution is an extension of A. F. Filippov’s definition in the deterministic case. When $\sigma$ is constant we prove one-sided uniqueness and approximation theorems under the assumption that $a$ satisfies a one-sided Lipschitz condition.