Stochastic differential equations with discontinuous diffusion coefficients

Abstract

We study one-dimensional stochastic differential equations of the form d X t = σ ( X t ) d Y t dX_t = \sigma (X_t)dY_t , where Y Y is a suitable Hölder continuous driver such as the fractional Brownian motion B H B^H with H > 1 2 H>\frac 12 . The innovative aspect of the present paper lies in the assumptions on diffusion coefficients σ \sigma for which we assume very mild conditions. In particular, we allow σ \sigma to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.