Stochastic Differential Equations with Discontinuous Diffusions

Abstract

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

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