Stochastic Differential Equations with Discontinuous Diffusions
arXiv:1908.03183
The paper investigates one-dimensional stochastic differential equations driven by Hölder continuous processes, allowing the diffusion coefficient to have discontinuities, and includes cases like fractional Brownian motion with Hurst parameter greater than 1/2.
Abstract
We study one-dimensional stochastic differential equations of form $dX_t = Ï(X_t)dY_t$, where $Y$ is a suitable Hölder continuous driver such as the fractional Brownian motion $B^H$ with $H>\frac12$. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients $Ï$ for which we assume very mild conditions. In particular, we allow $Ï$ to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.