Anton Yurchenko-Tytarenko

Abstract

In this research, we consider a stochastic differential equation with drift having singularity in zero and an additive noise being a fractional Brownian motion with an arbitrary Hurst parameter. Under some assumptions, existence, uniqueness and positivity of the solution are proved, finiteness of the moments of all orders (including the negative ones) is obtained. As examples, we provide fractional generalisations of the CIR and CEV processes and, using the finiteness of their negative moments, prove convergence of the corresponding backward Euler-type schemes.

Separately, we discuss possible extensions of the obtained results including replacing the fractional Brownian motion with general Holder-continuous Volterra process.