Anton Yurchenko-Tytarenko

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.

We study a stochastic differential equation with an unbounded drift and general Hölder continuous noise of an arbitrary order. The corresponding equation turns out to have a unique solution that, depending on a particular shape of the drift, either stays above some continuous function or has continuous upper and lower bounds. Under some additional assumptions on the noise, we prove that the solution has moments of all orders. Additionally, numeric schemes for the solution are considered. As illustrations of our approach, generalised CIR and CEV processes will be discussed. This talk is based on joint work with Giulia Di Nunno and Yuliya Mishura.