Singular Equations Driven by an Additive Noise and Applications

TL;DR

This paper studies singular stochastic differential equations driven by additive Gaussian noise, establishing ergodic properties, convergence of numerical schemes, and distributional regularity, with applications to a fractional Heston model.

Contribution

It introduces a framework for analyzing equations with singular drift driven by additive noise, including ergodic results, convergence of schemes, and distributional analysis using Malliavin calculus.

Findings

Proved ergodic theorem for the equations.

Established convergence of the implicit Euler scheme.

Analyzed the absolute continuity of the solution's distribution.

Abstract

In the pathwise stochastic calculus framework, the paper deals with the general study of equations driven by an additive Gaussian noise, with a drift function having an infinite limit at point zero. An ergodic theorem and the convergence of the implicit Euler scheme are proved. The Malliavin calculus is used to study the absolute continuity of the distribution of the solution. An estimation procedure of the parameters of the random component of the model is provided. The properties are transferred on a class of singular stochastic differential equations driven by a multiplicative noise. A fractional Heston model is introduced.