A class of stochastic differential equations with super-linear growth and non-Lipschitz coefficients

TL;DR

This paper investigates solutions to multidimensional stochastic differential equations with super-linear, non-Lipschitz coefficients, establishing pathwise uniqueness, stochastic flow properties, and large deviations under novel local conditions that extend existing theories.

Contribution

It introduces a new local condition ensuring uniqueness and flow properties for SDEs with super-linear, non-Lipschitz coefficients, surpassing previous assumptions and broadening applicability.

Findings

Pathwise uniqueness is guaranteed under the new local condition.

Solutions form a stochastic flow of continuous maps.

A large deviations principle of Freidlin-Wentzell type is established.

Abstract

The purpose of this paper is to study some properties of solutions to one dimensional as well as multidimensional stochastic differential equations (SDEs in short) with super-linear growth conditions on the coefficients. Taking inspiration from \cite{BEHP, KBahlali, Bahlali}, we introduce a new {\it{local condition}} which ensures the pathwise uniqueness, as well as the non-contact property. We moreover show that the solution produces a stochastic flow of continuous maps and satisfies a large deviations principle of Freidlin-Wentzell type. Our conditions on the coefficients go beyond the existing ones in the literature. For instance, the coefficients are not assumed uniformly continuous and therefore can not satisfy the classical Osgood condition. The drift coefficient could not be locally monotone and the diffusion is neither locally Lipschitz nor uniformly elliptic. Our conditions on the coefficients are, in some sense, near the best possible. Our results are sharp and mainly based on Gronwall lemma and the localization of the time parameter in concatenated intervals