On the existence, uniqueness and stability of solutions of SDEs with state-dependent variable exponent

TL;DR

This paper investigates the mathematical properties of a class of stochastic differential equations with state-dependent variable exponents, establishing existence, uniqueness, stability, and providing applications to complex financial models.

Contribution

It introduces a fixed-point method to prove existence and uniqueness for SDEs with state-dependent exponents, extending classical models like GBM and CEV.

Findings

Proved existence and uniqueness of solutions.

Analyzed higher-order moments and stability.

Provided a probabilistic representation for solutions to the Poisson equation.

Abstract

We study a time-inhomogeneous nonlinear SDE with drift and diffusion governed by state-dependent variable exponents. This framework generalizes models like the geometric Brownian motion (GBM) and the constant elasticity of variance (CEV), offering flexibility to capture complex dynamics while posing analytical challenges. Using a fixed-point approach, we prove existence and uniqueness, analyze higher-order moments, derive asymptotic estimates, and assess stability. Finally, we illustrate an application where the Poisson equation admits a probabilistic representation via a time-homogeneous nonlinear SDE with state-dependent variable exponents.