On the existence, uniqueness and stability of solutions of SDEs with state-dependent variable exponent
Mustafa Avci
TL;DR
This work introduces a time-inhomogeneous nonlinear SDE with state-dependent variable exponents in both the drift and diffusion terms, unifying GBM and CEV frameworks while enabling richer dynamics. Existence and uniqueness are established via a Banach fixed-point approach under a new class of exponents, with thorough analysis of higher-order moments, asymptotics, and stability. The paper also demonstrates a Poisson equation application by providing a probabilistic (Feynman-Kac) representation through a time-homogeneous SDE with state-dependent exponents. Together, these results advance the mathematical foundation for SDEs with state-responsive elasticity and provide tools for modeling complex state-driven stochastic phenomena.
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.