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

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.