On the geometric Brownian motion with state-dependent variable exponent diffusion term

TL;DR

This paper introduces a novel stochastic model with state-dependent variable exponent diffusion, generalizing classical models like GBM and CEV, with proven existence-uniqueness and error bounds, and compares interpretations.

Contribution

It presents a new flexible stochastic framework with state-dependent noise, extending classical models and providing theoretical and numerical analysis.

Findings

Proved existence and uniqueness of the model.

Derived upper-bound approximation for pathwise error.

Compared Itô and Stratonovich interpretations.

Abstract

We propose a new stochastic model involving state-dependent variable exponent $p(\cdot)$ which allows modeling of systems where noise intensity adapts to the current state. This new flexible theoretical framework generalizes both the geometric Brownian motion (GBM) and the Constant-Elasticity-of-Variance (CEV) models. We prove an existence-uniqueness theorem. We obtain an upper-bound approximation for the model-to-model pathwise error between our model and the GBM model as well as test its accuracy through analytical and numerical error estimates. A detailed comparison of the It\^o and Stratonovich interpretations for the proposed model is presented in the Appendix.