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

TL;DR

This work introduces a nonlinear SDE with a state-dependent diffusion exponent $p(X)$ that generalizes both the geometric Brownian motion and the CEV model. By defining a class of admissible exponents $p(\cdot)\in\mathcal{S}$, the authors prove existence-uniqueness of a positive strong solution and establish Lipschitz and linear-growth bounds for $x^{p(x)}$, along with a finite second-moment guarantee. They derive analytical and numerical error estimates that bound the pathwise distance to GBM in terms of $\sup_{x>0}|p(x)-1|$ and localization constants, and they validate these results via Monte Carlo simulations using the log-transformed Milstein scheme. The appendix compares Ito and Stratonovich formulations, showing drift corrections and moment-dynamics differences, and situates the framework as a flexible, computationally tractable tool for state-adaptive noise across finance, biology, and physics.

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 Itô and Stratonovich interpretations for the proposed model is presented in the Appendix.

Figures (3)

• Figure 1: $p_1(\cdot)$ vs GBM.
• Figure 2: $p_2(\cdot)$ vs GBM.
• Figure 3: Implied-volatility skews at maturity T=1 estimated from Monte Carlo call prices.

Theorems & Definitions (10)

• Remark 1
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Remark 2