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arXiv · 2508.07130

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

Mustafa Avci
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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.

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Mustafa Avci. 2025-08-10. On the geometric Brownian motion with state-dependent variable exponent diffusion term. https://arxiv.org/abs/2508.07130

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