Strong Solutions and Quantization-Based Numerical Schemes for a Class of Non-Markovian Volatility Models

Abstract

We investigate a class of non-Markovian processes that hold particular relevance in the realm of mathematical finance. This family encompasses path-dependent volatility models, including those pioneered by [Platen and Rendek, 2018] and, more recently, by [Guyon and Lekeufack, 2023]. Our study unfolds in two principal phases. In the first phase, we introduce a functional quantization scheme based on an extended version of the Lamperti transformation that we propose to handle the presence of a memory term incorporated into the diffusion coefficient. In the second phase, we study the problem of existence and uniqueness of a strong solution for the SDEs related to the examples that motivate our study, in order to provide a theoretical basis to correctly apply the proposed numerical schemes.