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], as well as an extension of the framework proposed by [Blanc et al., 2017]. 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. For scenarios involving a Brownian integral in the diffusion term, we propose alternative numerical schemes that leverage the power of marginal recursive quantization. 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.