Well-posedness and Stability Analysis of Two Classes of Generalized Stochastic Volatility Models

TL;DR

This paper establishes the well-posedness and stability of two classes of generalized stochastic volatility models, providing a rigorous mathematical foundation for advanced financial modeling involving path-dependence and barriers.

Contribution

It introduces and analyzes two new classes of stochastic volatility models with subdifferential operators, extending the theoretical understanding of their well-posedness and stability.

Findings

Proved strong solutions exist for the models.

Analyzed stability under small perturbations.

Covered models with multi-dimensional and path-dependent features.

Abstract

In this paper, to cope with the shortage of sufficient theoretical support resulted from the fast-growing quantitative financial modeling, we investigate two classes of generalized stochastic volatility models, establish their well-posedness of strong solutions, and conduct the stability analysis with respect to small perturbations. In the first class, a multidimensional path-dependent process is driven by another multidimensional path-dependent process. The second class is a generalized one-dimensional stochastic volatility model with H\"older continuous coefficients. What greatly differentiates those two classes of models is that both the process and its correlated driving process have their own subdifferential operators, whose one special case is the general reflection operators for multi-sided barriers. Hence, the models investigated fully cover various newly explored variants of stochastic volatility models whose well-posedness is unknown, and naturally serve as the rigorous mathematical foundation for new stochastic volatility model development in terms of multi-dimension, path-dependence, and multi-sided barrier reflection.