Stochastic Differential Geometry Analysis and Ricci Flow Dynamics in Financial Market Manifolds

Abstract

We develop an innovative framework for financial modeling by integrating stochastic differential geometry with Ricci flow dynamics. In this model, asset prices evolve on a Riemannian manifold, and volatility is governed by a stochastic Ricci flow equation, producing a dynamically evolving volatility surface influenced by geometric curvature and stochastic forcing. We establish rigorous theoretical results on the existence and uniqueness of stochastic flows and demonstrate their impact on option pricing. Numerical simulations illustrate volatility clustering, geometric deformation, and realistic asset price behavior under curvature-driven uncertainty. This approach extends classical stochastic volatility models by capturing intrinsic geometric features of market dynamics, offering a robust tool for modeling turbulence, clustering, and complex financial phenomena with enhanced fidelity.