Stochastic Analysis and Differential Geometry with Applications to Stochastic Volatility Models in Finance

Abstract

The thesis explores the interplay between stochastic analysis, differential geometry, partial differential equations (PDEs), and their applications in financial mathematics. It begins by establishing the fundamentals of stochastic processes, including martingales, stochastic integration, and the Itô formula, providing a comprehensive overview of stochastic calculus. It then introduces differential geometry concepts such as differentiable manifolds, tensors, and curvature, essential for understanding the geometric aspects of PDEs. The thesis also delves into PDEs, analyzing the heat kernel and differential operators within a geometric framework. Finally, it applies these theoretical frameworks to financial mathematics, focusing on the valuation of financial products. The thesis examines the stochastic volatility model lambda-SABR, together with its implementation and benchmarking. This work demonstrates how advanced mathematical concepts can be integrated to solve complex problems in finance.