Qiu, JinniaoWare, AntonyYang, Yang2024-09-242024-09-242024-09-19Yang, Y. (2024). A class of stochastic path-dependent models and controls (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.https://hdl.handle.net/1880/119838This thesis explores controlled stochastic path-dependent differential systems and their applications in stock and energy markets. We first focus on a class of path-dependent stochastic optimal control problems within an infinite-dimensional framework. Our primary contribution is proving that the value function serves as the unique viscosity solution to the corresponding infinite-dimensional stochastic path-dependent Hamilton-Jacobi equation. Then, we investigate the introduction of path-dependent features in stochastic volatility models. The price-storage dynamic is studied in natural gas markets. We propose a novel stochastic path-dependent volatility model that incorporates path-dependence into both price volatility and storage increments. Chapter 1 focuses on the stochastic optimal control problem for infinite-dimensional differential systems that incorporate both path-dependence and measurable randomness. Unlike deterministic path-dependent cases, the value function emerges as a random field over the path space, characterized by a stochastic path-dependent Hamilton-Jacobi (SPHJ) equation. We propose a notion of viscosity solution and demonstrate that the value function is the unique viscosity solution to the associated SPHJ equation. Chapter 2 and 3 explores price-storage dynamics in natural gas markets. We introduce a novel stochastic path-dependent volatility model that incorporates path-dependence in both price volatility and storage increments. Model calibrations are performed for both price and storage dynamics. Additionally, we address the pricing problem for discrete-time swing options using the dynamic programming principle and propose a deep learning-based method for numerical approximations. A numerical algorithm is presented, accompanied by a convergence analysis of the deep learning approach. Chapter 4 outlines future research directions in stochastic path-dependent Hamilton-Jacobi-Bellman (HJB) equations, stochastic volatility models, mean field games and stochastic path-dependent filtering problems.enUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.Stochastic path-dependent optimal control theoryStochastic path-dependent volatility modelsSwing option pricingMathematicsdoctoral thesis