Stochastic partial differential equations and their applications

Abstract

The research investigates the existence and uniqueness of solutions in investment forward performance stochastic partial di erential equations (SPDEs). It employs sophisticated mathematical methods, such as the Picard iteration approach, to establish the well-posedness of the SPDE and the convergence of iterative solution techniques. The study delves into advanced mathematical concepts to analyse the behaviour of these complex equations. The study drew upon the seminal works of researchers like Mykhaylo Shkolnikov, Ronnie Sircar, and Thaleia Zariphopoulou. They formulated equations for forward investment strategies and optimal feedback portfolios. Their contributions deepened the comprehension of optimal portfolio selection in incomplete market scenarios based on criteria for forward investment performance. The investigation put forth several key ideas to explain how certain processes function. These concepts, consistency, limits, continuity and memorability of complex processes, aimed to describe the conduct of the systems analyzed. The hypotheses played a vital role in understanding the intricate workings of the processes under examination. This work focuses solely on theoretical aspects concerning the existence and uniqueness of solutions to given equation, rather than covering estimation or calibration methods for determining the parameters of the SPED model, which would require specialized data, models, and statistical techniques. The choice of parameters will be random in certain applications discussed here. The core emphasis lies on exploring theorems and methodologies related to investigating the solution's existence and uniqueness for a given equation. The thesis emphasizes the importance of key assumptions and initial conditions for ensuring the well-posedness of the SPDE and the convergence of iterative solution methods. It also highlights the rigorous reasoning used in establishing the uniqueness of solutions with the strict application of mathematical techniques like Gronwall's lemma and Itô's formula. Modeling population dynamics and crack propagation in composite materials are two applications of the research's ndings.