Stochastic partial differential equations and their applications

This research investigates the existence and uniqueness of solutions in investment forward performance Stochastic Partial Differential Equations (SPDEs). The study employs the Picard iteration method to establish the well-posedness of the SPDE and the convergence of the iterative solution methods. It is built on the seminal works of (Mykhaylo, Ronnie, & Zariphopoulou, 2016) who introduced forward investment strategies and optimal feedback portfolios. Their groundbreaking contributions significantly advanced the understanding of optimal portfolio selection in incomplete markets, guided by forward investment performance criteria. The key concepts such as consistency, limits, continuity, and memorability are employed to elucidate the dynamics of the processes under examination. These notions play a crucial role in characterizing the intricate behavior of the systems and ensuring the validity of the proposed hypotheses. The main focus of this work is exclusively on the theoretical aspects of the existence and uniqueness of solutions, without delving into estimation or calibration methods for model parameters. The parameters are treated as random in certain applications discussed herein. Thus, the primary objective is to explore theorems and methodologies that guarantee the well-posedness of the given SPDE. Particular emphasis is placed on the role of key assumptions such as coercivity, monotonicity, boundedness, hemi continuity, Lipchitz continuity and growth condition as well as initial conditions in ensuring the convergence of iterative methods. Rigorous mathematical tools, including Gronwall's lemma and Itô's formula, are applied to establish uniqueness results. The findings have potential applications in diverse fields, such as population dynamics and Crack propagation in composite materials and others.