The Functional Ito Calculus and Applications to Causal Optimal Transport

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This research project seeks to study the methods arising from the Functional Ito's Calculus, developed by Cont et al.[1-3], and consider its application to Causal Optimal Transport problems introduced in [4]. The theory of optimal transport began as an engineering problem considered by Monge (and later by Kantorovich), on how to most efficiently transport mass from one location to another. Over the last few decades, the theory that spawned from this deceptively simple problem has experienced fervent development after connections to several different areas of mathematics were made[5]. The field of stochastic analysis, which is the study of processes that evolve randomly with respect to time, has been no exception to this[6,7]. A variant of the Optimal Transport problem, the Causal Optimal Transport problem and its associated notions are born out of a desire to capture the temporal structure of stochastic processes in the context of Optimal Transport and has already found use in answering questions from stochastic control[4,8] and mathematical finance[9]. However, Causal Optimal Transport is a relatively new concept and while some work has been done for the problem in both discrete time[10] and continuous time[8], there are still many open questions and possible points of inquiry. For example, one might ask which results in classical Optimal Transport theory have a "causal" counterpart. We intend to tackle these unanswered questions with the help of Functional Ito's Calculus - a non-anticipative functional calculus which generalizes several well-known results from classical Ito's Calculus[11] to path-dependent functionals of stochastic processes[1,12]. This presents a novel and promising avenue of research. Similar to the Causal Optimal Transport problem, Functional Ito's Calculus has also found use in the fields of stochastic control[13] and mathematical finance[14]. Indeed, Functional Ito's Calculus has, among other examples, already led to the formulation of a new class of partial differential equations - functional Kolmogorov equations[12], a generalization of the well-known Kolmogorov equations which characterizes Markov processes [11,15]. It is then hoped that the efficacy of Functional Ito's Calculus in advancing other areas of research will persist in the context of the Causal Optimal Transport problem and lead to fruitful results.This project falls within the EPSRC Mathematical Analysis research area.