UC Santa Barbara

Stochastic analysis, stochastic processes and machine learning of dynamical systems have depicted strong connection in many aspects. This thesis aims to study such re- lationship from two directions. The first direction is using signature and deep learning techniques to propose efficient algorithm learning Mean Field Games with common noises. The second direction deploy the distributional invariance property of directed chain stochastic differential equations to design a novel time series generator with excellent simulation ability.In the first part, we introduce signature, borrowed from Rough Paths theory, as an efficient feature extraction technique and propose a novel algorithm to address the curse of dimensionality issue. In the second part, we propose an application of signature. In the problem of learn- ing Mean Field Games with common noises, traditional algorithms admit a nested loop structure due to the appearance of individual and common noises. Our proposed algorithm (Sig-DFP), utilize the universality property of signatures, has only single loop, which improves the efficiency from quadratic to linear in both time and space complexity. In the third part, we first study smoothing property of directed chain stochastic differential equations via partial Malliavin calculus, and then propose a novel generative adversarial network based time series generator. We also point out the independence issue of this directed chain generator, and solve it via branching scheme.