Stochastic Analysis of Mean-Field Games, Portfolio Optimization and Low-Rank Matrix Approximation

Lai, Peiyao

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Stochastic Analysis of Mean-Field Games, Portfolio Optimization and Low-Rank Matrix Approximation

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This thesis employs stochastic analysis tools to address three distinct problems. Firstly, in Hybrid Linear Quadratic Gaussian (LQG) Mean Field Games (MFGs), we investigate the convergence rate of the N-player linear quadratic Gaussian game towards its asymptotic Mean Field Games, using an explicit coupling method. The two main results are as follows. With some assumptions, one is to characterize the Mean-Field game equilibrium path as well as the associated equilibrium measure. The other is to obtain the convergence rate from the N-player game to that from mean-field games in distribution. The second problem involves finding the robust relative performance maximizing portfolio in an incomplete information setting, where the objective is to find the optimal strategy for an investor maximizing her/his robust utility. In the third problem, we obtain tighter right-singular vector perturbation bounds for rectangular matrices perturbed by Gaussian random matrix noise, by analyzing the perturbed matrix as a Dyson-Bessel matrix-valued diffusion. Applications of the perturbation bounds include the subspace recovery problem and the rank-k matrix approximation problem.

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