Asymptotic Analysis for Portfolio Problems under Fast and Slow Stochastic Environments

The main objective of this dissertation is to use the multi-scale asymptotic analysis to study two-asset portfolio optimization problems under fast and slow stochastic environments. We first review the stochastic control theory and the classical Merton portfolio problem. Then we introduce the multi-scale asymptotic analysis for single-asset portfolio optimization problem [Fouque and Hu, 2017-2020]. We rigorously establish that, in the multi-scale case, the Merton type zeroth order optimal strategy recovers the first order approximation of the associated problem value. The asymptotic analysis can be extended to a multi-asset scenario. We consider two-asset portfolio problems with full and partial information, respectively. Combining the filtering theory and the asymptotic analysis, we show that similar results about the zeroth order optimal strategy hold for both full information case and partial information case under different assumptions.