Multiscale Asymptotic Analysis for Portfolio Optimization under Stochastic Environment

Empirical studies indicate the presence of multiscales in the volatility of underlying assets: A fast-scale on the order of days and a slow-scale on the order of months. In our previous works, we have studied the portfolio optimization problem in a Markovian setting under each single scale, the slow one in [J.-P. Fouque and R. Hu, SIAM J. Control Optim., 55 (2017), pp. 1990--2023], and the fast one in [R. Hu, Asymptotic optimal portfolio in fast mean-reverting stochastic environments, in Proceedings of the 2018 IEEE CDC, 2018, pp. 5771--5776]. This paper is dedicated to the analysis when the two scales co-exist in a Markovian setting. We study the terminal wealth utility maximization problem when the volatility is driven by both fast- and slow-scale factors. We first propose a zeroth order strategy, and rigorously establish the first order approximation of the associated problem value. This is done by analyzing the corresponding linear partial differential equation (PDE) via regular and singular perturbation techniques, as in the single-scale cases. Then, we show the asymptotic optimality of our proposed strategy by comparing its performance to admissible strategies of a specific form. Interestingly, we highlight that a pure PDE approach does not work in the multiscale case and, instead, we use the so-called epsilon-martingale decomposition. This completes the analysis of portfolio optimization in both fast mean-reverting and slowly varying Markovian stochastic environments.