Chebyshev wavelet-based method for solving various stochastic optimal control problems and its application in finance†

In this paper, a computational method based on parameterizing state and control variables is presented for solving Stochastic Optimal Control (SOC) problems. By using Chebyshev wavelets with unknown coefficients, state and control variables are parameterized, and then a stochastic optimal control problem is converted to a stochastic optimization problem. The expected cost functional of the resulting stochastic optimization problem is approximated by sample average approximation thereby the problem can be solved by optimization methods more easily. For facilitating and guaranteeing convergence of the presented method, a new theorem is proved. Finally, the proposed method is implemented based on a newly designed algorithm for solving one of the well-known problems in mathematical finance, the Merton portfolio allocation problem in finite horizon. The simulation results illustrate the improvement of the constructed portfolio return.