The Readout of Merton's Problem on Infinite Horizon - Stochastic Optimal Control Modelling

Abstract

Stochastic modeling is of the utmost importance in a world dominated by uncertainty. Many problems in economics, finance, and actuarial science naturally require decision-makers to undertake choices in stochastic environments. Classical methods for solving infinite horizon Stochastic Optimal Control Problems (SOCPs) primarily focus on deriving the solution by defining the value function through dynamic programming and the Hamilton-Jacobi-Bellman equation. However, obtaining a closed-form solution is generally challenging. This article proposes a hybrid method for solving SOCPs to address this issue and identify an optimal trajectory and control. This novel approach integrates the Multi-Step Stochastic Differential Transform Method (MSDTM) with an approximation technique that solves infinite horizon problems by leveraging a finite horizon. An applicable example diagram of the types of instances created from the simulation of the described approach and infinite horizon stochastic optimal control problem from management science is provided to show the method's effectiveness and efficiency, particularly in comparison with existing approaches.