Optimal Investment to Reach a Financial Goal: A Stochastic Control Framework

2026-10-07

AI Summary

This study investigates the problem of optimal timing and asset allocation for investors pursuing financial goals in stochastic markets. It proposes a novel class of stochastic control models based on time-discounting functions, optimizing investment strategies through expected utility maximization. The Hamilton–Jacobi–Bellman (HJB) equation, viscosity solution theory, and Howard’s algorithm are employed for numerical solution and analysis. Key contributions include establishing Bellman’s principle of optimality, fully characterizing the smooth solution structure under specific conditions, revealing non-convergence properties under high volatility, and identifying the counterintuitive phenomenon whereby the optimal strategy may decrease as the drift of the risky asset increases.

Abstract

We develop a framework for an investor who trades until she either reaches a financial goal or an exogenous deadline arrives. Analogous to utility functions over wealth, we measure satisfaction with the timing of reaching a goal by a discount function. For a continuous-time market where a stochastic factor drives the dynamics of stock prices and the financial goals, the investor maximizes the expected discount at the goal reaching time and the expected utility of the funding ratio if the goal remains unreached by the deadline. This setup leads to a new class of stochastic control problems. We establish Bellman's principle of optimality and characterize the value function as a viscosity solution to the associated Hamilton-Jacobi-Bellman equation. When the deadline is infinite and the goal is constant, the HJB equation reduces to a form related to the backward heat equation, for which we provide a complete characterization of smooth solutions. When analytical solutions are unavailable, we develop an approach based on Howard's algorithm to obtain numerical solutions. Our analysis shows that optimal investment policies for goal reaching problems can be decreasing in the drift of risky assets and need not converge to full risk-free investment even as volatility diverges to infinity.

Problem

• Optimal Investment
• Financial Goal
• Stochastic Control
• Hamilton-Jacobi-Bellman Equation
• Discount Function

Innovation

• Stochastic Control
• Hamilton-Jacobi-Bellman Equation
• Viscosity Solution
• Howard's Algorithm
• Goal-reaching Investment