Title
Optimal Investment to Reach a Financial Goal: A Stochastic Control Framework
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.
Based on joint work with Gechun Liang, Yuwei Wang, and Zhaojun Yang.
Bio
Moris Strub is an Associate Professor at the Warwick Business School. He is the inaugural Course Director for the MSc Financial Technology, a lead academic at the Gillmore Centre for Financial Technology, and lead academic for PropTech at the FutureFinance.AI Research Group. Moris’ research interests are in the areas of optimal investment, behavioural finance and economics, and financial technology. He obtained a PhD in Financial Engineering from the Chinese University of Hong Kong and holds a BSc in Mathematics and MSc in Applied Mathematics from ETH Zürich. Before joining WBS, Moris was an assistant professor at the Southern University of Science and Technology.