SCOP: Schrodinger Control Optimal Planning for Goal-Based Wealth Management

34 Pages Posted: 9 Sep 2023

Date Written: September 8, 2023

Abstract

We consider the problem of optimization of contributions of a financial planner such as a working individual towards a financial goal such as retirement. The objective of the planner is to find an optimal and feasible schedule of periodic installments to an investment portfolio set up towards the goal. Because portfolio returns are random, the practical version of the problem amounts to finding an optimal contribution scheme such that the goal is satisfied at a given confidence level. This paper suggests a semi-analytical approach to a continuous-time version of this problem based on a controlled backward Kolmogorov equation (BKE) which describes the tail probability of the terminal wealth given a contribution policy. The controlled BKE is solved semi-analytically by reducing it to a controlled Schrodinger equation and solving the latter using an algebraic method.

Numerically, our approach amounts to finding semi-analytical solutions simultaneously for all values of control parameters on a small grid, and then using the standard two-dimensional spline interpolation to simultaneously represent all satisficing solutions of the original plan optimization problem. Rather than being a point in the space of control variables, satisficing solutions form continuous contour lines (efficient frontiers) in this space.

Keywords: Wealth management, retirement planning, optimal control, Kolmogorov equation, Schrodinger equation

JEL Classification: C02, C61, D10, D14, D90, D91, G02, G11, G23, G28, H31

Suggested Citation

Halperin, Igor, SCOP: Schrodinger Control Optimal Planning for Goal-Based Wealth Management (September 8, 2023). Available at SSRN: https://ssrn.com/abstract=4566372 or http://dx.doi.org/10.2139/ssrn.4566372

Igor Halperin (Contact Author)

Fidelity Investments, Inc. ( email )

United States

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