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

TL;DR

This paper introduces a semi-analytical method for optimizing contribution schedules in retirement planning by solving a controlled Schrödinger equation, enabling efficient computation of feasible investment strategies with probabilistic guarantees.

Contribution

It develops a novel semi-analytical approach using a controlled backward Kolmogorov equation and Schrödinger equation reduction for continuous-time wealth management optimization.

Findings

Efficient semi-analytical solutions for all control parameters.

Representation of solutions as continuous contour lines (efficient frontiers).

Applicable to probabilistic wealth goal satisfaction in retirement planning.

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.