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

ArXiv ID: 2309.05926 “View on arXiv”

Authors: Unknown

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: contribution planning, backward Kolmogorov equation, controlled Schrodinger equation, efficient frontiers, personal finance

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper involves advanced mathematics including stochastic differential equations, backward Kolmogorov equations, reduction to Schrödinger equations, and semi-analytical solutions, placing it at high math complexity. However, the empirical rigor is low as the approach relies on theoretical derivations and numerical interpolation on a small grid without evidence of backtesting on real data, coding implementations, or statistical performance metrics.

  flowchart TD
    A["Research Goal: Optimal contribution plan<br>to meet financial goal at a given confidence level"] --> B{"Key Methodology"}
    B --> C["Semi-analytical approach using<br>Controlled Backward Kolmogorov Equation BKE"]
    C --> D["Reduce BKE to<br>Controlled Schrödinger Equation"]
    D --> E["Solve Schrödinger Eq<br>via algebraic method"]
    E --> F{"Computational Process"}
    F --> G["Compute solutions on fine grid<br>for all control parameters"]
    G --> H["Interpolate solutions using<br>2D Spline for continuous representation"]
    H --> I["Key Findings/Outcomes"]
    I --> J["Efficient Frontiers: Satisficing solutions<br>form continuous contours in control space"]
    I --> K["Objective: Find optimal & feasible<br>contribution schedules for goal-based planning"]