Paper: arXiv 2509.01744

Authors: Matthew Lorig

Abstract

We use classical tools from calculus of variations to formally derive necessary conditions for a Markov control to be optimal in a standard finite time horizon stochastic control problem. As an example, we solve the well-known Merton portfolio optimization problem.

Complexity vs Empirical Score

  • Math Complexity: 6.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This short pedagogical note provides an alternative, intuitive derivation of the Hamilton-Jacobi-Bellman equation using calculus of variations and Lagrangian multipliers instead of the dynamic programming principle. While mathematically clean and clearly written, it is purely theoretical with zero empirical or backtesting component, and applies known techniques to the classical Merton problem.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Derive necessary optimality conditions for Markov control via Calculus of Variations"] --> B["Inputs:<br>Finite time horizon stochastic control problem & Merton portfolio parameters"]
    B --> C["Methodology:<br>Apply classical Calculus of Variations to stochastic state dynamics"]
    C --> D["Computation:<br>Formulate Hamilton-Jacobi-Bellman (HJB) equations & solve stochastic differential equations"]
    D --> E["Outcomes:<br>Formal necessary conditions for optimal control & closed-form solution to Merton problem"]

Cited References (3)

  1. Continuous-time stochastic control and optimization with financial applications (2009)
  2. Lifetime portfolio selection under uncertainty: The continuous-time case (1969)
  3. Classical mechanics (1950)

0 of 3 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.