Paper: arXiv 2603.04880

Authors: Tiziano De Angelis, Erik Ekström

Abstract

We obtain a probabilistic solution to linear-quadratic optimal control problems with state constraints. Given a closed set $\mathcal{D}\subseteq [0,T]\times\mathbb{R}^d$, a diffusion $X$ in $\mathbb{R}^d$ must be linearly controlled in order to keep the time-space process $(t,X_t)$ inside the set $\mathcal{C}:=([0,T]\times\mathbb{R}^d)\setminus\mathcal{D}$, while at the same time minimising an expected cost that depends on the state $(t,X_t)$ and is quadratic in the speed of the control exerted. We find a probabilistic representation for the value function and an optimal control under a set of mild sufficient conditions concerning the coefficients of the underlying dynamics and the regularity of the set $\mathcal{D}$. The optimally controlled dynamics is in strong form, in the sense that it is adapted to the filtration generated by the driving Brownian motion. Fully explicit formulae are presented in some relevant examples.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework for stochastic control problems with state constraints using Doob’s h-transform and logarithmic transformations. While mathematically sophisticated and novel in establishing strong-form optimal dynamics, it is purely theoretical with no empirical data or backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve LQ Optimal Control<br/>with Time-Space State Constraints] --> B[Inputs: Controlled Diffusion X, Cost Function,<br/>and Forbidden Set D]
    B --> C[Methodology: Formulate Minimisation Problem<br/>to Keep Process in C = D^c]
    C --> D[Derive Probabilistic Representation<br/>for the Value Function]
    D --> E[Construct Optimal Control in Strong Form<br/>Adapted to Brownian Filtration]
    E --> F[Outcomes: Verified Mild Sufficient Conditions<br/>& Explicit Solutions for Key Examples]

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