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A class of stochastic control problems with state constraints

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.

A class of stochastic control problems with state constraints

Abstract

We obtain a probabilistic solution to linear-quadratic optimal control problems with state constraints. Given a closed set , a diffusion in must be linearly controlled in order to keep the time-space process inside the set , while at the same time minimising an expected cost that depends on the state 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 . 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.
Paper Structure (5 sections, 14 theorems, 135 equations, 3 figures)

This paper contains 5 sections, 14 theorems, 135 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Proof of Theorem \ref{['thm:main']}
  4. Sufficient conditions for the continuity of $u$
  5. Examples of regular sets $\mathcal{D}$

Key Result

Theorem 2.8

Let Assumptions ass:ZD, ass:L and ass:u hold. Then $u\in C^{1,2}(\mathcal{C}_{[0,T)})$ with $u>0$ in $\mathcal{C}$, $u=0$ in $\mathcal{D}$ and the value function of problem eq:P1 reads Setting the SDE admits a unique strong solution $X^*$ for any $(t,x)\in \mathcal{C}$. Moreover, $X^*$ is an optimally controlled dynamics in $\mathcal{C}$$($i.e, $(s,X^*_s)\in\mathcal{C}$ for all $s\in[t,T]$, $\m

Figures (3)

  • Figure 1: Optimal trajectories for Example \ref{['ex:1']} with $T=1$, obtained by standard Euler-Maruyama method with $X_0^*=-1.5$ and time-step $\Delta t=0.005$.
  • Figure 2: Optimal trajectories for Example \ref{['ex:2']} with $T=1$, obtained by standard Euler-Maruyama method with $X_0^*=0.2$ and time-step $\Delta t=0.005$.
  • Figure 3: Optimal trajectories for Example \ref{['ex:3']} with $T=1$, $x_0=-2$, $x_1=2$, $t_0=0.2$, obtained by standard Euler-Maruyama method with $X_0^*=0$ and time-step $\Delta t=0.005$.

Theorems & Definitions (39)

  • Definition 2.1: Admissible controls
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.8
  • Example 2.9
  • Example 2.10
  • Example 2.11
  • Remark 2.12
  • Remark 2.13
  • ...and 29 more