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Particle Approximation for Conditional Control with Soft Killing

Rene Carmona, Samuel Daudin

TL;DR

This work develops a particle-approximation scheme for Lions' conditional control with soft killing, where particle weights depend on all trajectories. By embedding the finite-N problem into a nonlocal mean-field framework and regularizing the limit value function via sup-convolution in $H^{-k}(\mathbb{T}^d)$, the authors obtain sharp convergence rates: $|v^N(t, {\bf x},{\bf a}) - \mathcal{V}(t, \mu^{N}_{{\bf x},{\bf a}})| \le C \sqrt{\sum_{i=1}^N (1/n^{i}[{\bf a}])^2}$. They prove the limit value $\mathcal{V}$ is Lipschitz and semiconcave in the measure argument and derive a coupled forward-backward PDE system characterizing optimality; they also establish the existence of optimal controls and provide explicit inequalities bounding the finite-N and mean-field values. This creates a rigorous bridge between finite particle systems with weights and the mean-field control problem, enabling precise quantitative rates and paving the way for analyzing hard-killing regimes. The approach blends projection arguments, regularization, and PDE techniques to address nonlocal nonlinear dynamics with conditioning.

Abstract

The aim of this paper is to develop a particle approximation for the conditional control problem introduced by P.-L. Lions during his lectures at the Collège de France in November 2016. We focus on a \textit{soft killing} relaxed version of the problem, which admits a natural counterpart in terms of stochastic optimal control for a large number of interacting particles. Each particle contributes to the overall population cost through a time-evolving weight that depends on the trajectories of all the other particles. Using recently developed techniques for the analysis of the value function associated with the limiting problem, we establish sharp convergence rates as the number of particles tends to infinity.

Particle Approximation for Conditional Control with Soft Killing

TL;DR

This work develops a particle-approximation scheme for Lions' conditional control with soft killing, where particle weights depend on all trajectories. By embedding the finite-N problem into a nonlocal mean-field framework and regularizing the limit value function via sup-convolution in , the authors obtain sharp convergence rates: . They prove the limit value is Lipschitz and semiconcave in the measure argument and derive a coupled forward-backward PDE system characterizing optimality; they also establish the existence of optimal controls and provide explicit inequalities bounding the finite-N and mean-field values. This creates a rigorous bridge between finite particle systems with weights and the mean-field control problem, enabling precise quantitative rates and paving the way for analyzing hard-killing regimes. The approach blends projection arguments, regularization, and PDE techniques to address nonlocal nonlinear dynamics with conditioning.

Abstract

The aim of this paper is to develop a particle approximation for the conditional control problem introduced by P.-L. Lions during his lectures at the Collège de France in November 2016. We focus on a \textit{soft killing} relaxed version of the problem, which admits a natural counterpart in terms of stochastic optimal control for a large number of interacting particles. Each particle contributes to the overall population cost through a time-evolving weight that depends on the trajectories of all the other particles. Using recently developed techniques for the analysis of the value function associated with the limiting problem, we establish sharp convergence rates as the number of particles tends to infinity.
Paper Structure (10 sections, 19 theorems, 189 equations)

This paper contains 10 sections, 19 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Statement of the problem and main result
  3. Analysis of the Limit value function
  4. Rate of convergence: formal argument
  5. The sup-convolution and its properties.
  6. Some preliminary facts about $H^{-k}(\mathbb{T}^d)$.
  7. Regularity Properties of $\mathcal{V}^{\epsilon}$.
  8. The hard inequality.
  9. The “easy" inequality.
  10. Existence of optimal controls and optimality conditions

Key Result

Theorem 2.1

Assume that $V: \mathbb{T}^d \rightarrow \mathbb R^+$ and $g: \mathbb{T}^d \rightarrow \mathbb R$ belong to $\mathcal{C}^k(\mathbb{T}^d)$ for some $k > d/2 +2$. There is a constant $C$ depending on $T$, $\left\Vert V\right\Vert_{\mathcal{C}^k(\mathbb{T}^d)}$ and $\left\Vert g\right\Vert_{\mathcal{C} where the weights $n^{i}[{\bf a}]$ were defined in fo:weights, and where the weighted empirical mea

Theorems & Definitions (34)

  • Theorem 2.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5: Dynamic Programming Principle
  • Proposition 3.6
  • ...and 24 more