Optimal resource allocation for maintaining system solvency

TL;DR

This paper analyzes an optimal resource allocation problem for Brownian agents to maximize survival probabilities, deriving PDEs, identifying thresholds, and evaluating the push-the-laggard rule's optimality.

Contribution

It formulates the control problem with PDEs, proves solution properties, and assesses the optimality of a specific allocation rule in different regimes.

Findings

Established existence and uniqueness of solutions to the PDEs.

Proved the push-the-laggard rule is optimal for the all-survive criterion in certain regimes.

Provided a counterexample showing the rule's suboptimality for the count-survivors criterion.

Abstract

We study an optimal allocation problem for a system of independent Brownian agents whose states evolve under a limited shared control. At each time, a unit of resource can be divided and allocated across components to increase their drifts, with the objective of maximizing either (i) the probability that all components avoid ruin, or (ii) the expected number of components that avoid ruin. We derive the associated Hamilton-Jacobi-Bellman equations on the positive orthant with mixed boundary conditions at the absorbing boundary and at infinity, and we identify drift thresholds separating trivial and nontrivial regimes. For the all-survive criterion, we establish existence, uniqueness, and smoothness of a bounded classical solution and a verification theorem linking the PDE to the stochastic-control value function. We then investigate the conjectured optimality of the push-the-laggard allocation rule: in the nonnegative-drift regime, we prove that it is optimal for the all-survive value function, while it is not optimal for the count-survivors criterion, by exhibiting a two-dimensional counterexample and then lifting it to all dimensions.