Optimal resource allocation for maintaining system solvency

Abstract

We study two optimal allocation problems 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 identify drift thresholds separating trivial and nontrivial regimes, and derive the associated Hamilton-Jacobi-Bellman equations on the positive orthant with mixed boundary conditions at the absorbing boundary and at infinity. We also establish the existence, uniqueness, and regularity of a bounded classical solution and a verification theorem linking the PDE to the stochastic control value function. Finally, we prove a conjecture on the optimality of a socialistic allocation rule : the push-the-laggard strategy. It is optimal for the all-survive value function, while it is suboptimal for the count-survivors criterion.