Exercising Control When Confronted by a (Brownian) Spider

TL;DR

This paper investigates the control problem of a Brownian spider, applying dynamic programming to establish bounds and optimality of control strategies for this stochastic process.

Contribution

It introduces a novel application of dynamic programming to analyze and establish bounds for the Brownian spider control problem.

Findings

Proved bounds for the Brownian spider using dynamic programming.

Established the optimality of control strategies via excessiveness.

Extended the theoretical understanding of stochastic control for complex processes.

Abstract

We consider the Brownian "spider," a construct introduced in \cite{Dubins} and in \cite{Pitman}. In this note, the author proves the "spider" bounds by using the dynamic programming strategy of guessing the optimal reward function and subsequently establishing its optimality by proving its excessiveness.