Stochastic scattering control of spider diffusion governed by an optimal diffraction probability measure selected from its own local-time

TL;DR

This paper develops a stochastic control framework for Walsh's spider diffusion, characterizing the optimal scattering control via a Hamilton-Jacobi-Bellman system with a novel boundary condition, using recent comparison theorems.

Contribution

It introduces a new stochastic control problem for spider diffusion, proving a dynamic programming principle and characterizing the value function with a unique HJB system on star-shaped networks.

Findings

Established the weak dynamic programming principle for spider diffusion.

Proved the value function is the unique solution to a HJB system with non-linear boundary conditions.

Extended results to simpler scattering control problems without local-time dependency.

Abstract

The purpose of this article is to study a new problem of stochastic control, related to Walsh's spider diffusion, named: stochastic optimal scattering control. The optimal scattering control of the spider diffusion at the junction point is governed by an appropriate and highly non-trivial condition of the Kirchhoff Law type, involving an optimal diffraction probability measure selected from the own local time of the spider process at the vertex. In this work, we prove first the weak dynamic programming principle in the spirit of [32], adapted to the new class of spider diffusion introduced recently in [37]-[38]. Thereafter, we show that the value function of the problem is characterized uniquely in terms of a Hamilton Jacobi Bellman (HJB) system posed on a star-shaped network, having a new boundary condition at the vertex called : non linear local-time Kirchhoff's transmission. The key main point is to use the recent comparison theorem obtained in [40], that has significantly unlocked the study of this type of problem. We conclude by discussing the formulation of stochastic scattering control problems, where there is no dependency w.r.t. the local-time variable, for which their well-posedness appear as a simpler consequence of the results of this work and the advances contained in [40].