Stochastic control on networks: weak DPP, and verification theorem

TL;DR

This paper investigates stochastic control problems on network junctions using weak control formulations, establishing the dynamic programming principle and verification theorem, with applications to PDEs and quadratic optimization at the junction.

Contribution

It introduces a weak control framework for junction problems, proves the DPP and verification theorem, and connects control solutions to PDEs with nonlinear boundary conditions.

Findings

Proved compactness of admissible control rules.

Established the dynamic programming principle for the problem.

Provided a verification theorem linking control and PDE solutions.

Abstract

The purpose of this article is to study a stochastic control problem on a junction, with control at the junction point. The problem of control is formulated in the weak sense, using a relaxed control, namely a control which takes values in the space of probability measures on a compact set. We prove first the compactness of the admissible rules and the dynamic programming principle (DPP). We complete this article by giving a verification Theorem for the value function of the problem, using some recent results on quasi linear non degenerate PDE posed on a junction, with non linear Neumann boundary condition at the junction point. An example is given, where the optimal control at the junction point is solution of a convex quadratic optimization problem with linear constraints.