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

Isaac Ohavi

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].

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

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].

Paper Structure

This paper contains 17 sections, 19 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. General Introduction:
  3. Review of literature:
  4. Organization of the paper
  5. Formulation of the problem and main results
  6. Notations and definitions
  7. Martingale problem and viscosity solutions:
  8. Main results
  9. Compactness of the admissible rules and Dynamic Programming Principle
  10. A criterion of compactness of the admissible rules at the junction point $\bf 0$.
  11. Some estimates and paths properties of the process
  12. Dynamic programming principle
  13. Proof of Theorem \ref{['th: PPD']}
  14. Comparison theorem for continuous viscosity backward parabolic Walsh's spider HJB equations having a non linear local-time Kirchhoff's boundary condition
  15. Unique characterization of the value function for the problem of stochastic scattering control
  16. ...and 2 more sections

Key Result

Theorem 2.4

Dynamic programming principle equation: Assume assumption $(\mathcal{H})$. For any $(\Psi_s)_{t\leq s \leq T}$ stopping time $\tau$, the value function $u$ defined in Definition def fonction valeur, satisfies for all $(i,t,x,l) \in[I]\times[0,T]\times[0,+\infty)^2$:

Theorems & Definitions (35)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 3.1
  • proof
  • ...and 25 more