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Zero-sum Stochastic Differential Games of Impulse Control with Random Intervention Costs

Magnus Perninge

Abstract

We consider a finite-horizon, zero-sum game in which both players control a stochastic differential equation by invoking impulses. We derive a control randomization formulation of the game and use the existence of a value for the randomized game to show that the upper and lower value functions of the original game coincide. The main contribution of the present work is that we can allow intervention costs that are functions of the state as well as time, and that we do not need to impose any monotonicity assumptions on the involved coefficients.

Zero-sum Stochastic Differential Games of Impulse Control with Random Intervention Costs

Abstract

We consider a finite-horizon, zero-sum game in which both players control a stochastic differential equation by invoking impulses. We derive a control randomization formulation of the game and use the existence of a value for the randomized game to show that the upper and lower value functions of the original game coincide. The main contribution of the present work is that we can allow intervention costs that are functions of the state as well as time, and that we do not need to impose any monotonicity assumptions on the involved coefficients.
Paper Structure (19 sections, 22 theorems, 202 equations)

This paper contains 19 sections, 22 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Problem formulation and main result
  3. Assumptions
  4. Notations
  5. Viscosity solutions
  6. Main result
  7. Truncations of the game
  8. A discretization of the control sets
  9. The dual game
  10. Additional definitions and notation
  11. A truncation
  12. Discretization of the impulse control for the randomized problem
  13. Proof of Proposition \ref{['prop:rel-rand-to-orig']}
  14. Part A
  15. Part B
  16. ...and 4 more sections

Key Result

Theorem 2.8

The game has a value and $v=\underline v=\bar{v}$ belongs to $\Pi^g_c$ and is the unique (in $\Pi^g$) viscosity solution to ekv:var-ineq.

Theorems & Definitions (34)

  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Corollary 2.9
  • Lemma 3.1
  • Proposition 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 24 more