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Numerical approximations of the value of zero-sum stochastic differential impulse controls game in finite horizon

Antoine Zolome, Brahim El Asri

Abstract

In this paper, we consider a differential stochastic zero-sum game in which two players intervene by adopting impulse controls in a finite time horizon. We provide a numerical solution as an approximation of the value function, which turns out to be the same for both players. While one seeks to maximize the value function, the other seeks to minimize it. Thus we find a single numerical solution for the Nash equilibrium as well as the optimal impulse controls strategy pair for both player based on the classical Policy Iteration (PI) algorithm. Then, we perform a rigorous convergence analysis on the approximation scheme where we prove that it converges to its corresponding viscosity solution as the discretization step approaches zero, and under certain conditions. We showcase our algorithm by implementing a two-player almost analytically solvable game in which the players act through impulse control and compete over the exchange rate.

Numerical approximations of the value of zero-sum stochastic differential impulse controls game in finite horizon

Abstract

In this paper, we consider a differential stochastic zero-sum game in which two players intervene by adopting impulse controls in a finite time horizon. We provide a numerical solution as an approximation of the value function, which turns out to be the same for both players. While one seeks to maximize the value function, the other seeks to minimize it. Thus we find a single numerical solution for the Nash equilibrium as well as the optimal impulse controls strategy pair for both player based on the classical Policy Iteration (PI) algorithm. Then, we perform a rigorous convergence analysis on the approximation scheme where we prove that it converges to its corresponding viscosity solution as the discretization step approaches zero, and under certain conditions. We showcase our algorithm by implementing a two-player almost analytically solvable game in which the players act through impulse control and compete over the exchange rate.

Paper Structure

This paper contains 23 sections, 14 theorems, 136 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Assumptions and Formulation of the problem
  3. Two player zero-sum stochastic impulse game
  4. The system of HJBI quasi-variational inequalities
  5. Example of application:
  6. Convergence Analysis
  7. Discretization and Numerical approximation
  8. Intervention operators
  9. Boundary and Terminal Conditions
  10. Boundary conditions
  11. Terminal condition
  12. The numerical approximation scheme
  13. Monotonicity
  14. Stability
  15. Non local consistency
  16. ...and 8 more sections

Key Result

Theorem 2.1

(Comparison theorem) The HJBQVI (eq:HJBI) satisfies a comparison principle in the set of bounded maps valued from $[0,T]\times \mathbb{R^n}$ (for any $n\geq 1$) (i.e., if $u$ is a bounded uniformly continuous viscosity subsolution of (eq:HJBI) and $w$ is a bounded uniformly continuous viscosity supe

Figures (3)

  • Figure 1: Value Function $V(t,x)$ (left) an $V(t=0,x)$ (right) at $t=0$.
  • Figure 2: Impulses actions for Player-$\xi$(top-left) and Player-$\eta$ (top-right), and game evolution (bottom) with no impulses.
  • Figure 3: Optimal Value Function for the initial state $x_0 = 2.5$ (left) and optimal states starting from $x_0 =2.5$(right).

Theorems & Definitions (33)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Remark 2.2
  • Theorem 2.1
  • proof
  • Proposition 2.1
  • Corollary 2.1
  • Remark 2.3
  • Definition 2.3
  • ...and 23 more