Numerical approximation of a Dynkin game with asymmetric information
A talk in the BI.discrete Workshop series by
Tsiry Randrianasolo from Leoben
| Abstract: | Zero-sum games of optimal stopping or Dynkin games have been proposed by Eugene Dynkin in 1967 as an extension of problems of optimal stopping.
The game consists of two players, A and B.
Each player can stop the game at any time to optimize a performance criterion.
If a player chooses to stop the game, B receives some rewards from A.
In such a game, B attempts to maximize his rewards, while A tries to minimize the payout.
The value of this game is a solution to a Hamilton--Jacobi--Bellman equation with two obstacles.
The game must also stop if its value hits one of these obstacles.
In 2013, Christine Grün considered the case where the performance criterion can take several configurations.
Before the game begins, the player A is informed about the configuration that is being played, while the other one only knows it up to a certain probability vector p.
In this configuration, a third obstacle appears. It enforces the value function to be convex with respect to the probability vector p.
In this joint work with Ľubomír Baňas and Giorgio Ferrari, we propose a fully discrete numerical scheme for the approximation of the value function of a Dynkin game with asymmetric information as described by Christine Grün.
The value function $u(t,\text{x}, \text{p})$ depends on three variables: the time $t\in [0,T]$, a spatial variable $\text{x}\in\mathbb{R}^d$, and a variable $\text{p}$ living in a simplex $\Delta(I)$ that represents the probability that a possible configuration of the game is played.
One part of the talk will be dedicated to the semi-discrete scheme with respect to $(t,\text{p})$, which preserves the convexity in $\text{p}$.
It relies on the probabilistic representation of the solution to the constrained PDE problem.
Following the Barles--Souganidis approach we show that the semi-discrete scheme converges in the viscosity sense toward the value function.
Another part of the talk will be dedicated to the fully discrete scheme, where we use a Feedforward Neural Network as a regression function and obtain an implementable scheme.
Under suitable conditions between the meshgrids for the variables $t$, $\text{x}$, $\text{p}$, and the parameters of the neural network, we obtain a uniform convergence on every compact toward the viscosity solution of the constrained PDE problem.
During the time left, we will go through a range of numerical studies to demonstrate the properties of the proposed scheme and compare it with the so-called Semi-Lagrangian scheme. Within the CRC this talk is associated to the project(s): A7, B3, B7 |