Numerical approximation of Dynkin games with asymmetric information

TL;DR

The paper addresses numerical approximation for a Dynkin game with asymmetric information, cast as a convexity-constrained obstacle PDE for the value function $u(t,x,p)$ with $p\in\Delta(I)$. It introduces a structure-preserving scheme that couples time discretization and convexity enforcement in $p$ with a spatial neural-network approximation, and proves convergence to the unique viscosity solution. The approach yields a practical, implementable method, demonstrated through numerical experiments that reproduce free-boundary structures akin to Israeli options and compare favorably to semi-Lagrangian schemes. The work contributes a provably convergent, convexity-preserving, neural-network–assisted framework for high-dimensional Dynkin games under asymmetric information, with potential applications in finance and stochastic control.

Abstract

We propose an implementable, neural network-based structure preserving probabilistic numerical approximation for a generalized obstacle problem describing the value of a zero-sum differential game of optimal stopping with asymmetric information. The target solution depends on three variables: the time, the spatial (or state) variable, and a variable from a standard $(I-1)$-simplex which represents the probabilities with which the $I$ possible configurations of the game are played. The proposed numerical approximation preserves the convexity of the continuous solution as well as the lower and upper obstacle bounds. We show convergence of the fully-discrete scheme to the unique viscosity solution of the continuous problem and present a range of numerical studies to demonstrate its applicability.

Figures (9)

• Figure 1: Left: plot of the function $u(t,\mathrm{x})= \cos(3\pi t)\cos(3\pi x)$. Middle: Graph of the solution at $t = 0.25$. Right: time evolution of the solution at $\mathrm{x} = 0.75$. (($-$) exact solution, ( $\Box$) neural network approximation, ($\bigcirc$) SL algorithm.
• Figure 2: Solution at $t = 0$ computed with the SL algorithm (left) and solution computed without the obstacle term $\lambda(\mathrm{p},\mathrm{D}^2_\mathrm{p} u)$ (right).
• Figure 3: Profiles of the solution computed using the FFN algorithm ( $+$) and the SL algorithm ($\circ$) at $(\mathrm{x},\mathrm{p}) = (0.75, 0.25)$ (left), $(t,\mathrm{p}) = (0.25, 0.25)$ (middle), $(t,\mathrm{x}) = (0.25, 0.75)$ (right)
• Figure 4: From left to right: plot of $\tfrac{1}{2}({\bar{v}^{\triangle}}_{\kappa,\mathcal{D}}(t,\mathrm{x},0) + {\bar{v}^{\triangle}}_{\kappa,\mathcal{D}}(t,\mathrm{x},1)) - {\bar{v}^{\triangle}}_{\kappa,\mathcal{D}}(t,\mathrm{x},0.5)$ (3d and top view) and of $\tfrac{1}{2}({\bar{u}^{\triangle}}_{\kappa,\mathcal{D}}(t,\mathrm{x},0) + {\bar{u}^{\triangle}}_{\kappa,\mathcal{D}}(t,\mathrm{x},1)) - {\bar{u}^{\triangle}}_{\kappa,\mathcal{D}}(t,\mathrm{x},0.5)$ (3d and top view).
• Figure 5: Results for the BR solver: 3d view (top row) and top view (bottom row) of the numerical solution $(t,\mathrm{x}) \rightarrow {\bar{u}^{\triangle}}_{\kappa,\mathcal{D}}(t,\mathrm{x},\mathrm{p})$ (from left to right) for $p=0,0.5,1$ and of the average $(t,\mathrm{x})\rightarrow \frac{1}{2} ({\bar{u}^{\triangle}}_{\kappa,\mathcal{D}}(t,\mathrm{x},0) + {\bar{u}^{\triangle}}_{\kappa,\mathcal{D}}(t,\mathrm{x},1))$ with obstacles respectively represented by $\frac{1}{2} f_1(\mathrm{x}) + \frac{1}{2}f_2(\mathrm{x})$, $\frac{1}{2} h_1(\mathrm{x}) + \frac{1}{2}h_2(\mathrm{x})$.

Theorems & Definitions (26)

• Remark 2.1
• Definition 2.2: Viscosity solution
• Theorem 2.3
• Theorem 3.1
• proof
• Remark 3.2
• Theorem 3.3
• proof
• Corollary 3.4
• proof