[Paper Review] Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions

Bruno Bouchard, Marcel Nutz|arXiv (Cornell University)|Jul 22, 2013

Stochastic processes and financial applications3 citations

TL;DR

This paper develops a geometric dynamic programming principle for stochastic target games under model uncertainty, where one player must ensure almost-sure target attainment despite adversarial controls. It introduces a novel regularization-based approach using smooth supersolutions to construct measurable almost-optimal Markovian strategies, enabling the characterization of the value function as the unique bounded viscosity solution of a nonlinear PDE.

ABSTRACT

We study a class of stochastic target games where one player tries to find a strategy such that the state process almost-surely reaches a given target, no matter which action is chosen by the opponent. Our main result is a geometric dynamic programming principle which allows us to characterize the value function as the viscosity solution of a non-linear partial differential equation. Because abstract mea-surable selection arguments cannot be used in this context, the main obstacle is the construction of measurable almost-optimal strategies. We propose a novel approach where smooth supersolutions are used to define almost-optimal strategies of Markovian type, similarly as in ver-ification arguments for classical solutions of Hamilton--Jacobi--Bellman equations. The smooth supersolutions are constructed by an exten-sion of Krylov's method of shaken coefficients. We apply our results to a problem of option pricing under model uncertainty with different interest rates for borrowing and lending.

Motivation & Objective

• To establish a geometric dynamic programming principle (GDP) for stochastic target games where the target must be almost-surely achieved under adversarial control.
• To overcome the fundamental challenge of constructing measurable almost-optimal strategies in non-Markovian, non-separable strategy spaces.
• To develop a novel regularization technique using Krylov’s shaking of coefficients to construct smooth supersolutions that generate Markovian strategies.
• To characterize the value function as the unique bounded viscosity solution of a nonlinear Hamilton–Jacobi–Bellman equation.
• To apply the framework to option pricing under model uncertainty with different borrowing and lending rates, capturing Knightian uncertainty.

Proposed method

• Introduce a geometric dynamic programming principle (GDP) that generalizes classical GDPs to stochastic target games with adversarial controls.
• Use smooth supersolutions of the HJB equation as a basis for constructing Markovian strategies that are almost-optimal via a regularization procedure.
• Apply Krylov’s method of shaking coefficients to construct smooth, uniformly bounded supersolutions that satisfy the required regularity and growth conditions.
• Establish the existence of a measurable selection of almost-optimal strategies by leveraging the smoothness and uniform bounds of the supersolutions.
• Prove that the value function is the unique bounded viscosity solution of a nonlinear PDE involving the infimum over controls of a Hamilton–Jacobi–Bellman operator.
• Use supermartingale arguments under equivalent measures (via Doléan-Dade exponential) to verify boundary conditions and continuity of the value function.

Experimental results

Research questions

• RQ1 Can a geometric dynamic programming principle be established for stochastic target games with adversarial controls when measurable selection arguments fail?
• RQ2 How can almost-optimal strategies be constructed in the absence of separability in the strategy space and irregular dependence on the opponent’s control?
• RQ3 Can Krylov’s shaking of coefficients method be extended to construct smooth supersolutions that yield measurable Markovian strategies in a game-theoretic setting?
• RQ4 What is the PDE characterization of the super-hedging price in a model with stochastic volatility and different borrowing/lending rates?
• RQ5 How does the value function behave under Knightian uncertainty, particularly when the drift and volatility are controlled by an adversary?

Key findings

• The value function is continuous and the unique bounded viscosity solution of a nonlinear PDE involving the infimum over controls of a Hamilton–Jacobi–Bellman operator.
• The geometric dynamic programming principle (GDP) is established, with GDP1 and GDP2 holding under the proposed regularization framework.
• Smooth supersolutions constructed via Krylov’s shaking of coefficients method yield measurable, Markovian almost-optimal strategies, overcoming the measurable selection problem.
• The super-hedging price under model uncertainty with different borrowing and lending rates is characterized as the solution to a nonlinear PDE with a specific HJB structure.
• The value function satisfies a supermartingale property under a family of equivalent measures, ensuring continuity and boundedness.
• The framework applies to financial problems with Knightian uncertainty, where the drift and volatility are controlled by an adversary, and the interest rate depends on the direction of trading.