Martingale approach to stochastic differential games of control and stopping

Abstract

We develop a martingale approach for studying continuous-time stochastic differential games of control and stopping, in a non-Markovian framework and with the control affecting only the drift term of the state-process. Under appropriate conditions, we show that the game has a value and construct a saddle pair of optimal control and stopping strategies. Crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities.

arXiv commentPublished in at http://dx.doi.org/10.1214/07-AOP367 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (16)

• Optimal stopping under adverse nonlinear expectation and related games
• Second order reflected backward stochastic differential equations
• A zero-sum game between a singular stochastic controller and a discretionary stopper
• Optimal Stopping for Dynamic Convex Risk Measures
• Nash equilibria for game contingent claims with utility-based hedging
• Dynamic Programming Principles for Optimal Stopping with Expectation Constraint
• Optimal Stopping for Non-linear Expectations
• Weak equilibria for time-inconsistent control: with applications to investment-withdrawal decisions
• Proving Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control
• How to detect a salami slicer: a stochastic controller-stopper game with unknown competition
• On the controller-stopper problems with controlled jumps
• Controller-and-Stopper Stochastic Differential Games with Regime Switching
• A problem of optimal switching and singular control with discretionary stopping in portfolio selection
• Optimal Capital Injections with the Risk of Ruin: A Stochastic Differential Game of Impulse Control and Stopping Approach
• Impulse Control of a Diffusion with a Change Point
• On controller-stopper problems with jumps and their applications to indifference pricing of American options