We consider a zero-sum stochastic differential controller-and-stopper game in which the state process is a controlled diffusion evolving in a multi-dimensional Euclidean space. In this game, the controller affects both the drift and diffusion terms of the state process, and the diffusion term can be degenerate. Under appropriate conditions, we show that the game has a value and the value function is the unique viscosity solution to an obstacle problem for a Hamilton-Jacobi-Bellman equation.Key Words: Controller-stopper games, weak dynamic programming principle, viscosity solutions, robust optimal stopping. over all choices of τ . At the same time, however, the controller plays against her by maximizing (1.1) over all choices of α.Ever since the game of control and stopping was introduced by Maitra & Sudderth [25], it has been known to be closely related to some common problems in mathematical finance, such as pricing American contingent claims (see e.g. [17,21,22]) and minimizing the probability of lifetime ruin (see [5]). The game itself, however, has not been studied to a great extent except certain particular cases. Karatzas and Sudderth [20] study a zero-sum controller-and-stopper game in which the state process X α is a one-dimensional diffusion along a given interval on R. Under appropriate conditions they prove that this game has a value and describe fairly explicitly a saddle point of Date: January 6, 2013. We would like to thank Mihai Sîrbu for his thoughtful suggestions. We also would like to thank the two anonymous referees whose suggestions helped us improve our paper.
This paper resolves a question proposed in Kardaras and Robertson [Ann. Appl. Probab. 22 (2012) 1576-1610]: how to invest in a robust growth-optimal way in a market where precise knowledge of the covariance structure of the underlying assets is unavailable. Among an appropriate class of admissible covariance structures, we characterize the optimal trading strategy in terms of a generalized version of the principal eigenvalue of a fully nonlinear elliptic operator and its associated eigenfunction, by slightly restricting the collection of nondominated probability measures.Comment: Published in at http://dx.doi.org/10.1214/12-AAP887 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org
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