On the Multidimensional Controller-and-Stopper Games

Abstract

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.

A stochastic approximation for fully nonlinear free boundary parabolic problems

“…Only very recently there has been some significant development in numerically solving these nonlinear PDEs using Monte Carlo methods, see, for examplpe, . When the control problem also contains a stopper, for example, in determining the super hedging price of an American option, see , or solving controller‐and‐stopper games, see , the nonlinear PDEs have free boundaries.…”

A Zero-Sum Stochastic Differential Game with Impulses, Precommitment, and Unrestricted Cost Functions

“…These are used to ensure that the strategies constructed in the proof of the DPP are strongly nonanticipative. The idea is similar to the definition of admissible stopping strategies in [4,Section 3], which serve a similar purpose.…”

“…The symbol w − ϕ should be understood subject to the conventionf A + f B := f A | B + f B whenever f A : A → R, f B : B → R, and B ⊂ A 4. We have abused notation slightly in writing F (t, x, w) since the derivatives (Du, D 2 u) do not appear on the boundary ∂ + O.…”

Exact and approximate Nash equilibria in discounted Markov stopping games with terminal redemption

“…This example can be formulated in terms of the present model using the market model proposed in Bielecki et al (1999). Recently, Bayraktar et al (2011) studied in continuous time some games related with risk measures, while Karatzas and Sudderth (2001) studied the kind of games presented in this note for linear diffusions; see also Zamifrescu (2005, 2008) and Lepeltier (1985). On the other hand, the optimal stopping problem has been studied separately for many authors, using different techniques such as the free boundary problems or excessive functions; see, for instance, the classical book by Shiryaev (1978).…”