Optimal Stopping of BSDEs with Constrained Jumps and Related Zero-Sum Games

In this paper, we introduce a non-linear Snell envelope which at each time represents the maximal value that can be achieved by stopping a BSDE with constrained jumps. We establish the existence of the Snell envelope by employing a penalization technique and the primary challenge we encounter is demonstrating the regularity of the limit for the scheme. Additionally, we relate the Snell envelope to a finite horizon, zero-sum stochastic differential game, where one player controls a path-dependent stochastic system by invoking impulses, while the opponent is given the opportunity to stop the game prematurely. Importantly, by developing new techniques within the realm of control randomization, we demonstrate that the value of the game exists and is precisely characterized by our non-linear Snell envelope

State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton-Jacobi-Isaacs Equations

In this paper, we consider state and control path-dependent stochastic zero-sum differential games, where the dynamics and the running cost include both state and control paths of the players. Using the notion of nonanticipative strategies, we define lower and upper value functionals, which are functions of the initial state and control paths of the players. We prove that the value functionals satisfy the dynamic programming principle. The associated lower and upper Hamilton-Jacobi-Isaacs (HJI) equations from the dynamic programming principle are state and control path-dependent nonlinear second-order partial differential equations. We apply the functional It\^o calculus to prove that the lower and upper value functionals are viscosity solutions of (lower and upper) state and control path-dependent HJI equations, where the notion of viscosity solutions is defined on a compact subset of an $\kappa$-H\"older space introduced in \cite{Tang_DCD_2015}. Moreover, we show that the Isaacs condition and the uniqueness of viscosity solutions imply the existence of the game value. For the state path-dependent case, we prove the uniqueness of classical solutions for the (state path-dependent) HJI equations.Comment: 29 page

Optimal stopping under adverse nonlinear expectation and related games

We study the existence of optimal actions in a zero-sum game $\inf_{\tau}\sup_PE^P[X_{\tau}]$ between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem $\inf_{\tau}\mathcal{E}(X_{\tau})$ for a class of sublinear expectations $\mathcal{E}(\cdot)$ such as the $G$-expectation. We show that the game has a value. Moreover, exploiting the theory of sublinear expectations, we define a nonlinear Snell envelope $Y$ and prove that the first hitting time $\inf\{t:Y_t=X_t\}$ is an optimal stopping time. The existence of a saddle point is shown under a compactness condition. Finally, the results are applied to the subhedging of American options under volatility uncertainty.Comment: Published at http://dx.doi.org/10.1214/14-AAP1054 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dynkin games with Poisson random intervention times

This paper introduces a new class of Dynkin games, where the two players are allowed to make their stopping decisions at a sequence of exogenous Poisson arrival times. The value function and the associated optimal stopping strategy are characterized by the solution of a backward stochastic differential equation. The paper further applies the model to study the optimal conversion and calling strategies of convertible bonds, and their asymptotics when the Poisson intensity goes to infinity

Path-dependent Hamilton-Jacobi equations in infinite dimensions

We propose notions of minimax and viscosity solutions for a class of fully nonlinear path-dependent PDEs with nonlinear, monotone, and coercive operators on Hilbert space. Our main result is well-posedness (existence, uniqueness, and stability) for minimax solutions. A particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle. One of the main difficulties, the lack of compactness in infinite-dimensional Hilbert spaces, is circumvented by working with suitable compact subsets of our path space. As an application, our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of (delay) evolution equations in the variational framework. Furthermore, differential games associated to such evolution equations can be investigated following the Krasovskii-Subbotin approach similarly as in finite dimensions.Comment: Final version, 53 pages, to appear in Journal of Functional Analysi