This article is dedicated to the study of mixed zero-sum two-player stochastic differential games in the situation when the player's cost functionals are modeled by doubly controlled reflected backward stochastic equations with two barriers whose coefficients have quadratic growth in Z. This is a generalization of the risk-sensitive payoffs. We show that the lower and the upper value function associated with this stochastic differential game with reflection are deterministic and they are also the unique viscosity solutions for two Isaacs equations with obstacles.
We introduce the notion of mild supersolution for an obstacle problem in an infinite dimensional Hilbert space. The minimal supersolution of this problem is given in terms of a reflected BSDEs in an infinite dimensional Markovian framework. The results are applied to an optimal control and stopping problem.
We prove some new results on reflected BSDEs and doubly reflected BSDEs driven by a multi-dimensional RCLL martingale. The goal is to develop a general multi-asset framework encompassing a wide spectrum of nonlinear financial models, including as particular cases the setups studied by Peng and Xu \cite{PX2009} and Dumitrescu et al. \cite{DGQS2018} who dealt with BSDEs driven by a one-dimensional Brownian motion and a purely discontinuous martingale with a single jump. Our results are not covered by existing literature on reflected and doubly reflected BSDEs driven by a Brownian motion and a Po…
We consider a robust impulse control problem in finite horizon where the underlying uncertainty stems from an impulsively and continuously controlled functional stochastic differential equation (FSDE) driven by Brownian motion. We assume that the controller acts upon the system by impulses while the adversary player (nature) acts through continuous controls. We look for a weak solution which leads us to consider a system of sequentially interconnected, obliquely reflected backward stochastic differential equations (RBSDEs) with stochastic Lipschitz coefficients. We show existence of solutions …
This paper studies the mixed zero-sum stochastic differential game problem. We allow the functionals and dynamics to be of polynomial growth. The problem is formulated as an extended doubly reflected BSDEs with a specific generator. We show the existence of solution for this doubly reflected BSDEs and we prove the existence of a saddle-point of the game. Moreover, in the Markovian framework we prove that the value function is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation.
We study a doubly reflected backward stochastic differential equation (BSDE) with integrable parameters and the related Dynkin game. When the lower obstacle $L$ and the upper obstacle $U$ of the equation are completely separated, we construct a unique solution of the doubly reflected BSDE by pasting local solutions and show that the $Y-$component of the unique solution represents the value process of the corresponding Dynkin game under $g-$evaluation, a nonlinear expectation induced by BSDEs with the same generator $g$ as the doubly reflected BSDE concerned. In particular, the first time when …
In this paper, an optimal switching problem is proposed for one-dimensional reflected backward stochastic differential equations (RBSDEs, for short) where the generators, the terminal values and the barriers are all switched with positive costs. The value process is characterized by a system of multi-dimensional RBSDEs with oblique reflection, whose existence and uniqueness are by no means trivial and are therefore carefully examined. Existence is shown using both methods of the Picard iteration and penalization, but under some different conditions. Uniqueness is proved by representation eithe…
We formulate a notion of doubly reflected BSDE in the case where the barriers $ξ$ and $ζ$ do not satisfy any regularity assumption and with a general filtration. Under a technical assumption (a Mokobodzki-type condition), we show existence and uniqueness of the solution. In the case where $ξ$ is right upper-semicontinuous and $ζ$ is right lower-semicontinuous, the solution is characterized in terms of the value of a corresponding $\mathcal{E}^f$-Dynkin game, i.e. a game problem over stopping times with (non-linear) $f$-expectation, where $f$ is the driver of the doubly reflected BSDE. In the g…
In this article we study the existence and the uniqueness of a solution for reflected backward stochastic differential equations in the case when the generator is logarithmic growth in the $z$-variable $(|z|\sqrt{|\ln(|z|)|})$, the terminal value and obstacle are an $L^p$-integrable, for a suitable $p > 2$. To construct the solution we use localization method. We also apply these results to get the existence of an optimal control strategy for the mixed stochastic control problem in finite horizon.
The aim of this short note is to fill in a gap in our earlier paper [16] on 2BSDEs with reflections, and to explain how to correct the subsequent results in the second paper [15]. We also provide more insight on the properties of 2RBSDEs, in the light of the recent contributions [13, 23] in the so--called $G-$framework.
In this paper, we analyze a real-valued reflected backward stochastic differential equation (RBSDE) with an unbounded obstacle and an unbounded terminal condition when its generator $f$ has quadratic growth in the $z$-variable. In particular, we obtain existence, comparison, and stability results, and consider the optimal stopping for quadratic $g$-evaluations. As an application of our results we analyze the obstacle problem for semi-linear parabolic PDEs in which the non-linearity appears as the square of the gradient. Finally, we prove a comparison theorem for these obstacle problems when th…
We consider quasi-variational inequalities (QVIs) with general non-local drivers and related systems of reflected backward stochastic differential equations (BSDEs) in a Brownian filtration. We show existence and uniqueness of viscosity solutions to the QVIs by first considering the standard (local) setting and then applying a contraction argument. In addition, the contraction argument yields existence and uniqueness of solutions to the related systems of reflected BSDEs and extends the theory of probabilistic representations of PDEs in terms of BSDEs to our specific setting.
We consider a non-Markovian optimal stopping problem on finite horizon. We prove that the value process can be represented by means of a backward stochastic differential equation (BSDE), defined on an enlarged probability space, containing a stochastic integral having a one-jump point process as integrator and an (unknown) process with a sign constraint as integrand. This provides an alternative representation with respect to the classical one given by a reflected BSDE. The connection between the two BSDEs is also clarified. Finally, we prove that the value of the optimal stopping problem is t…
We introduce a new type of reflected backward stochastic differential equations (BSDEs) for which the reflection constraint is imposed on its main solution component, denoted as $Y$ by convention, but in terms of its conditional expectation $\mathbb{E}[Y_t|\mathcal{G}_{t}]$ on a general sub-filtration $\{\mathcal{G}_{t}\}.$ We thus term such equation as conditionally reflected BSDE (for short, conditional RBSDE). Conditional RBSDE subsumes classical RBSDE with a pointwise reflection barrier, and the recent developed BSDE with a mean reflection constraint, as its two special and extreme cases: …
In this work we investigate regularity properties of a large class of Hamilton-Jacobi-Bellman (HJB) equations with or without obstacles, which can be stochastically interpreted in form of a stochastic control system which nonlinear cost functional is defined with the help of a backward stochastic differential equation (BSDE) or a reflected BSDE (RBSDE). More precisely, we prove that, firstly, the unique viscosity solution $V(t,x)$ of such a HJB equation over the time interval $[0,T],$ with or without an obstacle, and with terminal condition at time $T$, is jointly Lipschitz in $(t,x)$, for $t$…
In this paper we study continuous-time two-player zero-sum optimal switching games on a finite horizon. Using the theory of doubly reflected BSDEs with interconnected barriers, we show that this game has a value and an equilibrium in the players' switching controls.
We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingal…
We generalize the results of Fleming and Souganidis (1989) on zero sum stochastic differential games to the case when the controls are unbounded. We do this by proving a dynamic programming principle using a covering argument instead of relying on a discrete approximation (which is used along with a comparison principle by Fleming and Souganidis). Also, in contrast with Fleming and Souganidis, we define our pay-off through a doubly reflected backward stochastic differential equation. The value function (in the degenerate case of a single controller) is closely related to the second order doubl…
