Reflected Backward Stochastic Differential Equations Research Articles

Reflected and Doubly Reflected Backward Stochastic Differential Equations with Irregular Obstacles and a Large Set of Stopping Strategies

Reflected and Doubly Reflected Backward Stochastic Differential Equations with Irregular Obstacles and a Large Set of Stopping Strategies

Nonlinear BSDEs with Two Optional Doob’s Class Barriers Satisfying Weak Mokobodzki’s Condition and Extended Dynkin Games

We study reflected backward stochastic differential equations (RBSDEs) on the probability space equipped with a Brownian motion. The main novelty of the paper lies in the fact that we consider the following weak assumptions on the data: barriers are optional of class (D) satisfying weak Mokobodzki’s condition, generator is continuous and non-increasing with respect to the value-variable (no restrictions on the growth) and Lipschitz continuous with respect to the control-variable, and the terminal condition and the generator at zero are supposed to be merely integrable. We prove that under these conditions on the data there exists a solution to corresponding RBSDE. In the second part of the paper, we apply the theory of RBSDEs to solve basic problems in Dynkin games driven by nonlinear expectation based on the generator mentioned above. We prove that the main component of a solution to RBSDE represents the value process in corresponding extended nonlinear Dynkin game. Moreover, we provide sufficient conditions on the barriers guaranteeing the existence of the value for nonlinear Dynkin games and the existence of a saddle point.

Reflected Backward Stochastic Differential Equations Driven by G-Brownian Motion Under Monotonicity Condition

Reflected Backward Stochastic Differential Equations Driven by G-Brownian Motion Under Monotonicity Condition

Infinite horizon multi-dimensional BSDE with oblique reflection and switching problem

This paper studies a system of multi-dimensional reflected backward stochastic differential equations (RBSDEs) with oblique reflections in infinite horizon associated to switching problems. The existence and uniqueness of the adapted solution is obtained by using a method based on a combination of penalization, verification and contraction property.

Existence and Uniqueness of Solutions for Multi-dimensional Reflected Backward Stochastic Differential Equations with Diagonally Quadratic Generators

Existence and Uniqueness of Solutions for Multi-dimensional Reflected Backward Stochastic Differential Equations with Diagonally Quadratic Generators

Reflected Backward Stochastic Differential Equations Associated to Jump Markov Processes and Application to Partial Differential Equations

Reflected Backward Stochastic Differential Equations Associated to Jump Markov Processes and Application to Partial Differential Equations

Optimal stopping under g-Expectation with -integrable reward process

AbstractIn this paper we study a class of optimal stopping problems under g-expectation, that is, the cost function is described by the solution of backward stochastic differential equations (BSDEs). Primarily, we assume that the reward process is $L\exp\bigl(\mu\sqrt{2\log\!(1+L)}\bigr)$ -integrable with $\mu>\mu_0$ for some critical value $\mu_0$ . This integrability is weaker than $L^p$ -integrability for any $p>1$ , so it covers a comparatively wide class of optimal stopping problems. To reach our goal, we introduce a class of reflected backward stochastic differential equations (RBSDEs) with $L\exp\bigl(\mu\sqrt{2\log\!(1+L)}\bigr)$ -integrable parameters. We prove the existence, uniqueness, and comparison theorem for these RBSDEs under Lipschitz-type assumptions on the coefficients. This allows us to characterize the value function of our optimal stopping problem as the unique solution of such RBSDEs.

Sequential Systems of Reflected Backward Stochastic Differential Equations with Application to Impulse Control

We consider a system of finite horizon, sequentially interconnected, obliquely reflected backward stochastic differential equations (RBSDEs) with stochastic Lipschitz coefficients. We show existence of solutions to our system of RBSDEs by applying a Picard iteration approach. Uniqueness then follows by relating the limit to an auxiliary impulse control problem. Moreover, we show that the solution to our system of RBSDEs is connected to weak solutions of a stochastic differential game where one player implements an impulse control while the opponent plays a continuous control that enters the drift term. As all our arguments are probabilistic and hence hold in a non-markovian framework, we are able to consider the setting where the underlying uncertainty in the game stems from an impulsively and continuously controlled path-dependent stochastic differential equation driven by Brownian motion.

RBSDEs with optional barriers: monotone approximation

In this short note we consider reflected backward stochastic differential equations (RBSDEs) with a Lipschitz driver and barrier processes that are optional and right lower semicontinuous. In this case, the barrier is represented as a nondecreasing limit of right continuous with left limit (RCLL) barriers. We combine some well-known existence results for RCLL barriers with comparison arguments for the control process to construct solutions. Finally, we highlight the connection of these RBSDEs with standard RCLL BSDEs.

Doubly Reflected Backward Stochastic Differential Equations in the Predictable Setting

In this paper, we introduce a specific kind of doubly reflected backward stochastic differential equations (in short DRBSDEs), defined on probability spaces equipped with general filtration that is essentially non quasi-left continuous, where the barriers are assumed to be predictable processes. We call these equations predictable DRBSDEs. Under a general type of Mokobodzki’s condition, we show the existence of the solution (in consideration of the driver’s nature) through a Picard iteration method and a Banach fixed point theorem. By using an appropriate generalization of Ito’s formula due to Gal’chouk and Lenglart we provide a suitable a priori estimates which immediately implies the uniqueness of the solution.

Control-Stopping Games for Market Microstructure and Beyond

In this paper, we present a family of control-stopping games that arise naturally in equilibrium-based models of market microstructure as well as in other models with strategic buyers and sellers. A distinctive feature of this family of games is the fact that the agents do not have any exogenously given fundamental value for the asset, and they deduce the value of their position from the bid and ask prices posted by other agents (i.e., they are pure speculators). As a result, in such a game, the reward function of each agent at the time of stopping depends directly on the controls of other players. The equilibrium problem leads naturally to a system of coupled control-stopping problems (or, equivalently, reflected-backward stochastic differential equations), in which the individual reward functions (or reflecting barriers) depend on the value functions (or solution components) of other agents. The resulting system, in general, presents multiple mathematical challenges because of the nonstandard form of coupling (or reflection). In the present case, this system is also complicated by the fact that the continuous controls of the agents, describing their posted bid and ask prices, are constrained to take values in a discrete grid. The latter feature reflects the presence of a positive tick size in the market, and it creates additional discontinuities in the agents’ reward functions (or reflecting barriers). Herein we prove the existence of a solution to the associated system in a special Markovian framework, provide numerical examples, and discuss the potential applications.

Particles Systems for mean reflected BSDEs

In this paper, we study Reflected Backward Stochastic Differential Equations for which the constraint is not on the paths of the solution but on its law as introduced by Briand, Elie and Hu in Briand et al. (2018). We extend the recent work Briand et al. (2020) of Briand, Chaudru de Raynal, Guillin and Labart on the chaos propagation for mean reflected SDEs to the backward framework. When the driver does not depend on z, we are able to study general reflections for the particles system. We consider linear reflection when the driver depends also on z. In both cases, we get the rate of convergence of the particles system towards the square integrable deterministic flat solution to the mean reflected BSDE.

Reflected Backward Stochastic Differential Equation with Rank-Based Data

In this paper, we study reflected backward stochastic differential equation (reflected BSDE) with rank-based data in a Markovian framework; that is, the solution to the reflected BSDE is above a prescribed boundary process in a minimal fashion and the generator and terminal value of the reflected BSDE depend on the solution of another stochastic differential equation (SDE) with rank-based drift and diffusion coefficients. We derive regularity properties of the solution to such reflected BSDE and show that the solution at the initial starting time t and position x, which is a deterministic function, is the unique viscosity solution to some obstacle problem (or variational inequality) for the corresponding parabolic partial differential equation.

Reflected BSDEs with jumps in time-dependent convex càdlàg domains

In the first part of the paper, we study the unique solvability of multidimensional reflected backward stochastic differential equations (RBSDEs) of Wiener–Poisson type with reflection in the inward spatial normal direction of a time-dependent adapted càdlàg convex set D={Dt,t∈[0,T]}. The existence result is obtained by approximating the solutions of this class of RBSDEs by solutions of BSDEs with reflection in discretizations of D, while the uniqueness is established by using Itô’s formula. In the second part of the paper, we show that the solutions of our RBSDEs can be approximated via a non-standard penalization method.

Evaluating Optimal Stopping Problems Under Multivariate Settings and Model Uncertainty

In this article we study the evaluation of American options with stochastic volatility models and the optimal fish harvesting decision with stochastic convenience yield models, in the presence of drift ambiguity. From the perspective of an ambiguity averse agent, we transfer the problem to the solution of a reflected backward stochastic differential equations (RBSDE) and prove the uniform Lipschitz continuity of the generator. We then propose a numerical algorithm with the theory of RBSDEs and a general stratification technique, and an alternative algorithm without using the theory of RBSDEs. We test the accuracy and convergence of the numerical schemes. By comparing to the one dimensional case, we highlight the importance of the dynamic structure of the agent’s worst case belief. Results also show that the numerical RBSDE algorithm with stratification is more efficient when the optimal generator is attainable.

Optimal Stopping of Marked Point Processes and Reflected Backward Stochastic Differential Equations

We define a class of reflected backward stochastic differential equation (RBSDE) driven by a marked point process (MPP) and a Brownian motion, where the solution is constrained to stay above a given cadlag process. The MPP is only required to be non-explosive and to have totally inaccessible jumps. Under suitable assumptions on the coefficients we obtain existence and uniqueness of the solution, using the Snell envelope theory. We use the equation to represent the value function of an optimal stopping problem, and we characterize the optimal strategy.

Reflected Backward Stochastic Differential Equation with Jumps and Viscosity Solution of Second Order Integro-Differential Equation Without Monotonicity Condition: Case with the Measure of Lévy Infinite

We consider the problem of viscosity solution of integro-partial differential equation(IPDE in short) with one obstacle via the solution of reflected backward stochastic differential equations(RBSDE in short) with jumps. We show the existence and uniqueness of a continuous viscosity solution of equation with non local terms, if the generator is not monotonous and Levy’s measure is infinite.

Reflected and Doubly Reflected Backward Stochastic Differential Equations with Time-Delayed Generators

In this paper, we study reflected backward stochastic differential equations with two reflecting barriers and time-delayed generators (delay RBSDEs for short). We consider the case of Brownian noise as well as the case of Brownian and Poisson noise. For both cases, we show the existence and uniqueness of the solution and give a comparison theorem.

Reflected BSDEs with optional barrier in a general filtration

We consider reflected backward stochastic differential equations (RBSDE) with optional barrier and trajectories with left and right finite limits. We prove existence and uniqueness results and we show a comparison theorem for RBSDEs by using local time formula.

American options with asymmetric information and reflected BSDE

We consider an American contingent claim on a financial market where the buyer has additional information. Both agents (seller and buyer) observe the same prices, while the information available to them may differ due to some extra exogenous knowledge the buyer has. The buyer’s information flow is modeled by an initial enlargement of the reference filtration. It seems natural to investigate the value of the American contingent claim with asymmetric information. We provide a representation for the cost of the additional information relying on some results on reflected backward stochastic differential equations (RBSDE). This is done by using an interpretation of prices of American contingent claims with extra information for the buyer by solutions of appropriate RBSDE.