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 the same as the value of an auxiliary optimization problem where the intensity of the point process is controlled.
We consider Backward Stochastic Differential Equations in a setting where noise is generated by a countable state, continuous time Markov chain, and the terminal value is prescribed at a stopping time. We show that, given sufficient integrability of the stopping time and a growth bound on the terminal value and BSDE driver, these equations admit unique solutions satisfying the same growth bound (up to multiplication by a constant). This holds without assuming that the driver is monotone in y, that is, our results do not require that the terminal value be discounted at some uniform rate. We sho…
We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). We first provide general existence, uniqueness and comparison theorems for RBSDEs with jumps in the case of a RCLL adapted obstacle. We then show that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of an optimal stopping time is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, robust optimal stopping proble…
This article deals with the starting and stopping problem under Knightian uncertainty, i.e., roughly speaking, when the probability under which the future evolves is not exactly known. We show that the lower price of a plant submitted to the decisions of starting and stopping is given by a solution of a system of two reflected backward stochastic differential equations (BSDEs for short). We solve this latter system and we give the expression of the optimal strategy. Further we consider a more general system of $m$ ($m\geq 2$) reflected BSDEs with interconnected obstacles. Once more we show exi…
In this article, we deal with a multiple dimensional coupled Markovian BSDEs system with stochastic linear growth generators with respect to volatility processes. An existence result is provided by using approximation techniques.
We use the functional It{ô} calculus to prove that the solution of a BSDE with singular terminal condition is continuous at the terminal time. Hence we extend known results for a non-Markovian terminal condition.
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…
This article deals with the existence and the uniqueness of solutions to quadratic and superquadratic Markovian backward stochastic differential equations (BSDEs for short) with an unbounded terminal condition. Our results are deeply linked with a strong a priori estimate on $Z$ that takes advantage of the Markovian framework. This estimate allows us to prove the existence of a viscosity solution to a semilinear parabolic partial differential equation with nonlinearity having quadratic or superquadratic growth in the gradient of the solution. This estimate also allows us to give explicit conve…
In the first part of this paper, we study RBSDEs in the case where the filtration is not quasi-left continuous and the lower obstacle is given by a predictable process. We prove the existence and uniqueness by using some results of optimal stopping theory in the predictable setting, some tools from the general theory of processes as the Merten's decomposition of predictable strong supermartingale. In the second part we introduce an optimal stopping problem indexed by predictable stopping times with the non linear predictable $g$ expectation induced by an appropriate BSDE. We establish some use…
In [Stochastc Process. Appl., 122(9):3173-3208], the author proved the existence and the uniqueness of solutions to Markovian superquadratic BSDEs with an unbounded terminal condition when the generator and the terminal condition are locally Lipschitz. In this paper, we prove that the existence result remains true for these BSDEs when the regularity assumptions on the terminal condition is weakened.
In this Note we consider a quadratic backward stochastic differential equation (BSDE) driven by a continuous martingale $M$ and whose generator is a deterministic function. We prove (in Theorem \ref{theorem:main}) that if $M$ is a strong homogeneous Markov process and if the BSDE has the form \eqref{BSDE} then the unique solution $(Y,Z,N)$ of the BSDE is reduced to $(Y,Z)$, \textit{i.e.} the orthogonal martingale $N$ is equal to zero showing that in a Markovian setting the "usual" solution $(Y,Z)$ has not to be completed by a strongly orthogonal even if $M$ does not enjoy the martingale repres…
We study the stochastic control-stopping problem when the data are of polynomial growth. The approach is based on backward stochastic dierential equations (BSDEs for short). The problem turns into the study of a specic reected BSDE with a stochastic Lipschitz coecient for which we show existence and uniqueness of the solution. We then establish its relationship with the value function of the control-stopping problem. The optimal strategy is exhibited. Finally in the Markovian framework we prove that the value function is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman e…
We establish existence and uniqueness for a wide class of Markovian systems of backward stochastic differential equations (BSDE) with quadratic nonlinearities. This class is characterized by an abstract structural assumption on the generator, an a-priori local-boundedness property, and a locally-Hölder-continuous terminal condition. We present easily verifiable sufficient conditions for these assumptions and treat several applications, including stochastic equilibria in incomplete financial markets, stochastic differential games, and martingales on Riemannian manifolds.
(Working Paper) Using a purely probabilistic argument, we prove the global well-posedness of multidimensional superquadratic backward stochastic differential equations (BSDEs) without Markovian assumption. The key technique is the interplay between the local well-posedness of fully coupled path-dependent forward backward stochastic differential equations (FBSDEs) and backward iterations of the superquadratic BSDE. The superquadratic BSDE studied in this article includes quadratic BSDEs appear in stochastic differential game and price impact model. We also study the well-posedness of superquadr…
We introduce a new class of Backward Stochastic Differential Equations in which the $T$-terminal value $Y_{T}$ of the solution $(Y,Z)$ is not fixed as a random variable, but only satisfies a weak constraint of the form $E[Ψ(Y_{T})]\ge m$, for some (possibly random) non-decreasing map $Ψ$ and some threshold $m$. We name them \textit{BSDEs with weak terminal condition} and obtain a representation of the minimal time $t$-values $Y_{t}$ such that $(Y,Z)$ is a supersolution of the BSDE with weak terminal condition. It provides a non-Markovian BSDE formulation of the PDE characterization obtained fo…
We study the quantitative stability of the solutions to Markovian quadratic reflected BSDEs with bounded terminal data. By virtue of BMO martingale and change of measure techniques, we obtain stability estimates for the variation of the solutions with different underlying forward processes. In addition, we propose a truncated discrete-time numerical scheme for quadratic reflected BSDEs, and obtain the explicit rate of convergence by applying the quantitative stability result.
Two discretizations of a class of locally Lipschitz Markovian backward stochastic differential equations (BSDEs) are studied. The first is the classical Euler scheme which approximates a projection of the processes Z, and the second a novel scheme based on Malliavin weights which approximates the mariginals of the process Z directly. Extending the representation theorem of Ma and Zhang leads to advanced a priori estimates and stability results for this class of BSDEs. These estimates are then used to obtain competitive convergence rates for both schemes with respect to the number of points in …
This paper shows that penalized backward stochastic differential equation (BSDE), which is often used to approximate and solve the corresponding reflected BSDE, admits both optimal stopping representation and optimal control representation. The new feature of the optimal stopping representation is that the player is allowed to stop at exogenous Poisson arrival times. The convergence rate of the penalized BSDE then follows from the optimal stopping representation. The paper then applies to two classes of equations, namely multidimensional reflected BSDE and reflected BSDE with a constraint on t…
The aim of this paper is to study an optimal stopping problem for dynamic risk measures induced by backward stochastic differential equations with jumps and delayed generator. Firstly, we connect the value function of this problem to reflected BSDEs with jump and delayed generator. Furthermore, after establishing existence and uniqueness result for this reflected BSDE, we use its to address through a mixed/optimal stopping game problem for the previous dynamic risk measure in ambiguity case.
In this paper, we provide conditions which ensure that stochastic Lipschitz BSDEs admit Malliavin differentiable solutions. We investigate the problem of existence of densities for the first components of solutions to general path-dependent stochastic Lipschitz BSDEs and obtain results for the second components in particular cases. We apply these results to both the study of a gene expression model in biology and to the classical pricing problems in mathematical finance.
We study (backward) stochastic differential equations with noise coming from a finite state Markov chain. We show that, for the solutions of these equations to be `Markovian', in the sense that they are deterministic functions of the state of the underlying chain, the integrand must be of a specific form. This allows us to connect these equations to coupled systems of ODEs, and hence to give fast numerical methods for the evaluation of Markov-Chain BSDEs.
We establish the existence (and in an appropriate sense uniqueness) of Markovian solutions for ergodic BSDEs under a novel monotonicity condition. Our monotonicity condition allows us to prove existence even when the driver f has arbitrary (in particular super-quadratic) growth in z, which reveals an interesting trade-off between monotonicity and growth for ergodic BSDEs. The technique of proof is to establish a probabilistic representation of the derivative of the Markovian solution, and then use this representation to obtain a-priori estimates. Our study is motivated by applications to ergod…
In this paper, we study the stability of the solutions of Backward Stochastic Differential Equations (BSDE for short) with an almost surely finite random terminal time. More precisely, we are going to show that if $(W^n)$ is a sequence of scaled random walks or a sequence of martingales that converges to a Brownian motion $W$ and if $(τ^n)$ is a sequence of stopping times that converges to a stopping time $τ$, then the solution of the BSDE driven by $W^n$ with random terminal time $τ^n$ converges to the solution of the BSDE driven by $W$ with random terminal time $τ$.
This paper considers the problem of uniqueness of the solutions to a class of Markovian backward stochastic differential equations (BSDEs) which are also connected to certain nonlinear partial differential equation (PDE) through a probabilistic representation. Assuming that there is a solution to the BSDE or to the corresponding PDE, we use the probabilistic interpretation to show the uniqueness of the solutions, and provide an example of a stochastic control application.
We first study an optimal stopping problem in which a player (an agent) uses a discrete stopping time in order to stop optimally a payoff process whose risk is evaluated by a (non-linear) $g$-expectation. We then consider a non-zero-sum game on discrete stopping times with two agents who aim at minimizing their respective risks. The payoffs of the agents are assessed by g-expectations (with possibly different drivers for the different players). By using the results of the first part, combined with some ideas of S. Hamad{è}ne and J. Zhang, we construct a Nash equilibrium point of this game by a…
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 introduce a new formulation of reflected BSDEs and doubly reflected BSDEs associated with irregular obstacles. In the first part of the paper, we consider an extension of the classical optimal stopping problem over a larger set of stopping systems than the set of stopping times (namely, the set of split stopping times), where the payoff process $ξ$ is irregular and in the case of a general filtration. Split stopping times are a powerful tool for modeling financial contracts and derivatives that depend on multiple conditions or triggers, and for incorporating stochastic processes with jumps …
We consider the optimal stopping problem with non-linear $f$-expectation (induced by a BSDE) without making any regularity assumptions on the reward process $ξ$. and with general filtration. We show that the value family can be aggregated by an optional process $Y$. We characterize the process $Y$ as the $\mathcal{E}^f$-Snell envelope of $ξ$. We also establish an infinitesimal characterization of the value process $Y$ in terms of a Reflected BSDE with $ξ$ as the obstacle. To do this, we first establish a comparison theorem for irregular RBSDEs. We give an application to the pricing of American…
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…
In this paper, we suggest a useful technique based on time change to be effective for dealing with the backward stochastic differential equations. We show the relation between the BSDEs with stochastic Lipschtz coeffecients and the ones with uniformly Lipschtz coefficients and stopping terminal time.
In this paper we are concerned with one-dimensional backward stochastic differential equations (BSDE in short) of the following type: \[Y_t=ξ-\int_{t\wedge τ}^τY_r|Y_r|^q dr-\int_{t\wedge τ}^τZ_r dB_r,\qquad t\geq 0,\] where $τ$ is a stopping time, $q$ is a positive constant and $ξ$ is a $\mathcal{F}_τ$-measurable random variable such that $\mathbf{P}(ξ=+\infty)>0$. We study the link between these BSDE and the Dirichlet problem on a domain $D\subset \mathbb{R}^d$ and with boundary condition $g$, with $g=+\infty$ on a set of positive Lebesgue measure. We also extend our results for more general…
Standard Markovian optimal stopping problems are consistent in the sense that the first entrance time into the stopping set is optimal for each initial state of the process. Clearly, the usual concept of optimality cannot in a straightforward way be applied to non-standard stopping problems without this time-consistent structure. This paper is devoted to the solution of time-inconsistent stopping problems with the reward depending on the initial state using an adaptation of Strotz's consistent planning. More precisely, we give a precise equilibrium definition --- of the type subgame perfect Na…
We introduce a generalized Dynkin game problem with non linear conditional expectation ${\cal E}$ induced by a Backward Stochastic Differential Equation (BSDE) with jumps. Let $ξ, ζ$ be two RCLL adapted processes with $ξ\leq ζ$. The criterium is given by \begin{equation*} {\cal J}_{τ, σ}= {\cal E}_{0, τ\wedge σ} \left(ξ_τ\textbf{1}_{\{ τ\leq σ\}}+ζ_σ\textbf{1}_{\{σ<τ\}}\right) \end{equation*} where $τ$ and $ σ$ are stopping times valued in $[0,T]$. Under Mokobodski's condition, we establish the existence of a value function for this game, i.e. $\inf_σ\sup_τ {\cal J}_{τ, σ} = \sup_τ \inf_σ {\ca…
We provide a probabilistic solution of a not necessarily Markovian control problem with a state constraint by means of a Backward Stochastic Differential Equation (BSDE). The novelty of our solution approach is that the BSDE possesses a singular terminal condition. We prove that a solution of the BSDE exists, thus partly generalizing existence results obtained by Popier in [7] and [8]. We perform a verification and discuss special cases for which the control problem has explicit solutions.
In this paper, we study in the Markovian case the rate of convergence in the Wasserstein distance of an approximation of the solution to a BSDE given by a BSDE which is driven by a scaled random walk as introduced in Briand, Delyon and M{é}min (Electron. Comm. Probab. 6(2001),1-14).
We consider reflected backward stochastic different equations with optional barrier and so-called regulated trajectories, i.e trajectories with left and right finite limits. We prove existence and uniqueness results. We also show that the solution may be approximated by a modified penalization method. Application to an optimal stopping problem is given.
We study a class of nonlinear BSDEs with a superlinear driver process f adapted to a filtration F and over a random time interval [[0, S]] where S is a stopping time of F. The terminal condition $ξ$ is allowed to take the value +$\infty$, i.e., singular. Our goal is to show existence of solutions to the BSDE in this setting. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., attains the terminal values with probability 1. We consider three types of terminal values: 1) Markovian: i.e., $ξ$ is of the form $ξ$ = g($Ξ$ S) where $Ξ$ is a continuous Markovian di…
We consider a class of stochastic impulse control problems of general stochastic processes i.e. not necessarily Markovian. Under fairly general conditions we establish existence of an optimal impulse control. We also prove existence of combined optimal stochastic and impulse control of a fairly general class of diffusions with random coefficients. Unlike, in the Markovian framework, we cannot apply quasi-variational inequalities techniques. We rather derive the main results using techniques involving reflected BSDEs and the Snell envelope.
In this paper, we propose an extension of the forward improvement iteration algorithm, originally introduced in Irle (2006) and recently reconsidered in Miclo and Villeneuve (2021). The main new ingredient is a flexible window parameter describing the look-ahead distance in the improvement step. We consider the framework of a Markovian optimal stopping problem in discrete time with random discounting and infinite time horizon. We prove convergence and show that the additional flexibility may significantly reduce the runtime.
In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and uniqueness of the solutions to such RBSDEs in appropriate Banach spaces. The result is established by using some tools from the general theory of processes such as Mertens decomposition of optional strong (but not necessarily right-continuous) supermartingales, some tools from optimal stopping theory, as well as an appropriate generalization of It{ô}'s formula due…
This article deals with the numerical resolution of Markovian backward stochastic differential equations (BSDEs) with drivers of quadratic growth with respect to $z$ and bounded terminal conditions. We first show some bound estimates on the process $Z$ and we specify the Zhang's path regularity theorem. Then we give a new time discretization scheme with a non uniform time net for such BSDEs and we obtain an explicit convergence rate for this scheme.
In this paper, we study existence and uniqueness to multidimensional Reflected Backward Stochastic Differential Equation in an open convex domain, allowing for oblique directions of reflection. In a Markovian framework, combining \emph{a priori} estimates for penalised equations and compactness arguments, we obtain existence results under quite weak assumptions on the driver of the BSDEs and the direction of reflection, which is allowed to depend on both $Y$ and $Z$. In a non Markovian framework, we obtain existence and uniqueness result for direction of reflection depending on time and $Y$. W…
In this paper, we consider a reflected backward stochastic differential equation driven by a $G$-Brownian motion ($G$-BSDE), with the generator growing quadratically in the second unknown. We obtain the existence by the penalty method, and a priori estimates which implies the uniqueness, for solutions of the $G$-BSDE. Moreover, focusing our discussion at the Markovian setting, we give a nonlinear Feynman-Kac formula for solutions of a fully nonlinear partial differential equation.
In this paper we look at ergodic BSDEs in the case where the forward dynamics are given by the solution to a non-autonomous (time-periodic coefficients) Ornstein-Uhlenbeck SDE with Lévy noise, taking values in a separable Hilbert space. We establish the existence of a unique bounded solution to an infinite horizon discounted BSDE. We then use the vanishing discount approach, together with coupling techniques, to obtain a Markovian solution to the EBSDE. We also prove uniqueness under certain growth conditions. Applications are then given, in particular to risk-averse ergodic optimal control an…
In this note we prove existence of a solution to a system of Markovian BSDEs with interconnected obstacles. A key feature of our system, and the main novelty of this paper, is that we allow for the driver $f_i$ of the $i$-th component of the $Y$-process to depend on all components of the $Z$-process. This extends the existing theory on reflected BSDEs, which only addresses problems where $f_i$ depends on $Z^i$.
This article focuses on the mathematical problem of existence and uniqueness of BSDE with a random terminal time which is a general random variable but not a stopping time, as it has been usually the case in the previous literature of BSDE with random terminal time. The main motivation of this work is a financial or actuarial problem of hedging of defaultable contingent claims or life insurance contracts, for which the terminal time is a default time or a death time, which are not stopping times. We have to use progressive enlargement of the Brownian filtration, and to solve the obtained BSDE …
We consider reflected backward stochastic differential equations with two optional barriers of class (D) satisfying Mokobodzki's separation condition and coefficient which is only continuous and non-increasing. We assume that data are merely integrable and the terminal time is an arbitrary (possibly infinite) stopping time. We study the problem of existence and uniqueness of solutions, and their connections with the value process in nonlinear Dynkin games.
In this paper, we study the connections between three concepts - the reverse Hölder inequality for matrix-valued martingales, the well-posedness of linear BSDEs with unbounded coefficients, and the well-posedness of quadratic BSDE systems. In particular, we show that a linear BSDE with bmo (bounded mean oscillation) coefficients is well-posed if and only if the stochastic exponential of a related matrix-valued martingale satisfies a reverse Hölder inequality. Furthermore, we give structural conditions under which these two equivalent conditions are satisfied. Finally, we apply our results on l…
In this paper, we study the well-posedness of multi-dimensional backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs) with diagonal generators, the $z$ parts of whose $l$-th components only depend on the $l$-th columns. The existence and uniqueness of solutions are obtained via a contraction argument for $Y$ component and a backward iteration of local solutions. Furthermore, we show that, the solution of multi-dimensional $G$-BSDE in a Markovian framework provides a probabilistic formula for the viscosity solution of a system of nonlinear parabolic partial differ…
