Representation of non-Markovian optimal stopping problems by constrained BSDEs with a single jump *

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.


Introduction
Let (Ω, F, P) be a complete probability space and let F = (F t ) t≥0 be the natural augmented filtration generated by an m-dimensional standard Brownian motion W . For given T > 0 we denote L 2 T = L 2 (Ω, F T , P) and introduce the following spaces of processes. We suppose we are given f ∈ H 2 , h ∈ S 2 c , ξ ∈ L 2 T , satisfying ξ ≥ h T .
where T t (F) denotes the set of F-stopping times τ ≥ t. Thus, I is the value process of a non-Markovian optimal stopping problem with cost functions f, h, ξ. In [7] the process I is described by means of an associated reflected backward stochastic differential equation (BSDE), namely it is proved that there exists a unique (Y, Z, K) ∈ S 2 c × H 2 × A 2 c such that, P-a.s.
and that, for every t ∈ [0, T ], we have I t = Y t P-a.s. It is our purpose to present another representation of the process I by means of a different BSDE, defined on an enlarged probability space, containing a jump part and involving sign constraints. Besides its intrinsic interest, this result may lead to new methods for the numerical approximation of the value process, based on numerical schemes designed to approximate the solution to the modified BSDE. Some numerical methods for this class of BSDEs have already been proposed and analyzed, see [11], [12].
In the context of a classical Markovian optimal stopping problem, this may give rise to new computational methods for the corresponding variational inequality as studied in [2].
We use a randomization method, which consists in replacing the stopping time τ by a random variable η independent of the Brownian motion and in formulating an auxiliary optimization problem where we can control the intensity of the (single jump) point process N t = 1 η≤t . The auxiliary randomized problem turns out to have the same value process as the original one.
This method seems to be essentially different from other randomization procedures already introduced in the literature, for instance the classical relaxed controls technique used to ensure existence of optimal controls in the deterministic framework and extended to the stochastic case in [6], the randomized stopping method used to reduce optimal stopping problems to continuous optimal control problems (originally introduced in [15] and further studied in [9]), or the method developed in [16] to give a control-theoretic interpretation of various penalization schemes in terms of value functions of auxiliary optimization problems where intervention occurs only at arrival times of an exogenous Poisson process.
On the other hand, our approach is more closely connected to another randomization procedure that has been recently applied to several stochastic optimization problems and which is directly connected to a class of BSDEs with a sign constraint on one of its components. In [13] it has been shown that the solution to a constrained BSDE provides a representation formula for the value function of an impulse control problem for a controlled diffusion, coinciding with the solution to a quasi-variational inequality in the viscosity sense. In [4] and [5] a similar program has been carried out in the context of optimal switching problems. In [14] a BSDE representation has been constructed for the solution to a large class of integro-differential equations of Hamilton-Jacobi-Bellman type, including dynamic programming equations for the optimal control of a controlled diffusion with jumps. This result has been extended in [8] to a case of non-Markovian controlled diffusions. In all these papers the control process is randomized by means of an auxiliary Poisson random measure µ (hence preserving the Markovian character of the driving noise (W, µ)), whereas for our application to optimal stopping the one-jump process N seems to be the appropriate technical tool.

Statement of the main results
We are given (Ω, F, P), F = (F t ) t≥0 , W , T as before, as well as f, h, ξ satisfying (1.1). Let η be an exponentially distributed random variable with unit mean, defined in another probability space (Ω , F , P ). DefineΩ = Ω × Ω and let (Ω,F,P) be the completion of (Ω, F ⊗ F , P ⊗ P ). All the random elements W, f, h, ξ, η have natural extensions toΩ, denoted by the same symbols. In particular, η is independent of W . Define and letF = (F t ) t≥0 be theP-augmented filtration generated by (W, N ). UnderP, A is theF-compensator (i.e., the dual predictable projection) of N , W is anF-Brownian motion independent of N and (1.1) still holds provided H 2 , S 2 c , L 2 T (as well as A 2 etc.) are understood with respect to (Ω,F,P) andF as we will do. We also define We will consider the BSDĒ dA t (ω)P(dω) − a.s.
Since ν is bounded, L ν is anF-martingale on [0, T ] underP and we can define an equivalent probabilityP ν on (Ω,F) settingP ν (dω) = L ν t (ω)P(dω). By a theorem of Girsanov type (Theorem 4.5 in [10]) on [0, T ] theF-compensator of N underP ν is ECP 21 (2016), paper 3. t 0 ν s dA s , t ∈ [0, T ], and W remains a Brownian motion underP ν . We wish to characterize the value process J defined, for every t ∈ [0, T ], by Our second result provides a dual representation in terms of control intensity of the minimal solution to the BSDE with sign constraint.
Remark 2.5. Theorem 2.3 does not directly provide an optimal stopping rule in terms of the minimal solution (Ȳ ,Z,Ū ,K). However, the optimal stopping time is described in [7] in terms of the processes (Y, Z, K) solution to (1.2)-(1.3).
We would also like to raise the issue of the appropriate formulation of the auxiliary (randomized) control problem (2.5). As it is stated, we are unable to prove that it admits an optimal solution i.e., that the essential supremum is achieved in (2.5), and in fact we suspect it does not. One could try to embed the randomized problem into a larger class in order to achieve existence of a maximum, for example by relaxed control techniques as in [6]. In this sense, Theorem 2.3 should be viewed as a representation result for the value process rather than the solution of an auxiliary equivalent problem.

Proofs
Proof of Theorem 2.1. Uniqueness of the minimal solution is not difficult and it is established as in [14], Remark 2. since T tZ s dW s = η t Z s dW s P-a.s., h η −Ū η = Y η and, on the set {0 ≤ t < η < T }, Y t = Y t andK T −K t = K η − K t , this reduces to Y t + η t Z s dW s = η t f s ds + Y η + K η − K t ,P − a.s. Finally, on {0 < η < T, η ≤ t ≤ T } the verification of (2.1) is trivial, so we have proved that (Ȳ ,Z,Ū ,K) is indeed a solution.
Its minimality property will be proved later.