Optimal stopping time problem in a general framework

We study the optimal stopping time problem v ( S ) = ess sup θ ≥ S E [ φ ( θ ) |F S ] , for any stopping time S , where the reward is given by a family ( φ ( θ ) , θ ∈ T 0 ) of non negative random variables indexed by stopping times. We solve the problem under weak assumptions in terms of integrability and regularity of the reward family. More precisely, we only suppose v (0) < + ∞ and ( φ ( θ ) , θ ∈ T 0 ) upper semicontinuous along stopping times in expectation. We show the existence of an optimal stopping time and obtain a characterization of the minimal and the maximal optimal stopping times. We also provide some local properties of the value function family. All the results are written in terms of families of random variables and are proven by only using classical results of the Probability Theory.


Introduction
In the present work we study the optimal stopping problem in the setup of families of random variables indexed by stopping times, which is more general than the classical setup of processes.This allows technically simpler and clearer proofs, and also to solve the problem under weaker assumptions.
To the best of our knowledge, the most general result given in the literature is that of El Karoui (1981): existence of an optimal stopping time is proven when the reward is given by an upper semicontinuous non negative process of class D. For a classical exposition of the Optimal Stopping Theory, we also refer to Karatzas Shreve (1998) and Peskir Shiryaev (2005), among others.
Let T ∈ R * + be the terminal time and let (Ω, F, (F t ) 0≤t≤T , P ) be a filtered probability set which satisfies the usual conditions.
An optimal stopping problem can be naturally expressed in terms of families of random variables indexed by stopping times.Indeed, consider an agent who can choose a stopping time in T 0 .When she decides to stop at θ ∈ T 0 , she receives the amount φ(θ), where φ(θ) is a non negative F θ -measurable random variable.The family (φ(θ), θ ∈ T 0 ) of random variables indexed by stopping times is called the reward (or payoff) family.
It is identified with the map φ : θ → φ(θ) from T 0 into the set of random variables.
In the sequel, the reward family (φ(θ), θ ∈ T 0 ) is supposed to be an admissible family of non negative random variables.
At time 0, the agent wants to choose a stopping time θ * so that it maximizes E[φ(θ)] over the set of stopping times T 0 .The best expected reward at time 0 is thus given by v(0) := sup θ∈T0 E[φ(θ)], and is also called the value function at time 0. Similarly, for a stopping time S ∈ T 0 , the value function at time S is defined by v(S) := ess sup{ E[φ(θ) | F S ], θ ∈ T 0 and θ ≥ S a.s.}.
The family of random variables v = (v(S), S ∈ T 0 ) can be shown to be admissible, and characterized as the Snell envelope family of φ, also denoted by R(φ), defined here as the smallest supermartingale family greater than the reward family φ.The Snell envelope operator R : φ → R(φ) = v, thus acts on the set of admissible families of r.v.indexed by stopping times.
Solving the optimal stopping time problem at time S mainly consists to prove the existence of an optimal stopping times θ * (S), that is, such that v(S) = E[φ(θ * (S))|F S ] a.s.
Note that this setup of families of random variables indexed by stopping times is clearly more general than the setup of processes.Indeed, if (φ t ) 0≤t≤T is a progressive process, we set φ(θ) := φ θ , for each stopping time θ.Then, the family φ = (φ θ , θ ∈ T 0 ) is admissible.
The interest of such families has already been stressed, for instance, in the first chapter of El Karoui (1981).However, in that work as well as in the classical litterature, the optimal stopping time problem is set and solved in the setup of processes.In this case, the reward is given by a progressive process (φ t ) and the associated value function family (v(S), S ∈ T 0 ) is defined as above but does not a priori correspond to a progressive process.An important step of the classical approach consists in aggregating this familly, that is in finding a process (v t ) 0≤t≤T such that, for each stopping time S, v(S) = v S a.s.This aggregation problem is solved by using some fine results of the General Theory of Processes.Now, it is well known that this process (v t ) is also characterized as the Snell envelope process of the reward process (φ t ).Consequently, the previous aggregation result allows to define the Snell envelope process operator R : (φ t ) → R[(φ t )] := (v t ) which acts here on the set of progressive processes.The second step then consists, by a penalization method introduced by Maingueneau (1978), and under some right regularity conditions on the reward process, in showing the existence of ε-optimal stopping time.Next, under additional left regularity conditions on the reward process, the minimal optimal stopping time is characterized as a hitting time of processes, namely decomposition.This decomposition is then used to characterize the maximal optimal stopping time as well as to obtain some local properties of the value function (v t ).The proofs of these properties thus rely on strong and sophisticated results of the General Theory of Processes (see the second chapter of El Karoui (1981) for details).It is not the case in the framework of admissible families.
In the present work, which is self-contained, we study the general case of a reward given by an admissible family φ = (φ(θ), θ ∈ T 0 ) of non negative random variables, and we solve the associated optimal stopping time problem only in terms of admissible families.Using this approach, we avoid the aggregation step as well as the use of Mertens' decomposition.
Moreover, we only make the assumption v(0) = sup θ∈T0 E[φ(θ)] < +∞, which is, in the case of a reward process, weaker than the assumption (φ t ) of class D, required in the previous literature.
Furthermore, the existence ε-optimal stopping times is obtained when (φ(θ), θ ∈ T 0 ) is right upper semicontinuous along stopping times in expectation, that is, for each stopping time θ, and, for each non decreasing sequence (θ n ) of stopping time tending to θ, ].This condition is, in the case of a reward process, a bit wilder than the usual assumption "(φ t ) right upper semicontinuous and of class D".
Then, under the additional assumption that the reward family is left upper semicontinuous along stopping times in expectation, we show the existence of optimal stopping times and we characterize the minimal optimal stopping time θ * (S) for v(S) by θ * (S) = ess inf{ θ ∈ T 0 , θ ≥ S a.s. and u(θ) = φ(θ) a.s.}.
Let us emphasize that θ * (S) is no longer defined as a hitting time of processes but as an essential infimum of a set of stopping times.This formulation is a key tool to solve the optimal stopping time problem in the unified framework of admissible families.
Furthermore, we introduce the following random variable θ(S) := ess sup{ θ ∈ T 0 , θ ≥ S a.s. and and show that it is the maximal optimal stopping time for v(S).Some local properties of the value function family v are also investigated.To that purpose, some new local notions for families of random variables are introduced.We point out that these properties are proved using only classical probability results.In the case of processes, these properties correspond to some known results shown, using very sophisticated tools, by Dellacherie and Meyer (1980) and El Karoui (1981), among others.
At last, let us underline that the setup of families of random variables indexed by stopping time was used by Kobylanski et al. (2011), in order to study optimal multiple stopping.This setup is particularly relevant in that case.In particular, it avoids the aggregation problems, which, in the case of multiple stopping times, appear to be particularly knotty and difficult.The setup of families of random variables is also used in Kobylanski et al. (2012) to revisit the Dynkin game problem and provides a new insight on this well-known problem.
Let F = (Ω, F, (F t ) 0≤t≤T , P ) be a probability space which filtration (F t ) 0≤t≤T satisfies the usual conditions of right continuity and augmentation by the null sets of F = F T .We suppose that F 0 contains only sets of probability 0 or 1.The time horizon is a fixed constant T ∈]0, ∞[.We denote by T 0 the collection of stopping times of F with values in [0, T ].More generally, for any stopping times S, we denote by T S (resp.T S + ) the class of stopping times θ ∈ T 0 with θ ≥ S a.s.(resp.θ > S a.s. on {S < T } and θ = T a.s. on {S = T }).
For S, S ∈ T 0 , we also define T [S,S ] the set of θ ∈ T 0 with S ≤ θ ≤ S a.s. and T ]S,S ] the set of θ ∈ T 0 with S < θ ≤ S a.s.
Similarly, the set "T ]S,S ] on A" denotes the set of θ ∈ T 0 with S < θ ≤ S a.s. on A.
We use the following notation: for real valued random variables X and X n , n ∈ N, "X n ↑ X" stands for "the sequence (X n ) is nondecreasing and converges to X a.s.".

First properties
In this section we prove some results about the value function families v and v + when the reward is given by an admissible family of random variables indexed by stopping times.Most of these results are, of course, well-known in the case of processes.Definition 1.1.We say that a family φ = (φ(θ), θ ∈ T 0 ) is admissible if it satisfies the following conditions 1. for all θ ∈ T 0 φ(θ) is a F θ -measurable non negative random variable, 2. for all θ, θ ∈ T 0 , φ(θ) = φ(θ ) a.s. on {θ = θ }.
Remark 1.2.By convention, the non negativity property of a random variable means that it takes its values in R + .Also, it is always possible to define a admissible family associated with a given process.More precisely, let (φ t ) be a non negative progressive process.Set φ(θ) := φ θ , for each θ ∈ T 0 .Then, the family φ = (φ θ , θ ∈ T 0 ) is clearly admissible.
Let (φ(θ), θ ∈ T 0 ) be an admissible family called reward.For S ∈ T 0 , the value function at time S is defined by  where T S + is the class of stopping times θ ∈ T 0 with θ > S a.s. on {S < T } and θ = T a.s. on {S = T }.Note that v + (S) = φ(T ) a.s. on {S = T }.
Note that the essential supremum of a family X of non negative random variables, denoted "ess sup X ", is a well defined, almost surely unique random variable.Moreover, if X is stable by pairwise maximization (that is X ∨ X ∈ X for all X and X ∈ X ), then there exists a sequence (X n ) in X such that X n ↑ (ess sup X ).We refer to Neveu (1975) for a complete and simple proof (Proposition VI-1.1.p 121).Proof.The arguments are the same for (v(S), S ∈ T 0 ) and (v + (S), S ∈ T 0 ).We prove the property only for (v + (S), S ∈ T 0 ).Property 1 of admissibility for (v + (S), S ∈ T 0 ) follows from the existence of the essential supremum (see Neveu (1975)).Take S, S ∈ T 0 and let is increasing and such that v(S) Proof.Again, the arguments are the same for (v(S), S ∈ T 0 ) and (v + (S), S ∈ T 0 ).We prove the property only for (v + (S), S ∈ T 0 ).For each S ∈ T 0 , one can show that the set The result follows by a classical result on essential suprema (Neveu (1975)).
An admissible family (h(θ), θ ∈ T 0 ) is said to be a supermartingale family (resp.a martingale family) if for any θ, θ ∈ T 0 such that θ ≥ θ a.s., We now prove that both v and v + are supermartingale families and that the value function v is characterized as the Snell envelope family associated with the reward φ.

More precisely:
Proposition 1.5.The two following properties hold.
• The value function family (v(S), S ∈ T 0 ) is characterized as the Snell envelope family associated with (φ(S), S ∈ T 0 ), that is the smallest supermartingale family which is greater (a.s.) than (φ(S), S ∈ T 0 ) Proof.Let us prove the first point for v + .Fix S ≥ S a.s..By Proposition 1.4, there exists an optimizing sequence (θ n ) for v + (S).
Taking the supremum over θ ∈ T S , we have v (S) ≥ v(S) a.s.
The following proposition, known as the optimality criterion, gives a characterization of optimal stopping times for the v(S).
Proof.Let us show that 1) implies 2).Suppose 1) is satisfied.Since the value function v is a supermartingale family greater that φ, we have clearly Since equality (1.3) holds, this implies that the previous inequalities are actually equalities.
Proof.Note first that v(S) ≥ v + (S) a.s. and that v(S) ≥ φ(S) a.s., which yields the inequality v(S) ≥ φ(S) ∨ v + (S) a.s.It remains to show the other inequality.Fix θ ∈ T S .
First, the following inequality holds: (1.4) Indeed, since the random variable θ defined by θ := θ By taking the essential supremum over θ ∈ T S , we derive that v(S) ≤ φ(S) ∨ v + (S) a.s. and the proof is ended.

Right continuity property of the strict value function
Definition 1.10.An admissible family (φ(θ), θ ∈ T 0 ) is said to be right continuous along stopping times in expectation (RCE) if for any θ ∈ T 0 and for any sequence of stopping times The following localization property holds.
Proof.Note that if (φ(θ), θ ∈ T 0 ) is an admissible family, then for each S ∈ T 0 and A ∈ F S , the family (φ(θ)1 A , θ ∈ T S ) can easily be shown to be S-admissible, that is, to satisfy properties 1) and 2) of Definition 1.1 with T 0 replaced by T S .Fix θ ∈ T S .Let (θ n ) n∈N be a nonincreasing sequence of stopping times such that θ n ↓ θ.
We now show that the strict value function (v + (S), S ∈ T 0 ) is RCE (without any regularity assumption on the reward φ).This result is close to Proposition D.3 in Karatzas and Shreve (1998).
Proposition 1.12.(RCE property for v + ) Let (φ(θ), θ ∈ T 0 ) be an admissible family.The associated strict value function family In particular, the RCE property of (v + (θ)1 A , θ ∈ T S ) at S gives that for each non increasing sequence of stopping times (S n ) n∈N such that S n ↓ S, we have ] is a non increasing function of stopping times.Suppose it is not RCE at θ ∈ T 0 .We first consider the case when E[v + (θ)] < ∞.Then there exists a constant α > 0 and a sequence of stopping times (θ n ) n∈N such that θ n ↓ θ and One can easily show, by using an optimizing sequence of stopping time for v + (θ Let us first consider the simpler case where θ < T a.s.In this case, θ ∈ T θ + implies that θ > θ a.s.; one has {θ > θ} Hence, there exists n 0 such that Define the stopping time θ := θ 1 {θ >θn 0 } + T 1 {θ ≤θn 0 } .One has θ > θ n0 a.s.which gives by the positivity of (1.6) Electron.J. Probab.17 (2012), no.72, 1-28.ejp.ejpecp.org Optimal stopping in a general framework which gives the expected contradiction.
Let us now consider a general θ ∈ T This with (1.5) implies that there exists n 0 such that . Finally, we derive again (1.6) which gives the expected contradiction.
In the case where E[v + (θ)] = ∞, by similar arguments, one can show that when θ n ↓ θ the limit We now state a useful lemma.Lemma 1.14.Let (φ(θ), θ ∈ T 0 ) be an admissible family.For each θ, S ∈ T 0 , we have Proof.Recall that there exists an optimizing sequence of stopping times By taking the conditional expectation, we derive that a.s. on {θ > S}, where the second equality follows from the monotone convergence theorem for conditional expectation.Now, on {θ > S}, since θ n ≥ θ > S a.s., by inequality (1.4), we have E[φ(θ n )|F S ] ≤ v + (S) a.s.Passing to the limit in n and using the previous equality gives that E[v(θ)|F S ] ≤ v + (S) a.s. on {θ > S}.

Optimal stopping times
The main aim of this section is to prove the existence of an optimal stopping time under some minimal assumptions.We stress on that the proof of this result is short and only based on the basic properties shown in the previous sections.
We use a penalization method as the one introduced by Maingueneau (1978) in the case of a reward process.
More precisely, suppose that v(0) < ∞ and fix S ∈ T 0 .In order to show the existence of an optimal stopping time for v(S), we first construct for each ε ∈]0, 1[, an ε-optimal stopping time θ ε (S) for v(S), that is such that The existence of an optimal stopping time is then obtained by letting ε tend to 0.

Existence of epsilon-optimal stopping times
In the following, in order to simplify notation, we make the change of variable λ := 1−ε.We now show that if the reward is right upper semicontinuous over stopping times in expectation, then, for each λ ∈]0, 1[, there exists an (1 − λ)-optimal stopping time for v(S).
Proof.The set T λ S is clearly stable by pairwise minimization.Therefore, there exists a minimizing sequence (θ n ) in T λ S such that θ n ↓ θ λ (S).In particular, θ λ (S) is a stopping time and θ λ (S) ≥ S a.s.
Remark 2.6.We stress on that the right upper semicontinuity along stopping times in expectation of the reward family φ is sufficient to ensure this key property.The proof relies on the definition of θ λ (S) as an essential infimum of a set of stopping times and on the RCE property of the strict value function family v + .
Recall that there exists a minimizing sequence (θ n ) in T λ S .Hence, θ λ = lim n→∞ ↓ θ n and, as v + ≤ v, we have that for each n, Let us consider the first term of the right member of this inequality and let us now use the RCE property of the strict value function family v + .More precisely, by applying Remark 1.13 to the stopping time θ λ and to the set {v(θ λ ) > φ(θ λ )} ∩ A, we obtain the following equality By inequality (2.3), it follows that Consequently, using equality (2.4), we derive that where for each n, θ Note that (θ n ) is a non increasing sequence of stopping times such that θ n ↓ θ λ .
We now state the second lemma: Lemma 2.7.Let (φ(θ), θ ∈ T 0 ) be an admissible family with v(0) < ∞.Proof.The proof consists to adapt the classical penalization method, introduced by Maingueneau (1978) in the case of a continuous process, to our more general framework.It appears that it is clearer and simpler in the setup of families of random variables than in the setup of processes.Let us define for each S ∈ T 0 , the random variable ) is a supermartingale family and since θ λ (S) ≥ S a.s., we have that It remains to show the reverse inequality.This will be done in two steps.
Hence, E[J λ (S ) Consequently, which ends the proof of step 1.
In the next subsection, under the additional assumption of left USCE property of the reward, we derive from this theorem that the (1 − λ)-optimal stopping times θ λ (S) tend to an optimal stopping time for v(S) as λ ↑ 1.

Left continuity property of the value function family
Note first that, without any assumption on the reward family, the value function is right USCE.Indeed, from the supermartingale property of (v(θ), θ ∈ T 0 ), we clearly have the following property: for each S ∈ T 0 and each sequence of stopping times (S n ) such that S n ↓ S, Proof.Let S ∈ T 0 and let (S n ) be a sequence of stopping times such that S n ↑ S. Let us show that lim n→∞ Suppose now by contradiction that lim Now, for each n, θ * (S n ) ≥ S n a.s.By letting n tend to ∞, it clearly follows that θ ≥ S a.s., which provides the expected contradiction.
Consequently, the following corollary holds.
Fix S ∈ T 0 , and suppose that θ is an optimal stopping time for v(S), then, as a consequence of the optimality criterion (Remark 1.7), the family v(τ ), τ ∈ T [S,θ] is a martingale family.Consider the set A natural candidate for the maximal optimal stopping time for v(S) is thus the random variable θ(S) defined by θ(S) := ess sup A S . (2.7) Note that if v(0) < ∞, we clearly have: θ(S) = ess sup{θ ∈ T S , E[v(θ)] = E[v(S)] }.
Proposition 2.13.For each S ∈ T 0 , the random variable θ(S) is a stopping time.
This proposition is a clear consequence of the following lemma Lemma 2.14.For each S ∈ T 0 , the set A S is stable by pairwise maximization.
In particular there exists a nondecreasing sequence (θ n ) in A S such that θ n ↑ θ(S).
Proof.Let S ∈ T 0 and θ 1 , θ 2 ∈ A S .Let us show that θ 1 ∨ θ 2 belongs to A S .Note that this property is intuitive since if v(τ ), τ ∈ T [S,θ1] and v(τ ), τ ∈ T [S,θ2] are martingale families, then it is quite clear that v(τ ), τ ∈ T [S,θ1∨θ2] is a martingale family.For the sake of completeness, let us show this property.We have clearly that a.s.
The second point of the lemma fellows.In particular, θ(S) is a stopping time.
For each S ∈ T 0 , θ(S) is the maximal optimal stopping time for v(S).
Remark 2.17.In the previous works, in the setup of processes, the maximal optimal stopping time is given, when the Snell envelope process (v t ) is a right continuous supermartingale process of class D, by using the Doob Meyer decomposition of (v t ) and, in the general case, by using the Mertens decomposition of (v t ) (see El Karoui (1981)).
Thus fine results of the General Theory of Processes are needed.
In comparison, our definition of θ(S) as an essential supremum of a set of stopping times relies on simpler tools of Probability Theory.
Proof of Theorem 2.15.Fix S ∈ T 0 .To simplify the notation, in the following, the stopping time θ(S) will be denoted by θ.
Step 1 : Let us show that θ ∈ A S .By Lemma 2.14, there exists a nondecreasing sequence (θ n ) in A S such that θ n ↑ θ.
, and therefore θ ∈ A S .
Step 3 : Let us show that θ is the maximal optimal stopping time for v(S).By Proposition 1.6, we have that each θ which is optimal for v(S) belongs to A S and hence is smaller than θ (since θ = ess sup A S ).This gives step 3.
By using localization techniques (see below), one can prove more generally that, for

Localization and case of equality between the reward and the value function family
Recall that we have shown that for all S ∈ T 0 , v(S) = φ(S) ∨ v + (S) a.s.(see Proposition 1.9).Thus, one can wonder if it possible to have some conditions which ensure that v(S) = φ(S) almost surely on Ω (or even locally, that is on a given subset A ∈ F S ).
Thisi s be the object of this section.
We first provide some useful localization properties.

Localization properties
Let (φ(θ), θ ∈ T 0 ) be an admissible family.Let S ∈ T 0 and A ∈ F S .Let (v A (θ), θ ∈ T S ) be the value function associated with the admissible reward (φ(θ)1 A , θ ∈ T S ), defined for each θ ∈ T S by v A (θ) = ess sup and let (v + A (θ), θ ∈ T S ) be the strict value function associated with the same reward, defined for each θ ∈ T S by Note first that the families (v A (θ), θ ∈ T S ) and (v + A (θ), θ ∈ T S ) can easily be shown to be S-admissible.
We now state the following localization property: Proposition 3.1.Let {φ(θ), θ ∈ T 0 } be an admissible family.Let θ ∈ T S and let A ∈ F S .The value functions v A and v A + defined by (3.1) and (3.2) satisfy the following equalities Proof.Thanks to the characterization of the essential supremum (see Neveu (1975)), one can easily show that v(θ)1 A coincides a.s. with ess sup The proof is the same for the strict value function v + .
Remark 3.2.Let θ * ,A (S) and θA (S) be respectively the minimal and the maximal optimal stopping times for v A .One can easily show that θ * ,A (S) = θ * (S) a.s. on A and θA (S) = θ(S) a.s. on A.

When does the value function coincide with the reward?
We will now give some local strict martingale conditions on v which ensure the a.s.equality between v(S) and φ(S) for a given stopping time S.
We introduce the following notation: let X, X be real random variables and let A ∈ F. We say that X ≡ X a.s. on The family u is said to be a martingale family on the right at S on A if there exists S ∈ T 0 with (S ≤ S and S ≡ S ) a.s. on A such that u(τ ), τ ∈ T [S,S ] is a martingale family on A.
The family u is said to be a strict supermartingale family on the right at S on A if it is not a martingale family on the right at S on A.
We now provide a sufficient condition to locally ensure the equality between v(S) and φ(S) for a given stopping time S. Theorem 3.4.Suppose (φ(θ), θ ∈ T 0 ) is right USCE and such that v(0) < ∞.Let S ∈ T 0 and A ∈ F S be such that (S ≤ T and S ≡ T ) a.s. on A.
If the value function (v(θ), θ ∈ T 0 ) is a strict supermartingale on the right at S on A, then v(S) = φ(S) a.s. on A.
Proof.Note that, in the case where there exists an optimal stopping time for v(S) and where A = Ω, the above property is clear.Indeed, by assumption, the value function is a strict supermartingale on the right at S on Ω.Also, thanks to the optimality criterion, we derive that S is the only one optimal stopping time for v(S) and hence v(S) = φ(S)  By letting λ tend to 1, we derive that v(S) ≤ φ(S) a.s. on A. Since v(S) ≥ φ(S) a.s., it follows that v(S) = φ(S) a.s. on A, which completes the proof.

Additional regularity properties of the value function
We first provide some regularity properties which hold for any supermartingale family.where A[(S n )] = {S n ↑ S and S n < S for all n }.

Regularity properties of supermartingale families
Recall some definitions and notation.Suppose that S ∈ T 0 + .A non decreasing sequence of stopping times (S n ) n∈N is said to announce S on A ∈ F if S n ↑ S a.s. on A and S n < S a.s. on A.
The stopping time S is said to be accessible on A if there exists a non decreasing sequence of stopping times (S n ) n∈N which announces S on A.
The set of accessibility of S, denoted by A(S) is the union of the sets on which S is accessible.
Lemma 4.2.Let S ∈ T 0 + .There exists a sequence of sets (A k ) k∈N in F S − such that for each k, S is accessible on A k , and A(S) = ∪ k A k a.s.
It follows that, in Definition 4.1, the left limit φ(S − ) is unique on A(S) and the family (φ(S − )1 A(S) , S ∈ T 0 ) is admissible.
This result clearly follows from the result of Dellacherie and Meyer (1977) quoted above together with the following lemma.Lemma 4.4.Let (u(θ), θ ∈ T 0 ) be a supermartingale family.Let S be a stopping time in T 0 + .Suppose that S is accessible on a measurable subset A of Ω.
There exists an F S − -measurable random variable u(S − ), unique on A (up to the equality a.s.), such that, for any non decreasing sequence (S n ) n∈N announcing S on A, one has Proof.Let S be stopping time accessible on a set A ∈ F and let (S n ) be a sequence announcing S on A. It is clear that (u(S n )) n∈N is a discrete non negative supermartingale relatively to the filtration (F Sn ) n∈N .By the well-known convergence theorem for discrete supermartingales, there exists a random variable Z such that (u(S n )) n∈N converges a.s. to Z.If u(0) < +∞, then Z is integrable.Set u(S − ) := Z.
It remains to show that this limit, on A, does not depend on the sequence (S n ).Let (S n ) be a sequence announcing S on A. Again, by the supermartingales convergence theorem, there exists a random variable Z such that (u(S n )) n∈N converges a.s. to Z .We will now prove that Z = Z a.s. on A.
For each n and each ω, consider the reordered terms S (0 It is easy to see that for each n, Sn is a stopping time and that the sequence ( Sn ) announces S on A. Again, by the supermartingales convergence theorem, there exists a random variable Z such that (u( Sn )) n∈N converges a.s. to Z. Let us show that Z = Z a.s. on A.
For almost every ω ∈ A, as S n (ω) < S(ω) and S n (ω) < S(ω) for all n, the sequence ( Sn (ω)) describes all the values taken by both the sequences (S n (ω)) and (S n (ω)) on A. Hence, by construction, for each k, almost surely.Without loss of generality, we can suppose that this equality is satisfied everywhere.Also, by the admissible property of the value function, for each k, n ∈ N, the following equality holds almost surely.Again, without loss of generality, we can suppose that for each k, n ∈ N, this equality is satisfied everywhere on the set { Sn = S k }.Also, we can suppose that the sequences (u(S n )) and (u( Sn )) converge to Z and Z everywhere on Ω.
We have thus shown that u(S − )(= Z), on A, does not depend on the sequence (S n ).
Electron.J. Probab.17 (2012), no.72, 1-28.ejp.ejpecp.orgIt remains to show that u(S − ) can be chosen F S − -measurable.Indeed, the above part of the proof still holds with A replaced by A[(S n )] = {S n ↑ S and S n < S for all n }, which contains A, and with u(S The proof of the lemma is thus complete.Definition 4.5.Let S ∈ T 0 .An admissible family (φ(θ), θ ∈ T 0 ) is said to be right limited along stopping times (RL) at S if there exists an F S -measurable random variable φ(S + ) such that, for any non increasing sequence of stopping times (S n ) n∈N , such that S n ↓ S and S n > S for each n, one has φ(S + ) = lim n→∞ φ(S n ).
By a convergence theorem for discrete supermartingales indexed by non positive integers, and uniformly bounded in L 1 (see chap.V, Thm.30 in Dellacherie and Meyer (1980)), there exists an integrable random variable Z such that the sequence (Z n ) n≤0 converges a.s. and in L 1 to Z.We then define u(S + ) by u(S + ) := Z.It remains to show that this limit does not depend on the sequence (S n ).The proof is not detailed since it is similar to that of the previous theorem.
The left continuity along stopping times (LC) property is defined in a similar way.Proposition 4.8.Let (u(θ), θ ∈ T 0 ) be a uniformly integrable supermartingale family .
Let us prove the first point.Thanks to the RL property of u, we have u(S + ) = lim n→∞ u(S n ) a.s.The supermartingale property of u yields that E[u(S n ) | F S ] ≤ u(S) a.s.for each n.By letting n tend to ∞ and using the uniform integrability property of (u(S n )), we get E[u(S + ) | F S ] ≤ u(S) a.s.Since u(S + ) is F S -measurable, we get u(S + ) ≤ u(S) a.s.
Let us now prove the second point.Thanks to the RL property of u and the uniform integrability property of (u(S n )), we have E[u(S + )] = lim n→∞ E[u(S n )] = E[u(S)], where the last equality follows from the RCE property of u.Now, by the first point, we have u(S + ) ≤ u(S) a.s.This with the previous equality leads to the desired result.
The last point is clear.Proposition 4.9.Let (u(θ), θ ∈ T 0 ) be a supermartingale family with u(0) < +∞.Let S ∈ T 0 + and (S n ) be a non decreasing sequence in T 0 such that S n ↑ S a.s.which corresponds to Th. 14 Chap VI in Dellacherie and Meyer (1980).This inequality is linked to the jumps of the predictable non decreasing process (A t ) associated to the decomposition of (u t ) (see equality B.4).Consider now a general stopping time S ∈ T 0 + .Recall that there exists a sequence of sets (A k ) k∈N in F S − such that for each k, S is accessible on A k , and A(S) = ∪ k A k a.s.One can easily show that for each k, there exists a predictable stopping time τ k such that S = τ k on A k a.s.(see for example Lemma 4.7 in [6]).It follows that for each S ∈ T 0 + , ∆u(S) = ∆u(τ k ) on A k a.s.
Remark 4.13.When u is defined via a martingale process (u t ), the last assertion implies that, if the filtration is left quasicontinuous, the martingale (u t ) has only totally inaccessible jumps.
All the above properties hold for the value functions families v and v + since they are supermartingale families.

Complementary properties of the value function
First, the value functions families v and v + satisfy the following property.
We now provide some local properties of the value function at a stopping time on the left.
First, in the case where the reward is supposed to be USCE, we have the following property.
Suppose that the event A = A[(θ n )] := {θ n < θ, for all n } is non empty.Then, we have where φ C (θ) := ess sup and A(θ, C) is the set of non decreasing sequences in T 0 which announce θ on C.
We stress the importance of this corollary, which allows us to compute the jumps ∆A d t of the predictable non decreasing process (A t ) associated to the decomposition of (u t ) (see Proposition B.11).
Proof of Theorem 4.16.
Also, A and B belong to F θ − .Hence, φ C (θ) is clearly well defined and F θ − -measurabme.
Let us show the second one.For this, it is sufficient to prove that this property holds for f := 1 B , where B ∈ F S − .Let G := {B ∈ F S − , B ∩ A ∈ ∨ n F Sn }.First, G is a σ-algebra.Recall that F S − is the σ-algebra generated by the set C := {C ∩ {t < S}, C ∈ F t and t ∈ R + }.Now, by using the assumption lim n→∞ S n = S a.s., one can show that if B ∈ C, then B ∩ A ∈ ∨ n F Sn .It follows that G is a σ-algebra which contains C, which yields that G = F S − .Hence, the second assertion holds.
It remains to show the third one.By the first assertion, E Hence, equality (A.1) follows.

B Case of a reward process
In this section, we consider the particular case where the reward is given by a progressive process (φ t ) 0≤t≤T .By using the results provided in this paper and naturally some fine results of the General Theory of processes, we derive the corresponding results in the case of processes.
The proof of the third point is thus complete.
It remains to show the last point.Now, when the filtration is left quasicontinuous, the martingale M only admits inaccessible jumps (as seen in Proposition 4.12).Consequently, if θ is a jump time for (v t ) which is a predictable stopping time, then it corresponds to a jump of the nondecreasing predictable process (A d t ).We thus have which makes the proof ended.
Remark B.13. Suppose now that the reward φ is given by a right upper semicontinuous and left USCE progressive process (φ t ) of class D. One can prove that for each S ∈ T 0 , the maximal optimal stopping time θ(S) satisfies that for almost every ω, θ(S)(ω) = inf{t ≥ S(ω) , v t (ω) = M t (ω)} ∧ T.

Remark 2 . 3 .
Note that it is clear that if an admissible family (φ(θ), θ ∈ T 0 ) is right (resp.left) USCE, then, for each S ∈ T 0 and each A ∈ F S , (φ(θ)1 A , θ ∈ T S ) is right (resp.left) USCE.The arguments to show this property are the same as those used in Lemma 1.11.

Proposition 2 . 11 .
Define now the property of left continuity in expectation along stopping times (LCE property) similarly to the RCE property (see Definition 1.10) with θ n ↑ θ instead of θ n ↓ θ .Using the monotonicity property of θ * with respect to stopping times (see Remark 2.10), we derive the following regularity property of the value function: If (φ(θ), θ ∈ T 0 ) is USCE and v(0) < ∞, then (v(S), S ∈ T 0 ) is left continuous in expectation along stopping times (LCE).

4. 1 . 1 Definition 4 . 1 .
Left and right limits of supermartingale families along stopping times Let S ∈ T 0 .An admissible family (φ(θ), θ ∈ T 0 ) is said to be left limited along stopping times (LL) at S if there exists an F S − -measurable random variable φ(S − ) such that, for any non decreasing sequence of stopping times (S n ) n∈N , φ(S − ) = lim n→∞ φ(S n ) a.s. on A[(S n )],
)] is equivalent to the fact that the family (v(θ), θ ∈ T [S,θ * ] ) is a martingale family, that is for all * ) = φ(θ * ) a.s., and E[v(S)] = E[v(θ * )]. 3. The following equality holds: E[v(S)] = E[φ(θ * )].* Sn (defined by(2.6)) is optimal for v(S n ).It follows that for each n, E[φ(θ * (S n ))] ≥ E[v(S)] + α.Now, the sequence of stopping times (θ * (S n )) is clearly non decreasing.Let θ := lim n→∞ ↑ θ * (S n ).The random variable θ is clearly a stopping time.Using the USCE property of φ, we obtain Then, there exists α > 0 such that for all n, one has E[v(S n )] ≥ E[v(S)] + α.By Theorem 2.9, for each n, the stopping time θ * (S n ) ∈ T Proof.First, by the second point of Proposition 4.8, and since v + is RCE, v + is RC.Let (S n ) n∈N in T S + such that S n ↓ S. One has v(S n ) ≥ v + (S n ) a.s.for each n.Passing to the limit, we have v(S + ) ≥ v + (S) a.s.Also, since S n > S a.s., by Lemma 1.14, we have v + (S) ≥ E[v(S n ) | F S ] a.s.for each n.
which, with equality (4.8), yields that B ∩ A ∩ {θ λ (θ p ) ≥ θ} = ∅} a.s.Electron.J. Probab.17 (2012), no.72, 1-28.ejp.ejpecp.orgIt follows that for each p, θ p ≤ θ λ (θ p ) < θ a.s. on B ∩ A and θ λ (θ p ) ↑ θ a.s. on B ∩ A. In other words, the sequence (θ λ (θ p )) announces θ on B ∩ A. By a property of the (1 − λ)-optimal stopping times (see Lemma 2.5), for each λ ∈ [0, 1[ and for each p ∈ N, Dellacherie and Meyer (1980)ork, we have only made the assumption v(0) < +∞, which is, in the case of a reward process, weaker than the assumption (φ t ) of class D, required in the previous literature.Note that according to the terminology ofDellacherie and Meyer (1980), the process (v t ) a strong supermartingale that is, a supermartingale such that the family (v(θ), θ ∈ T 0 ) is a supermartingale family.By a fine result of Dellacherie and Meyer (1980) (see Theorem 4 p408), it follows that there exists a right limited and left limited version of (v t ), which we still denote by (v t ).Note that s. for all θ ∈ T 0 .(B.1)This follows from a classical result (see for instance Therem 3.13 in Karatzas and Shreeve (1994) and Proposition 4.1 in Kobylanski et al. (2011)).Define the process (v t ) by v t := φ t ∨ v + t .(B.2)By Proposition 1.9, we clearly have Proposition B.1.Suppose that the reward is given by a progressive process (φ t ) such that the associated value function satisfies v(0) < ∞.Then, the adapted process (v t ) defined by (B.2) aggregates the value function family (v(S), S ∈ T 0 ), that is for all S ∈ T 0 , v(S) = v S a.s.Remark B.2.