Optimal stopping: Bermudan strategies meet non-linear evaluations

We address an optimal stopping problem over the set of Bermudan-type strategies $\Theta$ (which we understand in a more general sense than the stopping strategies for Bermudan options in finance) and with non-linear operators (non-linear evaluations) assessing the rewards, under general assumptions on the non-linear operators $\rho$. We provide a characterization of the value family V in terms of what we call the ($\rho$,$\Theta$) -Snell envelope of the pay-off family. We establish a Dynamic Programming Principle. We provide an optimality criterion in terms of a ($\rho$,$\Theta$) -martingale property of V on a stochastic interval. We investigate the ($\rho$,$\Theta$)-martingale structure and we show that the ''first time'' when the value family coincides with the pay-off family is optimal. The reasoning simplifies in the case where there is a finite number n of pre-described stopping times, where n does not depend on the scenario $\omega$. We provide examples of non-linear operators entering our framework.


Introduction
In the recent years, optimal stopping problems with non-linear evaluations have gained an increasing interest in the financial mathematics literature and in the stochastic control literature.
In the linear case, a classical reference are the notes by El Karoui (1981).A presentation based on families of random variables indexed by stopping times can be found in Quenez and Kobylanski (2012).In discrete time, a non-linear optimal stopping with dynamic monetary utilities was studied in Krätschmer and Schoenmakers (2010), and with g-evaluations (induced by Backward SDEs with Lipschitz driver g) -in Grigorova and Quenez (2016).In continuous time, a non-linear optimal stopping with dynamic convex risk measures was studied in Bayraktar et al. (2010); with the so-called F -expectations -in Bayraktar and Yao-Part I (2011) and Bayraktar and Yao-Part II (2011); with g-evaluations-in e.g., Quenez and Sulem (2014), Grigorova et al. (2015), Grigorova et al. (2020), Klimsiak and Rzymowski (2021); with a focus on applications to American options in complete or incomplete non-linear financial markets -in Kim et al. (2021), Grigorova et al. (2021); with suprema of linear/affine operators over sets of measures-in, e.g.Ekren et al. (2014), Nutz and Zhang (2015).
In the present paper, we address an optimal stopping problem with Bermudantype strategies and with general non-linear operators (non-linear evaluations) assessing the rewards.
Our purpose is two-fold: 1. We consider a modelling framework which is in-between the discrete-time and the continuous-time framework, by focusing on what we call in this paper the Bermudan-type stopping strategies1 .
-In the discrete-time framework with finite terminal horizon T ą 0, the agent is allowed to stop at a finite number only of pre-described determinstic times, and gain/loss processes are indexed by these pre-described determinsitic times.If we denote by t0 " t 0 ď t 1 ď ... ď t n " T u the predefined finite deterministic grid of n `1 time points, the stopping strategies of the agent are of the form τ " ř n k"0 t k 1 A k , where pA k q kPt0,1,...nu is a partition, such that A k is F t k -measurable, for each k P t0, 1, . . ., nu.Thus, for almost each scenario ω, the agent is allowed to stop only at a finite number of times (provided they do that in a non-anticipative way), where both the number of time instants (here, n `1) and the time instants themselves (here, the t k 's), are the same, whatever the scenario ω.
-In the continuous-time framework (with finite horizon T ą 0), the agent is allowed to stop continuously at any time instant t P r0, T s, and the gain/loss processes are indexed by t P r0, T s.The set of the agent's stopping strategies is the set of all stopping times valued in r0, T s.Thus, for almost each scenario ω, the agent can stop at any time instant (provided they do that in a nonanticipative way).
-In the intermediate modelling framework of the current paper (with finite terminal horizon T ą 0), in (almost) every scenario ω, the agent is allowed to stop at a finite number of times or infinite countable number of times (provided they do that in a non-anticipative way), where both the number of time instants and the time instants themselves are allowed to depend on the scenario ω.More specifically, we are given a non-decreasing sequence of stopping times pθ k q kPN such that lim kÑ8 θ k " T. This countable stopping grid being given, the agent's stopping strategies τ are thus of the form τ " ř `8 k"0 θ k 1 A k `T 1 Ā, where tpA k q kPN , Āu is a partition, such that A k is F θ k -measurable, for each k P N, and Ā is F T -measurable.We call τ of this form a Bermudan stopping strategy, and we denote by Θ the set of Bermudan stopping strategies.The gain/losses are then "naturally" defined via families of random variables indexed by the stopping times τ of this form.This modelling framework is thus closer to the real-life situations where the number of possible decision points depends on the scenario/state of nature, and so do the decision times themselves, but where the agents do not necessarily act continuously in time.
2. The second purpose of the paper is to allow for gains/losses being assessed by general non-linear evaluations ρ " pρ S,τ r¨sq, while imposing minimal assumptions on the non-linear operators, under which the results hold.
We note also that, in the above framework, working with families of random variables φ " pφpτ qq indexed by Bermudan stopping times τ , allows for an exposition in which it is not necessary to invoke any results from the theory of stochastic processes.
After formulating the non-linear optimal stopping with Bermudan-style strategies, we provide a characterization of the value family in terms of a suitably defined non-linear Snell envelope.A dynamic programming principle is established under suitable assumptions on the non-linear evaluations.An optimality criterion is proven and existence of optimal stopping times is investigated; it is shown in particular that the first "hitting time" is optimal.Examples of non-linear operators, well-known in financial mathematics and in stochastic control, entering our framework, are given, such as the non-linear evaluations induced by Backward SDEs, the non-linear F -expectations introduced by Bayraktar and Yao, as well as the dynamic concave utilities (from the risk measurement literature).In the appendix, we consider the particular case of finite number of pre-described stopping times 0 " θ 0 , θ 1 , ..., θ n " T , where n does not depend on the scenario ω, and provide an explicit construction of the non-linear Snell envelope by backward induction, as well as a simpler proof of the optimality of the first hitting time.
The remainder of the paper is organized as follows: In Subsection 2.1, we set the framework and the notation.In Subsection 2.2, we formulate the optimisation problem.In Subsection 2.3, we characterize the value family of the problem in terms of the pΘ, ρq-Snell envelope family of the pay-off family.In Subsection 2.4, we show a Dynamic Programming Principle.In Subsection 2.5, we investigate the question of optimal stopping times: we provide some technical lemmas regarding the pΘ, ρq-martingale property, we provide an optimality criterion, as well as some useful consequences of the DPP, and we show that, under suitable assumptions, "the first time" ν k when the value family hits the pay-off family is optimal.Section 3 is dedicated to three examples of non-linear operators from the literature, entering our framework.The Appendix is dedicated to the particular case where in almost each scenario ω P Ω, there are n `1 pre-described opportunities for stopping, where n does not depend on ω P Ω.In this case, we have an explicit construction of the pΘ, ρq-Snell envelope by backward induction and a simplified proof of the optimality of ν k (requiring "less continuity" on ρ).
2 Optimal stopping with non-linear evaluations and Bermudan strategies

The framework
Let T ą 0 be a fixed finite terminal horizon.Let pΩ, F , P q be a (complete) probability space equipped with a right-continuous complete filtration F " tF t : t P r0, T su.
In the sequel, equalities and inequalities between random variables are to be understood in the P -almost sure sense.Equalities between measurable sets are to be understood in the P -almost sure sense.
Let N be the set of natural numbers, including 0. Let N ˚be the set of natural numbers, excluding 0. Let (θ k q kPN be a sequence of stopping times satisfying the following properties: (a) The sequence pθ k q kPN is non-decreasing, i.e. for all k P N, θ k ď θ k`1 , a.s.
Moreover, we set θ 0 " 0. We note that the family of σ-algebras pF θ k q kPN is non-decreasing (as the sequence pθ k q is non-decreasing).We denote by Θ the set of stopping times τ of the form where tpA k q `8 k"0 , Āu form a partition of Ω such that, for each k P N, A k P F θ k , and Ā P F T .
The set Θ can also be described as the set of stopping times τ such that for almost all ω P Ω, either τ pωq " T or τ pωq " θ k pωq, for some k " kpωq P AE.
Note that the set Θ is closed under concatenation, that is, for each τ P Θ and each A P F τ , the stopping time τ 1 A `T 1 A c P Θ.More generally, for each τ P Θ, τ 1 P Θ and each A P F τ ^τ 1 , the stopping time τ 1 A `τ 1 1 A c is in Θ.The set Θ is also closed under pairwise minimization (that is, for each τ P Θ and τ 1 P Θ, we have τ ^τ 1 P Θ) and under pairwise maximization (that is, for each τ P Θ and τ 1 P Θ, we have τ _ τ 1 P Θ).Moreover, for each non-decreasing (resp.non-increasing) sequence of stopping times pτ n q nPN P Θ AE , we have lim nÑ`8 τ n P Θ.
We note also that all stopping times in Θ are bounded from above by T .
Remark 2.1.We have the following canonical writing of the sets in (1): From this writing, we have: For each τ P Θ, we denote by Θ τ the set of stopping times ν P Θ such that ν ě τ a.s.The set Θ τ satisfies the same properties as the set Θ. We will refer to the set Θ as the set of Bermudan stopping strategies, and to the set Θ τ as the set of Bermudan stopping strategies, greater than or equal to τ (or the set of Bermudan stopping strategies from time τ perspective).
Definition 2.1.We say that a family φ " pφpτ q, τ P Θq is admissible if it satisfies the following conditions 1. for all τ P Θ, φpτ q is a real valued random variable, which is F τmeasurable.
The following remark is worth noting, as a consequence of the admissibility.
We show this by the following reasoning: for each fixed n P N, let C n :" tτ n " τ u.
For each fixed m P N, let A m :" X něm C n " X něm tτ n " τ u.Note that the set A m might be empty.We have Y mPN A m " Ω.Moreover, by the admissibility of φ, we have, for each fixed n P N, φpτ n q " φpτ q, on C n " tτ n " τ u.Hence, for each fixed m P N, for all n ě m, φpτ n q " φpτ q on A m " X něm C n . (3) Let ω P Ω.By assumption, there exists n 0 " n 0 pωq such that ω P A n 0 .By property (3) (applied with m " n 0 ), for all n ě n 0 , φpτ n qpωq " φpτ qpωq, which is the desired conclusion.

The optimisation problem
Let p P r1, `8s be fixed.Let ξ " pξpτ q, τ P Θq be p-integrable admissible family modelling an agent's dynamic financial position.
Remark 2.3.For example, the family ξ can be defined via a given progressive process pξ t q tPr0,T s , corresponding to a given dynamic financial position process.For each τ P Θ, we set ξpτ q :" ξ τ .The family of random variables ξ " pξpτ q, τ P Θq can be shown to be admissible.If for each k P AE, ξ θ k P L p , and ξ T P L p , then the admissible family ξ is p-integrable.The financial interpretation of this example is as follows: the agent can choose his/her strategy only among the stopping times in Θ, that is, among the stopping times which, for almost each ω, have values in the finite grid t0, θ 1 pωq, . . ., θ npωq pωq " T u, where npωq depends on ω, or in the infinite countable grid t0, θ 1 pωq, . . ., θ n pωq, θ n`1 pωq, . . ., T u.In this example, the financial position which is actually taken into account in the problem corresponds to the values of the process pξ t q only at times 0, θ 1 , ..., θ n , θ n`1 , ..., T .
Let p P r1, `8s.We introduce the following properties on the non-linear operators ρ S,τ r¨s, which will appear in the sequel.
Fatou property is often assumed in the literature on risk measures (particularly in the case where p " `8).
Let us emphasize that no assumptions of convexity (or concavity) or translation invariance of the non-linear operators ρ are made.

pΘ, ρq-Snell envelope family and optimal stopping
As is usual in optimal control, we embed the above optimization problem (5) in a larger class of problems by considering for each ν P Θ, the random variable V pνq, where V pνq :" ess sup τ PΘν ρ ν,τ rξpτ qs.
We note that, if ρ satisfies the property of knowledge preservation (property (iii)), then V pT q " ρ T,T rξpT qs " ξpT q.
Lemma 2.1.(Admissibility of V ) Under the assumption of admissibility (ii) and "generalized zero-one law" (vi) on the non-linear operators, the family of random variables V :" pV pνq, ν P Θq defined in (6) is admissible in the sense of Definition 2.1.
The proof uses arguments similar to those of Lemma 8.1 in Grigorova et al. (2020), combined with some properties of the non-linear operators ρ.
Proof.Property 1. of the definition of admissibility follows from the definition of the essential supremum, the random variables of the family pρ ν,τ rξpτ qs, τ P Θ ν q being F ν -measurable.Let us prove Property 2. Let ν and ν 1 be two stopping times in Θ.We set A :" tν " ν 1 u and we show that V pνq " V pν 1 q, P -a.s. on A. We have where we have used the admissibility property on ρ for the last equality.Let τ P Θ ν .We set τ A :" τ I A `T I A c .We note that τ A P Θ ν 1 and τ A " τ p.s. on A. Using this, the admissibility of the family ξ, and the generalized zero-one law property of ρ, we get I A ρ ν 1 ,τ rξpτ qs " I A ρ ν 1 ,τ A rξpτ A qs ď I A V pν 1 q.As τ P Θ ν is arbitrary, we conclude that ess sup τ PΘν I A ρ ν 1 ,τ rξpτ qs ď I A V pν 1 q.Combining this inequality with (7) gives I A V pνq ď I A V pν 1 q.We obtain the converse inequality by interchanging the roles of ν and ν 1 .
Under the assumptions of the above lemma, the following remark holds true.
Remark 2.4.As a consequence of the admissibility of the value family V , we have: for each k P N, it holds V pνq " V pθ k q a.s. on tν " θ k u and V pνq " V pT q a.s. on tν " T u.Hence, under the assumptions of Lemma 2.1, for ν P Θ of the form ν " Remark 2.5. 1.Under the assumption of knowledge preservation (iii) on ρ, we have V pθ k q ě ξpθ k q, for each k P N. Indeed, V pθ k q " ess sup τ PΘ θ k ρ θ k ,τ rξpτ qs ě ρ θ k ,θ k rξpθ k qs, and by the property (iii) of the non-linear operators, we have ρ θ k ,θ k rξpθ k qs " ξpθ k q.Hence, V pθ k q ě ξpθ k q.
2. If, moreover, ρ satisfies the properties of admissibility (ii) and "generalized" zero-one law (vi), then, for each τ P Θ, V pτ q ě ξpτ q.This follows from the first statement of the remark, and from the admissibility of ξ and that of V (cf.Lemma 2.1 and Remark 2.4).Now, let us introduce the notion of pΘ, ρq-(super)martingale family.
We introduce the following integrability assumption on V , which is assumed in the sequel.
Assumption 2.1.For each ν P Θ, the random variable V pνq is in L p .
We will see in Section 3, concrete examples for which this integrability assumption on V is satisfied.
Theorem 2.1.1. (pΘ, ρq-supermartingale) Under the assumptions of admissibility (ii), consistency (v), "generalized zero-one law" (vi) and monotone Fatou property with respect to the terminal condition (vii) on the non-linear operators, the value family V is a pΘ, ρq-supermartingale family.
2. (pΘ, ρq-Snell envelope) If moreover the non-linear operators also satisfy the properties of knowledge preservation (iii) and monotonicity (iv), the value family V is equal to the pΘ, ρq-Snell envelope of the family ξ, that is, the smallest pΘ, ρqsupermartingale family dominating the family ξ " pξpτ q, τ P Θq.
To prove this theorem, we first state a useful lemma.
Lemma 2.2.(Maxmizing sequence lemma) Under the assumption of "generalized zero-one law" (vi) on the non-linear operators, there exists a maximizing sequence for the value V pSq of problem (6).
The proof of this lemma is similar to that of Lemma 2.3 in Grigorova et al. (2020) and is given for the convenience of the reader.
Proof.It is sufficient to show that the family pρ S,τ rξpτ qsq τ PΘ S is stable under pairwise maximization.The result then follows by a well-known property of the essential supremum.Let τ P Θ S and τ 1 P Θ S .Set A :" tρ S,τ 1 rξpτ 1 qs ď ρ S,τ rξpτ qsu and ν :" τ I A `τ 1 I A c .Trivially, A P F S .Moreover, ν P Θ S (cf.properties of the set Θ S ).Also, ν " τ on A, ν " τ 1 on A c .By the "generalized zero-one law" of the non-linear operators ρ, we get This shows the stability under pairwise maximization of the value family (indexed by Θ S ).
Let us now show the theorem.The idea of the proof is similar to that of Theorem 8.2 in Grigorova et al. (2020).The properties on ρ being weakened here, we give the proof for clarity and completeness.
Proof of Theorem 2.1.By Lemma 2.1, the value family V is admissible.By Assumption 2.1, the value family V is p-integrable.
Let now S P Θ and τ P Θ S .To show the pΘ, ρq-supermartingale property of the value family, it remains to show ρ S,τ rV pτ qs ď V pSq a.s.By the maximizing sequence lemma (Lemma 2.2), there exists a sequence pτ p q P pΘ τ q AE , such that ρ S,τp rξpτ p qs ď V pSq, the last inequality being due to Θ τ Ă Θ S .We conclude that ρ S,τ rV pτ qs ď V pSq.
Hence, the value family V is a pΘ, ρq-supermartingale family.This proves Statement 1 of the theorem.

2.4
The strict value family and the Dynamic Programming Principle (DPP) Definition 2.3 (Dynamic Programming Principle).We say that an admissible p-integrable family satisfies the Dynamic Programming Principle (abridged DPP), if the following property holds true: For all k P N, and φpT q " ξpT q.
The purpose of this sub-section is to investigate under which assumptions on ρ, the DPP holds.To do this, we are first interested in "what happens on the right of V pθ k q", for each k P AE.
Let k P AE be fixed.We define and we define the strict value V `pθ k q at θ k by: Indeed, let τ P Θ θ k .Then τ can be written as: where tpA i q iěk`1 , Āu is a partition of Ω such that for each i ě k `1, A i P F θ i , and We have tpB i q iěk , Bu form a partition of Ω; Conversely, let τ P Θ θ k`1 ; then, τ can be written as: Hence, τ P Θ θ k `.
Due to this remark, we get Lemma 2.3.Under the assumption of "generalized zero-one law" (vi) on the nonlinear operators, there exists a maximizing sequence for V `pθ k q.
Proof.The proof of this lemma is similar to the proof of the existence of a maximizing sequence for V pθ k q, and is left to the readers.(We also refer to the proof of Lemma 2.3 in Grigorova et al. (2020) for similar arguments).
The following proposition establishes that the strict value V `pθ k q at θ k is equal to the non-linear evaluation from θ k perspective of the value V pθ k`1 q.
Proposition 2.1.Under the assumptions of monotonicity (iv), consistency (v), "generalized zero-one law" (vi) and monotone Fatou property with respect to the terminal condition (vii) on the non-linear operators, we have By Lemma 2.3, there exists a maximizing sequence pτ m q P pΘ θ k`1 q AE such that Now, by using the consistency property of the non-linear evaluations, we get For each m P AE, we have Then, by the monotonicity property of ρ θ k ,θ k`1 r¨s, we get Hence, we have We conclude, combining this with (11), that Now, let us show the converse inequality.By Lemma 2.2, there also exists a maximizing sequence pτ 1 Hence, We first use the monotone Fatou property with respect to the terminal condition of the non-linear operator ρ θ k ,θ k`1 r¨s; then, we apply the consistency property of the non-linear operators to get: where we have used Eq. ( 10) to obtain the last equality.Hence, The proof is complete.
Proposition 2.2.Under the assumptions (iii) and "generalized zero-one law" (vi) on the non-linear operators, we have Proof.By Remark 2.5, first statement, which can be applied as ρ satisfies property (iii), we have V pθ k q ě ξpθ k q.On the other hand, since Θ θ k`1 Ă Θ θ k , we have V pθ k q ě V `pθ k q.By combining these two inequalities, we get V pθ k q ě ξpθ k q _ V `pθ k q.It remains to show the converse inequality.Let τ P Θ θ k .We define τ " τ ½ tτ ąθ k u `T ½ tτ ďθ k u .As τ P Θ θk , we have Hence, we have (12) Moreover, on the set tτ ą θ k u, we have τ " τ , so the "generalized zero-one law" gives By combining ( 12) and ( 13), we get On the other hand, as τ P Θ θ k , we have By using the "generalized zero-one law" and property (iii) of the non-linear operator ρ θ k ,τ r¨s, we get From Eqs. ( 14) and ( 15), we get Now, by taking the essential supremum over τ P Θ θ k , we get V pθ k q ď ξpθ k q _ V `pθ k q.Hence, the proof is complete.
We refer to Quenez and Kobylanski (2012) for a similar approach the one in the above proposition in the linear case.
By combining Proposition 2.1 and Proposition 2.2, we get: Theorem 2.2 (DPP).Under the assumptions of knowledge preservation (iii), monotonicity (iv), consistency (v), "generalized zero-one law" (vi) and monotone Fatou property with respect to the terminal condition (vii) on the non-linear operators, the value family V satisfies the DPP: for each k P AE, V pθ k q " ξpθ k q _ ρ θ k ,θ k`1 rV pθ k`1 qs, and V pT q " ξpT q.

Optimal stopping times
For each k, let us define the random variable ν k ν k :" ess inf A k where A k :" t τ P Θ θ k : V pτ q " ξpτ q a.s.u.
As T ă 8, under property (iii) on ρ, the set A k is clearly non-empty (as V pT q " ξpT q in this case).Moreover, it is clearly stable by pairwise minimization.Hence, by classical properties of the essential infimum, there exists a non increasing sequence pτ n q in A k such that lim nÑ`8 τ n " ν k a.s.In particular, ν k is a stopping time and T ě ν k ě θ k a.s., and ν k P Θ θ k (by stability of Θ θ k when passing to a monotone limit).
In the following theorem, we show that, under suitable assumptions, the stopping time ν k defined in ( 16) is optimal for the optimization problem (6) at time ν " θ k .
We introduce the following assumption on the value family V .
Assumption 2.2.We assume that the value family V is left-upper-semicontinuous (LUSC) along the sequence pθ n ^νk q nPN , that is, Remark 2.8.Assumption 2.2 is trivially satisfied in the following particular case on Θ: Besides the assumptions paq and pbq on Θ, the additional assumption (c) is imposed, namely: (c)For almost all ω, there exits n 0 " n 0 pωq (depending on ω) such that θ n pωq " T , for all n ě n 0 .In other words, for almost all ω, there exists at most a finite number of time points θ n pωq such that θ n pωq ă T .In this case, for all n after a certain rank n " npωq, we have pθ n ^νk qpωq " ν k pωq.Hence, as V is admissible, we have, by Remark 2.2, for all n ě npωq, V pθ n ^νk qpωq " V pν k qpωq.Hence, Assumption 2.2 holds true.
We will see later on a further discussion on Assumption 2.2 in the case of the general Θ, and conditions (on ρ and on the pay-off family ξ) under which this assumption is satisfied.
Theorem 2.3 (Optimality of ν k ).Let k P AE and let ν k be the stopping time defined by (16).Let Assumption 2.2 on V be satisfied.Let ρ satisfy the properties of admissibility (ii), knowledge preservation (iii), monotonicity (iv), consistency (v), "generalized zero-one law" (vi), and monotone Fatou property with respect to the terminal condition (vii).We assume additionally that ρ satisfies the following property: • (left-upper-semicontinuity (LUSC) along Bermudan stopping times with respect to the terminal condition and the terminal time at ν k ), that is, for each non-decreasing sequence pτ n q P Θ N S such that lim nÑ`8 Ò τ n " ν k a.s., and for each p-integrable admissible family φ such that sup nPN |φpτ n q| P L p .
Note that in the case where ρ " pρ S r¨sq SPΘ does not depend on the second time index, the above additional property reduces to the LUSC of ρ S r¨s (with respect to the terminal condition) along Bermudan stopping sequences.
2.5.1 The pΘ, ρq-martingale property on a stochastic interval Before proving the theorem, we give several useful technical lemmas.
The first two clarify the pΘ, ρq-martingale structure on a stochastic interval in a more "handy" way.The third one deals with an "if -condition" (optimality criterion) and an "only if -condition" for optimality.
Definition 2.4.(Strictly monotone operator) Let S P Θ, τ P Θ S .We say that ρ S,τ is strictly monotone if the following two conditions hold: 1. ρ S,τ is monotone.
Lemma 2.5.We assume that the non-linear operators satisfy the properties of monotonicity (iv) and consistency (v).Assume moreover that the non-linear operators ρ are strictly monotone.Let φ " pφpνqq be a given p-integrable admissible family).Let S P Θ and τ P Θ be such that S ď τ a.s.We assume that the two conditions hold: 1. φ is a pΘ, ρq-supermartingale family on rS, τ s; 2. φpSq " ρ S,τ rφpτ qs a.s.
Due to the additional assumption, the non-linear operators ρ are strictly monotone.
1. (Optimality criterion) If i) and ii) are satisfied, then ν k is optimal for problem (6).
2. If, moreover, the non-linear operator ρ θ k ,ν k is assumed to be strictly monotone and satisfies the assumptions of admissibility (ii), knowledge preservation (iii), consistency (v), "generalized zero-one law" (vi) and monotone Fatou property (vii), then the converse statement is also true.
Remark 2.9.We note that the property V pν k q " ξpν k q a.s.implies that ρ θ k ,ν k rV pν k qs " ρ θ k ,ν k rξpν k qs a.s.The converse implication is true under the additional assumption: ρ θ k ,ν k is strictly monotone.
Proof.First, let us show statement 1.
Let ν k P Θ θ k be such that the two conditions i) and ii) introduced above are satisfied.By condition ii) the family pV pν ^νk qq νPΘ θ k is a pΘ, ρq-martingale family.
In particular, for ν " ν k , we get From this, together with condition i), we have which implies that the stopping time ν k is an optimal stopping time for problem (6).Now, let us show statement 2. Let ν k P Θ θ k be an optimal stopping time for problem (6).Hence, we have By the first part of Theorem 2.1 (which is applicable as ρ satisfies the assumptions), the value family V is a pΘ, ρq-supermartingale family.Thus, by the pΘ, ρqsupermartingale property of V , and as ν k P Θ θ k , we have On the other hand, due to the fact that ξ ď V (cf.Remark 2.5, Statement 2) and to the monotonicity of the non-linear operator ρ θ k ,ν k , it holds Moreover, since V pθ k q " ρ θ k ,ν k rV pν k qs, by applying Lemmas 2.4 and 2.5 (the latter is applicable as ρ is assumed to be strictly monotone) with S " θ k , τ " ν k , we conclude that V is a pΘ, ρq-martingale on rθ k , ν k s.
The proof is complete.

Two useful consequences of the DPP
The following two results hold, if a given admissible p-integrable family φ satisfies the (DPP) from Eq.( 9), and if νk is defined by νk :" ess inf Ãk where Ãk :" t τ P Θ θ k : φpτ q " ξpτ q a.s.u.
The following lemma is a consequence of the definition of νk and of the DPP.
Proof.Let l P tk, k `1, ...u.By the definition of νk , on the set tν k ą θ l u, we have φpθ l q ą ξpθ l q.From this and from the DPP, we conclude that on the set tν k ą θ l u, φpθ l q " ρ θ l ,θ l`1 rφpθ l`1 qs.
Proof.First, we show Statement 1 of the Lemma.
We now prove Statement 2 of the Lemma.
For the second summand of Eq.( 27), by Statement 1 of the Lemma, we get For the first summand of Equation ( 27), we apply the "generalized zero-one law" (as the set tν k ď θ l u is F θ l ^ν k -measurable and on the set tν k ď θ l u, we have θ l`1 ^ν k " θ l ^ν k ), we have where we have used property (iii) on ρ to obtain the last equality.Finally, we get ρ θ l ^ν k ,θ l`1 ^ν k rφpθ l`1 ^ν k qs " ½ tν k ďθ l u φpθ l ^ν k q `½tν k ěθ l`1 u φpθ l ^ν k q " φpθ l ^ν k q.

The proof of optimality of ν k
We are now ready to prove Theorem 2.3 on the optimality of the Bermudan stopping time ν k , defined by ( 16).We will need also the following remark: Remark 2.10.Any admissible family pφpτ q, τ P Θq in our framework is rightcontinuous along Bermudan stopping times, that is, for all τ P Θ, and for all non-increasing sequences of Bermudan stopping times pτ n q P Θ AE such that τ n Ó τ , it holds lim nÑ`8 φpτ n q " φpτ q.
Proof of Theorem 2.3.By Lemma 2.6, in order to show that ν k , defined in ( 16), is optimal for problem (6), it is enough to show the following two conditions: (ii) The family pV pνqq is a pΘ, ρq-martingale on rθ k , ν k s.
We start our proof by showing the second condition first.
By Lemma 2.1, V is an admissible family, and it is also p-integrable by Assumption 2.1.By Lemma 2.4 (on the pΘ, ρq-martingale property), in order to show the second condition, it is enough to show that: for each σ P Θ, such that θ k ď σ ď ν k , Let σ P Θ θ k .Then σ is of the form σ " ř měk θ m ½ Am `T ½ Ā, where tpA m q mPN , Āu form a partition of Ω; A m is F θm -measurable for each m, and Ā P F T .
Hence, the sequence of random variables pρ θm^ν k ,θ m`n ^νk rV pθ m`n ^νk qsq nPAE does not depend on n and is constantly equal to the random variable V pθ m ^νk q.

nd
Step: As V is left-upper-semicontinuous (LUSC) along the sequence pθ m`n νk q nPN by Assumption 2.2, and as ρ is LUSC along Bermudan stopping strategies with respect to terminal condition and terminal time at pν k q, we have lim sup where we have used the monotonicity of ρ and Assumption 2.2 on V to obtain the last inequality.Hence, V pθ m ^νk q ď ρ θm^ν k ,ν k rV pν k qs.
We conclude, by Lemma 2.4, that V is a pΘ, ρq-martingale on the stochastic interval rθ k , ν k s.This shows condition (ii) in the optimality criterion of Lemma 2.6.
It remains for us to show condition (i) in the optimality criterion.Let us recall that ν k " ess inf A k , where A k " tτ P Θ θ k : V pτ q " ξpτ q a.s.u.Let pτ n q be a non-increasing sequence in A k , such that lim nÑ`8 Ó τ n " ν k .As τ n is in A k , we have V pτ n q " ξpτ n q.By passing to the limit in this equality and by using that both families V and ξ are right-continuous along the sequence of Bermudan stopping strategies pτ n q (cf.Remark 2.10), we obtain which proves condition (i).This concludes the proof of the optimality of ν k .

Assumption 2.2 on V : Discussion
Let us now check under which conditions Assumption 2.2 on V holds true.
Under the assumptions paq, pbq on Θ, the set tω P Ω : ν k pωq " T, θ l pωq ă T, for all l P Nu might be non-empty.We will show the following lemma.
Indeed, for each τ P Θ θm , we have

Examples
In this section we provide some examples of non-linear operators ρ, known from the stochastic control and mathematical finance literature, which enter into our framework.

Non-linear operators induced by BSDEs
In this example, p " 2.

The g-evaluations
Peng ( 2004) and El Karoui and Quenez (1997) introduced a type of non-linear evaluation, now known as g-evaluation, via a non-linear backward stochastic differential equation (BSDE) with a driver g.
Let T ą 0 be a fixed time horizon.We place ourselves in the Brownian framework (for simplicity).Let pΩ, F , Èq be a complete probability space, endowed with a d-dimensional Brownian motion pW t q tPr0,T s , and let pF t q tPr0,T s be the (augmented) natural filtration of the Brownian motion.
Let g " gpω, t, y, zq : Ω ˆr0, T s ˆÊ ˆÊd Ñ Ê be Lipschitz driver, that is, a function satisfying the following conditions: ➀ For each y P Ê, z P Ê d , gp¨, ¨, y, zq P L 2 pΩ ˆr0, T sq and g is progressively mea- surable; ➁ There exists C ą 0 such that for each y 1 , y 2 P Ê, and for each z 1 , z 2 P Ê d , |gpω, t, y 1 , z 1 q ´gpω, t, y 2 , z 2 q| ď Cp|y 1 ´y2 | `}z 1 ´z2 }q, uniformly for a.e.pω.tq,where } ¨} denotes the Euclidean norm on Ê d ; Let us consider the following 1-dimensional BSDE with terminal time t and terminal condition η, defined on the interval r0, ts, given that 0 ď t ď T and η P L 2 pF t q: y s " η `ż t s gpr, y r , z r qdr ´ż t s z r dB r , s P r0, ts.
Definition 3.1.(g-evaluation) For each 0 ď s ď t ď T and η P L 2 pF t q, we define E g s,t rηs :" y s .
The family of operators E g s,t r¨s : L 2 pF t q Ñ L 2 pF s q, 0 ď s ď t ď T are called g-evaluation.
We recall (cf.El Karoui et al. (1997)) that if the terminal time is given by a stopping time τ valued in r0, T s and if η is F τ -measurable, the solution of the BSDE with terminal time τ , terminal condition η and Lipschitz driver g is defined as the solution of the BSDE with (deterministic) terminal time T , terminal condition η and Lipschitz driver g τ defined by g τ pt, y, zq :" gpt, y, zq½ ttďτ u .The first component of this solution at time t is equal to E g τ t,T pηq, also denoted by E g t,τ pηq.We have E g t,τ pηq " η a.s. on the set tt ě τ u.
The following result summarizes some of the well-known properties of the gevaluations.
Proposition 3.1.Let g satisfy ➀ and ➁.Let S, τ, θ be stopping times.Then the g-evaluation satisfies the following properties: rηs " η, for all S, τ , such that S ď τ , for all η P L 2 pF S q.
(A5) (Continuity with respect to terminal time and terminal condition) Let pτ n q nPAE be a sequence of stopping times in T S,τ , such that lim nÑ8 τ n " τ a.s.Let pη n q nPAE be a sequence of random variables, such that η n P L 2 pF τn q, sup n η n P L 2 and lim nÑ8 η n " η a.s.Then, we have lim nÑ8 E g S,τn rη n s " E g S,τ rηs a.s.
Remark 3.1.For Property (A4) we refer, e.g., to Grigorova and Quenez (2016).Property (A5) was proven in Quenez and Sulem (2013) (in the case of jumps).In Peng (2004), the additional assumption gp¨, 0, 0q " 0 is made ensuring that the g-evaluation satisfies the usual "zero-one law": for each S ď τ , for each A P F S , E g S,τ r½ A ηs " ½ A E g S,τ rηs.Moreover, the g-evaluation E g satisfies the property (18) in Theorem 2.3.Indeed, we have lim sup nÑ`8 E g S,τn rφpτ n qs ď E g S,τ ‹ rlim sup nÑ`8 φpτ n qs, for each nondecreasing sequence τ n such that lim nÑ`8 τ n " τ ‹ , and for each square-integrable admissible family φ, such that sup n |φpτ n q| P L 2 .For a proof of this property, based on property (A5) of the g-evaluations, we refer to Lemma A.5 in Dumitrescu et al. (2016).
Moreover, in the Brownian framework, the first component py t q of the solution of the BSDE with Lipschitz driver g, terminal time T , and terminal condition η P L 2 pF T q, has continuous trajectories (in t).Hence, for any non-decreasing sequence pτ n q, such that lim nÑ`8 τ n " T , lim nÑ`8 E g τn,T pηq " lim nÑ`8 y τn " y T " E g T,T pηq " η.
Thus, property (33) from Lemma 2.9 and Proposition 2.3 is satisfied.Hence, the g-evaluation satisfies all the properties of the non-linear operators ρ used in our results.
It remains for us to argue that the integrability Assumption 2.1 (with p " 2) on the value family V is satisfied, under some suitable assumptions on ξ.The following remark is an application of Remark 2.6 to the framework of a complete financial market model with imperfections encoded in the driver g of the dynamics of self-financing portfolios.
Remark 3.2.Let us place ourselves in a complete financial market model with possible imperfections (such as e.g. trading constraints, different interest rates for borrowing and lending, different repo rates, etc.).Let g be the driver from the dynamics of self-financing portfolios in this market.If the family ξ " pξpτ qq τ PΘ is assumed to be square-integrable and super-replicable by a self-financing portfolio with wealth Xpτ q at time τ , then, the family pXpτ qq τ PΘ is a pΘ, E g q-martingale.By Remark 2.6 the value V " pV pτ qq τ PΘ is square-integrable (that is, satisfies Assumption 2.1 with p " 2).

Peng's g-expectation
Peng's g-expectation is a particular case of the previous example, introduced in Peng (1997).In this case, the driver g is assumed to satisfy the conditions ➀, ➁, and condition ➂' (gp¨, y, 0q " 0, for all y P Ê).In this particular case, the non-linear operators do not depend on the second index, but on the first index only.Moreover, they satisfy the usual zero-one law.More precisely, let g satisfy conditions ➀, ➁ and ➂'.The operators E g r¨s and E g r¨|F S s are defined by E g r¨s :" E g 0,τ r¨s, E g r¨|F S s :" E g S,τ r¨s.The family of non-linear operators pE g r¨|F S sq, indexed by the stopping times S, is called g-expectation.
The g-expectation pE g r¨|F S sq satisfies all the properties of the g-evaluation and additionally the following property: (usual zero-one law) For stopping times S, τ such that S ď τ , for A P F S , E g r½ A η|F S s " ½ A E g rη|F S s.

Bayraktar -Yao's non-linear expectations
Let T ą 0 and let pΩ, F , Èq be a complete probability space endowed with a filtration pF t q tPr0,T s , satisfying the usual conditions.A non-linear expectation, called F -expectation, depending on one time index only, was introduced in the work by Bayraktar and Yao-Part I (2011), and optimal stopping problems in continuous time with F -expectations were studied in Bayraktar and Yao-Part II (2011).For simplicity of the exposition, we will consider the case where the domain of the F -expectation is the whole space L 8 .The non-linear operators (F -expectation) in Bayraktar and Yao-Part I (2011) are defined first for deterministic times, then extended to stopping times valued in a finite deterministic grid, then, extended to general stopping times valued in r0, T s.We will not repeat the construction here, but will recall the basic properties only (cf.Bayraktar and Yao-Part I (2011) for explanations and details; in particular, cf.Propositions 2.7, 2.8, and 2.9 therein2 ).
(C3) (Usual zero-one law) Er½ A η|F S s " ½ A Erη|F S s, for any A P F S .
(C4) (Translation invariance) Erη `X|F S s " Erη|F S s `X, if X P L 8 pF S q.
(C6) (Local property) Erη½ A `η1 ½ A c |F S s " Erη½ A |F S s `Erη 1 ½ A c |F S s, for any A P F S and for any η, ξ P L 8 pF T q.
(C7) (Fatou property) Let pη n q be a sequence in L 8 , satisfying inf n η n ě c, a.s.for some constant c P R, and such that lim nÑ8 η n " η, where η P L 8 .Then, Erη|F S s ď lim inf nÑ`8 Erη n |F S s.
(C8) (Dominated convergence) Let pη n q be a sequence, such that inf n η n ě c, a.s.for some constant c P R, and such that lim nÑ8 η n " η.If there exists η P L 8 such that η n ď η for all n P AE, then the limit η P L 8 , and lim nÑ8 Erη n |F S s " Erη|F S s.
Note that the knowledge preserving property is called constant preserving in Bayraktar and Yao-Part I (2011).
Remark 3.3.The F -expectation satisfies property (18) of Theorem 2.3.Indeed, let pτ n q be a non-decreasing sequence of stopping times, such that lim nÑ`8 Ò τ n " τ .Let φ be an admissible L 8 -integrable family such that sup nPN |φpτ n q| P L 8 .As the F -expectation does not depend on the second index, to show property (18), we need to show lim sup nÑ`8 Erφpτ n q|F S s ď Erlim sup nÑ`8 φpτ n q|F S s.
For each n P AE, φpτ n q ď sup pěn φpτ p q. Hence, by monotonicity of Er¨|F S s, we have Erφpτ n q|F S s ď Ersup pěn φpτ p q|F S s.
The sequence η n defined by η n " sup pěn φpτ p q is non-increasing and tends to η :" lim sup nÑ`8 φpτ n q.Moreover, φpτ n q|F S s, which is the desired property.
The assumptions imposed on the pay-off in Bayraktar -Yao (Bayraktar and Yao-Part I (2011) and Bayraktar and Yao-Part II (2011)) ensure that the value in their case belongs to the domain of the operator.Hence, if we consider the Bermudan style version of their problem, Assumption 2.1 on V will also be satisfied.

Dynamic Concave Utilities
The dynamic concave utilities are among the examples of non-linear operators depending on two time indices.
In this example the space is L 8 (that is p " `8).
We place ourselves again in the Brownian framework.A representation result, with an explicit form for the penalty term, for dynamic concave utilities was established in Delbaen et al. (2010).The optimal stopping problem with dynamic concave utilities was studied by Bayraktar, Karatzas and Yao in Bayraktar et al. (2010), where the authors rely on the representation result from Delbaen et al. (2010). 3e recall the following definition from Delbaen et al. (2010).
Moreover, in Delbaen et al. (2010) and Bayraktar et al. (2010) the following properties on the dynamic concave utilities are assumed: (D5) (Time consistency) for any stopping time σ P T S,τ , we have u S,σ pu σ,τ pηqq " u S,τ pηq; (D6) (Continuity from above) for any non-increasing sequence pη n q Ă L 8 pF τ q with η " lim nÑ8 Ó η n P L 8 pF τ q, we have lim nÑ8 Ó u S,τ pη n q " u S,τ pηq (D7) (Local property) u S,τ pη½ A `ξ½ A c q " u S,τ pηq½ A `uS,τ pξq½ A c , for any A P F S and for any η, ξ P L 8 pF τ q; (D8) E P rη|F t s ě 0 for any η P L 8 pF T q, such that u t,T pηq ě 0.
By the results of Delbaen et al. (2010), any functional ρ satisfying properties (D1) -(D8), has the following representation: u S,τ pηq " ess inf where the function f is such that f p¨, ¨, xq is predictable for any x; f is a proper, convex function in the space variable x, and valued in r0, `8s, and the process pψ Q t q is the process from the Doleans-Dade exponential representation for the density process pZ Q t q, where Z Q t " dQ dP | Ft , and Remark 3.5.A close inspection of the proof of the duality result in (Bion-Nadal (2009) and Delbaen et al. (2010)) reveals that the dynamic concave utilities depend on the second index only via their penalty term.
It has been noted in Delbaen et al. (2010) that property (D8) is equivalent to c t,T pP q " 0, for all t P r0, T s. (37) The dynamic concave utilities u S,τ do not enter directly into the framework of the present paper, as they are defined only for S, τ such that S ď τ a.s.(cf.Delbaen et al. (2010) and Bayraktar et al. (2010)).There is, however, a "natural" extension of u S,τ in view of the representation property (36).This extension is as follows: For S and τ stopping times, and η P L 8 pF τ q, we define ½ tSąτ u u S,τ pηq " ½ tSąτ u ˆη. (38) Remark 3.6.By properties (D4) and (D7) of the dynamic concave utilities, we get that the "generalized zero-one law" is satisfied.Indeed, let A P F S , and let τ, τ 1 P Θ S be such that τ " τ 1 on A. Let η P L 8 pF τ q.By applying property (D7) with ξ " 0, we get u S,τ p½ A ηq " ½ A u S,τ pηq `½A c u S,τ p0q.As u S,τ p0q " 0 due to the normalisation property (D4), we obtain u S,τ p½ A ηq " ½ A u S,τ pηq.Hence, the "generalized zero-one law" is satisfied.
Remark 3.8.The dynamic concave utilities satisfy property (18) in Theorem 2.3.Indeed, let pτ n q be a non-decreasing sequence of stopping times such that τ n Ò τ , and let φ be an L 8 -integrable admissible family such that sup n |φpτ n q| P L 8 .Then, for each n P N, where, for the inequality, we have used that f is valued in r0, `8s, and τ n ď τ.As ess inf QPQ S E Q r¨|F S s is non-decreasing, we get: for each n P N, We have η n Ó η, where η :" lim sup nÑ`8 φpτ n q `şτ S f pu, ψ Q u qdu.As ess inf QPQ S E Q r¨|F S s is continuous from above, we deduce Hence, from Eqs. (40) and (41), we get .This is the desired property (18).
Remark 3.9.The dynamic concave utilities satisfy property (33) from Lemma 2.9 and Proposition 2.3.Indeed, for τ n Ò T , u τn,T pηq " ess inf where we have used that for each n P N, c τn,T pP q " 0 by Eq.( 37).Hence, u τn,T pηq ď E P rη|F τn s.The sequence pE P rη|F τn sq being a uniformly integrable P -martingale, with terminal value E P rη|F T s " η, we get lim sup nÑ8 u τn,T pηq ď lim nÑ8 E P rη|F τn s " η, which is the desired property.
To finish, the pay-off process pξ t q in Bayraktar et al. ( 2010) is assumed to be bounded.Hence, by the monotonicity and the knowledge preservation of u, if we consider the Bermudan-style version of the problem studied in Bayraktar et al. (2010), then the value V satisfies the integrability Assumption 2.1 (that is, for each S P Θ, V pSq P L 8 ).
4 Appendix: The case of a finite number of predescribed stopping times In this appendix, we treat the particular case where pθ k q kPN 0 is constant from a certain term, independent of ω, onwards.More precisely, we place ourselves in the situation where there exists n P N˚(independent of ω) such that for each m ě n, θ m " T Theorem 4.1.Let φ " pφpτ q, τ P Θq be a p-integrable admissible family.Under the assumptions of knowledge preservation (iii) and "generalized zero-one law" (vi) on the non-linear operators, if φ satisfies then, for all τ P Θ, we have Proof.Let k P N 0 .We have We note that on the set tτ ď θ k u, θ k`1 ^τ " θ k ^τ .Hence, by the "generalized zero-one law", we have As θ k ^τ ď θ k , by property (iii) of the non-linear evaluation ρ, we get ρ θ k ,θ k ^τ rφpθ k ^τ qs " φpθ k ^τ q.
Hence, we have For the second term on the right-hand side of Equation ( 45), we note that τ ^θk`1 " θ k`1 on tτ ą θ k u.Hence, by the "generalized zero-one law" of the non-linear evaluation ρ, we have This, together with Equation ( 43) on φ and the admissibility of φ, gives By plugging in ( 46) and (47) in Equation ( 45), we get ρ θ k ,θ k`1 ^τ rφpθ k`1 ^τ qs ď I tτ ďθ k u φpθ k ^τ q `Itτąθ k u φpθ k ^τ q " φpθ k ^τ q.
This ends the proof.
We establish a characterization of pΘ, ρq-supermartingale (resp.pΘ, ρq-martingale) families in the particular case where the sequence pθ k q kPN 0 is constant from a certain term, independent of ω, onwards.
Proof.We prove the result for the case of a pΘ, ρq-supermartingale family; the case of a pΘ, ρq-martingale family can be treated similarly.Let σ, τ in Θ be such that σ ď τ a.s.As σ P Θ, we have σ " ř n k"0 θ k 1 A k , where pA k q kPt0,...,nu is a partition of Ω such that A k P F θ k .We notice that in order to prove ρ σ,τ rφ τ s ď φ σ , it is sufficient to prove the following property: ρ θ k ^τ,τ rφpτ qs ď φpθ k ^τ q, for all k P t0, 1, . . ., nu. (48) Indeed, this property proven, we will have ρ σ,τ rφpτ qs " ρ σ^τ,τ rφpτ qs " I A k φpθ k ^τ q " φpσ ^τ q " φpσq, where we have used the admissibility of ρ to show the second equality.This will conclude the proof.Let us now prove property (48).We proceed by backward induction.For k " n, we have (recall that θ n " T ) ρ θn^τ,τ rφpτ qs " ρ T ^τ,τ rφpτ qs " ρ τ,τ rφpτ qs " φpτ q " φpT ^τ q, where we have used property (iii) to obtain the last but one equality.
For this, we first show an easy lemma, based on Proposition 4.1 and on the definition of the family U.
The following proof of Theorem 4.2 is a combination of Lemma 4.1 and of a proof of the minimality property of U.
The reasoning by backward induction is thus finished and the minimality property of U shown.We conclude that the family U is equal to the smallest pΘ, ρqsupermartingale family dominating the family ξ, that is, to the pΘ, ρq-Snell envelope of the family ξ.
Let us now show the converse inequality: νk ě inftθ l P tθ k , ..., θ n u : Upθ l q " ξpθ l q a.s.u.As the set Āk is stable by pairwise minimization, there exists a sequence pτ pmq q mPAE , such that: pτ pmq q is non-decreasing; for each m, τ pmq P Āk ; and lim mÑ`8 τ pmq " νk .
Proof.As U satisfies the DPP (which it does by definition of U), we have, by Lemma 2.8, for each l P AE, Upθ l ^ν k q " ρ θ l ^ν k ,θ l`1 ^ν k rUpθ l`1 ^ν k qs.