BSDEs with weak reflections and partial hedging of American options
aa r X i v : . [ m a t h . O C ] A ug BSDEs with weak reflectionsand partial hedging of American options
Roxana Dumitrescu ∗ Romuald Elie † Wissal Sabbagh ‡ Chao Zhou § August 22, 2017
Abstract
We introduce a new class of
Backward Stochastic Differential Equations with weakreflections whose solution (
Y, Z ) satisfies the weak constraint E [Ψ( θ, Y θ )] ≥ m, for allstopping time θ taking values between 0 and a terminal time T , where Ψ is a randomnon-decreasing map and m a given threshold. We study the wellposedness of suchequations and show that the family of minimal time t -values Y t can be aggregated bya right-continuous process. We give a nonlinear Mertens type decomposition for lowerreflected g -submartingales, which to the best of our knowledge, represents a new resultin the literature. Using this decomposition, we obtain a representation of the minimaltime t -values process. We also show that the minimal supersolution of a such equationcan be written as a stochastic control/optimal stopping game , which is shown to admit,under appropriate assumptions, a value and saddle points. From a financial point ofview, this problem is related to the approximative hedging for American options. Key words :
BSDEs with weak reflections, Partial hedging, American options, Optimalcontrol, Optimal stopping, Stochastic game, Stochastic target.
AMS 1991 subject classifications :
The theory of reflected backward stochastic differential equations (RBSDEs for short) wasfirst introduced by El Karoui et al. [13]. In this context, the first component of the BSDEsolution is forced to stay above a given - so-called obstacle or reward - stochastic process. ∗ Department of Mathematics, King’s College London, United Kingdom, email: [email protected] . † LAMA, Universit´e Paris Est Marne-la-Vall´ee, France, email: [email protected] ‡ LAMME, Universit´e Evry, France, email: [email protected] . The research of WissalSabbagh benefited from the support of the “Chair Markets in Transition”, F´ed´eration Bancaire Fran¸caise,and of the ANR 11-LABX-0019. § Department of Mathematics, National University of Singapore, Singapore, [email protected]
1n order to maintain the solution above the obstacle, the BSDE dynamics contains anextra increasing component, which is part of the solution. The uniqueness property forsuch equations is due to a Skorokhod type minimality condition. The first application ofRBSDEs was related to pricing and hedging concerns of American options. Since then, alarge number of applications to optimal stopping, optimal switching or stochastic gamesgave rise to a vast literature on this topic.The valuation of an American option with payoff process ( L t ) ≤ t ≤ T requires to determineits optimal selling time and corresponding hedging strategy. Nevertheless, in realistic andhereby imperfect financial markets, a replicating strategy is often inaccessible. From thepoint of view of the seller, who wants to protect himself against his contractual obligation, aconservative approach consists in superhedging the American option, via the constructionof an investment strategy generating enough capital to cover the payoff at any possiblestopping time chosen by the option holder. Solving such problem consists in finding aninitial data Y , a control Z and an additional increasing process K such that Y Zt = Y + Z t g ( s, Y Zs , Z s ) ds − Z t Z s dW s + K t (1.1) Y Zτ ≥ L τ , P − a.s. for all stopping time τ ∈ [0 , T ] , (1.2) Z T ( Y Zt − L t ) dK t = 0 . (1.3)The driver function g contains in particular the discounting factors as well as some imper-fections of the financial market. It may be non linear, whenever for example the lending andborowing rates are different. Y t interprets as the super-replication price of the Americanoption at time t , whereas Z corresponds to the optimal sur-replication strategy. Observethat the Skorkhod condition (1.3) enforces to choose the minimal super-replicating price.From a practical point of view, however, the cost of superhedging is fairly too high, sothat the option seller needs to accept to take some risk. One alternative approach mainlydeveloped so far for European options consists in replacing the too strong super-replicating P -a.s. terminal condition by a weaker one. Namely, for European options, Y ZT ≥ L T is replaced by E [ ℓ ( Y ZT − L T )] ≥ m , where m stands for a given success threshold and ℓ represents a non-decreasing loss function.From a financial point of view, this approach is referred to as quantile or efficient hedg-ing, and was first discussed by F¨ollmer and Leukert [14, 15]. In particular, they explainedhow the so-called quantile hedging price for European option can be computed explicitlyin a complete market, using duality arguments and Neyman-Pearson lemma. In a generalMarkovian setting, Bouchard et al. [4] provided a direct dynamic approach to tackle thisquestion, via the introduction of an additional well-chosen state variable. Even in incom-plete markets and for general loss functions, they characterize the pricing function as thesolution of a non-linear parabolic second order differential equation, using tools developedin the context of stochastic target problems by Soner and Touzi [23]. Recently, Bouchard,2lie and Reveillac [3] extended this approach to a possibly non-Markovian setting and in-troduced a new class of BSDEs, namely BSDEs with weak terminal condition, in which theterminal value Y T of the portfolio is required to satisfy a weak constraint of the form (1).This approach has been extended by Dumitrescu [10], allowing for the consideration of nonlinear risk measure constraints.The seller of an American option using a quantile efficient hedging approach is herebyrequired to solve a BSDE with dynamics (1.1), but shall replace the too strong constraint(1.2) by a weaker one of the form E [ ℓ ( Y Zτ − L τ )] ≥ m, for all stopping time τ ∈ [0 , T ] . (1.4)The main objective of this paper is to derive the well-posedness and main properties of BS-DEs with such type of constraint (1.4) and discuss its connection with the efficient hedgingof American options. Up to our knowledge, we provide the first dynamic probabilistic rep-resentation for the efficient price of American Options in continuous time. Let now mentionsome related works in the literature. P´erez [22] or Mulinacci [18] discuss the existence ofan efficient hedge in such context. Dolinsky and Kifer [9] focus on the partial hedging ofgame options in a discrete time setting with transaction costs. In a Markovian setting, anobstacle version of the geometric dynamic programming principle of Soner and Touzi [23]is given in [5], and Bouchard et al [2] provided a probabilistic numerical algorithm for thecomputation of the quantile hedging of Bermudean options, using duality representations.Recently, Briand et al [6] followed a very different approach to study BSDEs of the form(1.1) together with a weaker version of (1.4) where the constraint only hold for deterministictimes on [0 , T ]. In such a framework, no dynamic programming principle is available andthe derived solution relates to stochastic differential equations of McKean-Vlasov type.Trevino [24] considered the problem that the seller of an American option aims to con-trol the shortfall risk by using a partial hedge. He is interested in the problems of partialhedging and of optimal exercise of an American option in an incomplete market in con-tinuous time. In particular, Trevino [24] proposed an optimization problem which involvesminimization over a family of stochastic integrals and maximization over the family of stop-ping times.In this paper, we first formulate the notion of BSDEs with weak reflections , whoseconstraint takes the following general form E τ [Ψ( θ, Y Zθ )] ≥ µ, for all stopping time θ ∈ [ τ, T ] , (1.5)where µ ∈ L ( F τ ) is the target success ratio at a given stopping date τ and Ψ is a possiblyrandom non-decreasing map. This representation allows of course to encompass the efficientpricing of American options presented above. We first observe that the minimal solution tothis BSDE rewrites as the infimum over a family of solutions to classical reflected BSDEswith appropriate obstacle, that is inf { Y α , α ∈ V } , where ( Y αt ) is the first composant of3he solution of the reflected BSDE associated with obstacle Φ( t, M αt ), with M α a martingaleprocess. As it is usual in stochastic control, we study the dynamical counterpart Y α ( τ ) = ess inf { Y α ′ τ , α ′ ∈ V s . t . α ′ = α on [[0 , τ ]] } . (1.6)We derive a dynamic programming principle for this family, from which we deduce thatthe value ( Y α ) is a R ef g,α -submartingale family, where R ef g,α is the nonlinear operatorinduced by the solution of the lower reflected BSDE with obstacle Φ( t, M αt ) and driver g .Using some fine results from the general theory of processes, we show that the value family( Y α ) can be aggregated by a right-continuous and left limited process ( Y αt ). Moreover, weshow that any strong R g,α -submartingale admits a E g -Mertens decomposition, which to thebest of our knowledge, represents a new result in the literature. We propose an originalproof, which does not use the classical penalization approach. Taking advantage of thisdecomposition, we show that, for each α , the value process ( Y αt ) has a backward SDE rep-resentation. Moreover, ( Y αt ) corresponds to the upper value of a stochastic control/optimalstopping game, which is shown to admit, under appropriate assumptions, a value and asaddle point.The outline of the paper is the following. After presenting notations, we introduce thedefinition of supersolution of BSDE with weak reflections in Section 2. In Section 3, wespecialize our discussion to the minimal supersolution of the
BSDE with weak reflections .We first prove a dynamic programming principle and that we can aggregate the value familyby a c`adl`ag process. In this Section we also provide a nonlinear Mertens decompositionof R ef g,ξ -submartingale processes, which is then used in order to give a representation ofthe value process. In Sections 4, we study a related stochastic control/optimal stoppingproblem, which is shown to admit a value and a saddle point. Notations
We first introduce a series of notations that will be used throughout the paper.Let d ≥ T > W := ( W t ) t ∈ [0 ,T ] a d -dimentional Brownianmotion defined on a probability space (Ω , F , P ) with P - augmented natural filtration F =( F t ) t ∈ [0 ,T ] . The notation E will stand for the expectation with respect to P . Hereafter, wedefine the following spaces: • L p ( U, G ) is the set of p integrable G -measurable random variables with values in U , p ≥ U a Borel set of R n for some n ≥ G ⊂ F . When U and G can be clearlyidentified by the context, we omit them. This will be in particular the case when G = F . • H is the set of R d -valued F -predictable processes φ = ( φ t ) t ∈ [0 ,T ] such that k φ k H := E (cid:20)Z T | φ t | dt (cid:21) < ∞ . • S is the set of real-valued optional processes φ = ( φ t ) t ∈ [0 ,T ] such that k φ k S := E [ess sup ≤ τ ≤ T | φ τ | )] < ∞ . K is the set of real-valued non decreasing RCLL and F -predictable processes K =( K t ) t ∈ [0 ,T ] with K = 0 and E [ K T ] < ∞ . • T denotes the set of F -stopping times τ such that τ ∈ [0 , T ] a.s. The notation E τ [ . ]stands for the conditional expectation given F τ , τ ∈ T . • For θ in T , T θ is the set of stopping times τ ∈ T such that θ ≤ τ ≤ T P -a.s. Let us introduce the new mathematical object.
Definition 2.1 (BSDEs with weak reflections) Given a measurable map
Ψ : [0 , T ] × R × Ω → U , with U ⊂ R ∪ {−∞} and µ ∈ L ( U, F τ ) , we say that ( Y, Z ) ∈ S × H is asupersolution of the BSDE with generator g : Ω × [0 , T ] × R × R d → R and weak reflectionsif for any ≤ t ≤ s ≤ T , Y t ≥ Y s + Z st g ( s, Y s , Z s ) ds − Z st Z s dW s (2.7) E τ [Ψ( θ, Y θ )] ≥ µ for all θ ∈ T τ . (2.8)We would like to emphasize that the terminology BSDEs with weak reflections is due to thefact that, given a stopping time τ ∈ T and a threshold µ ∈ L , the first composant of thesolution of the above BSDE, denoted here by ( Y t ), satisfies the condition E τ [Ψ( θ, Y θ )] ≥ µ for all θ ∈ T τ . The wellposedness of this BSDE is discussed in Remark 2.1.Throughout the paper, we assume that g satisfies Assumption 2.1 g is a measurable map from Ω × [0 , T ] × R × R d to R and g ( ., y, z ) is F -predictable, for each ( y, z ) ∈ R × R d . There exists a constant K g > and a randomvariable χ g ∈ L ( R + ) , such that | g ( t, , | ≤ χ g P − a.s. | g ( t, y , z ) − g ( t, y , z ) | ≤ K g ( | y − y | + | z − z | ) P − a.s. ∀ ( t, y i , z i ) ∈ [0 , T ] × R × R d , i = 1 , . We also recall the definition of the conditional g -expectation. Definition 2.2 (Conditional g -expectation) We recall that if g is a Lipschitz driverand if ξ is a square-integrable F T -measurable random variable, then there exists a uniquesolution ( X, π ) ∈ S × H to the following BSDE X t + Z Tt g ( s, X s , π s ) ds − Z Tt π s dW s for all t ∈ [0 , T ] a . s . or t ∈ [0 , T ] , the non-linear operator E gt,T : L ( F T ) L ( F t ) which maps a given terminalconditon ξ ∈ L ( F t ) to the first component at time t of the solution of the above BSDE(denoted by X t ) is called conditional g -expectation at time t . It is also well-knwon that thisnotion can be extended to the case where the (deterministic) terminal time T is replaced bya general stopping time τ ∈ T and t is replaced by a stopping time S such that S ≤ τ a.s. We now give the assumption on the map Ψ.
Assumption 2.2
For Leb × d P -a.e. ( t, ω ) ∈ [0 , T ] × Ω , the map y ∈ R Ψ( t, ω, y ) isnondecreasing, right-continuous, valued in [0 , ∪ {−∞} and its left-continuous inverse Φ( t, ω, · ) satisfies Φ : [0 , T ] × Ω × [0 , [0 , is measurable. By left-continuous inverse we mean the left-continuous map defined for ( t, ω ) fixed byΦ( t, ω, x ) := inf { y ∈ R , Ψ( t, ω, y ) ≥ x } which satisfies Φ( t, ω, Ψ( t, ω, x )) ≤ x ≤ Ψ( t, ω, Φ( t, ω, x )) . Remark 2.1
Let us discuss the wellposedness of the BSDE with weak reflections. Let ξ be a square integrable F T -measurable random variable such that E τ [ ξ ] = µ a.s. Due tothe martingale representation theorem, there exists β ∈ H such that M βT = ξ a.s., where M βT = µ + R Tτ β s dW s . The solution ( Y βt , Z βt ) of the reflected BSDE associated with the driver g and obstacle (Φ( t, M βt )) (which exists under the above assumptions) is a supersolution ofthe BSDE with weak reflections. Note that, due to the weak constraints, we do not haveuniqueness of the solution. We now introduce the set Θ( τ, µ ) of ( τ, µ )-initial supersolutions, which is defined asfollows: Θ( τ, µ ) := { Y τ : ( Y, Z ) is a supersolution of(2.7) and (2.8) } . The aim of this paper is to study the lower bound of the set Θ( τ, µ ), that is ess inf Θ( τ, µ ).We would like to emphasize once again the relation between this quantity and the priceof an American corresponding to an approximative hedging, under the risk constraint E τ [Ψ( θ, Y θ )] ≥ µ a.s., for all θ ∈ T τ . Our main purpose now is to show that we can reformulate the problem into an equivalent onewith ”strong” constraints, similar to the case of the partial hedging problem for Europeanoptions ( we refer to Bouchard, Elie, Touzi [4] in the Markovian framework or Bouchard,Elie, Reveillac [3] in the Non-Markovian setting).For this, let V τ,µ denote the set elements α ∈ H such that M τ,µ,α := µ + Z τ ∨ .τ α s dW s takes values in [0 , . (2.9)The main difficulty in our case is represented by the fact that, a priori, we can obtain anequivalent formulation in which the controlled martingale depends on the stopping time θ ,6hat is for each θ ∈ T τ there exists α θ ∈ H such that E τ [Φ( θ, Y θ )] ≥ µ is equivalent to Y θ ≥ Φ( θ, M α θ θ ) a.s.We see in the next Lemma that we can overcome this issue and obtain the existence ofa controlled martingale independent on the stopping time θ . Lemma 2.3
Let ( Y t ) be an optional process belonging to S and satisfying (2.7) - (2.8) , τ astopping time belonging to T and µ a F τ -measurable random variable. Then the condition E [Ψ( θ, Y θ ) |F τ ] ≥ µ for all θ ∈ T τ is equivalent to the existence of a control α ∈ V τ,µ suchthat Y θ ≥ Φ( θ, M τ,µ,α θ ) a.s. for all θ ∈ T τ . Proof.
For each σ ∈ T , we define the F σ -measurable random variable V ( σ ) := ess inf τ ∈T σ E [Ψ( τ, Y τ ) |F σ ] . (2.10)By classical results of the general theory of processes, the family ( V ( σ ) , σ ∈ T ) is asubmartingale family, which can be aggregated by an optional process ( V t ) admitting theMertens decomposition: V t := N t + A t + C t − , where N is a square integrable martingale, A is an increasing RCLL predictable process suchthat A = 0 and C is a right-continuous adapted process, purely discontinuous satisfying C − = 0.Let us first show the first implication, that is: E [Ψ( θ, Y θ ) |F τ ] ≥ µ for all θ ≥ τ implies the existence of a control α ∈ H such that Y θ ≥ Φ( θ, M ( τ,µ ) ,αθ ) for all θ ≥ τ. Since for all θ ∈ T τ we have E [Ψ( θ, Y θ ) |F τ ] ≥ µ a.s., we get that ess inf θ ≥ τ E [Ψ( θ, Y θ ) |F τ ] ≥ µ a.s. Hence, by using the definition of V (see (2.10)), we obtain V τ = N τ + A τ + C τ − ≥ µ a . s . (2.11)We now fix θ ≥ τ . We have Ψ( θ, Y θ ) = E [Ψ( θ, Y θ ) |F θ ] ≥ ess inf σ ≥ θ E [Ψ( σ, Y σ ) |F θ ] = V θ a . s . This observation together with (2.11) implyΨ( θ, Y θ ) ≥ N θ + A θ + C θ − = N τ + A τ + C τ − + Z θτ α s dW s + A θ − A τ + C θ − − C τ − . Using the above inequality, (2.11) and the fact the processes A and C are non-decreasing,we obtain Ψ( θ, Y θ ) ≥ M τ,µ,α θ a . s . By applying now the map Φ which is non-decreasing in its last variable, we finally derive Y θ ≥ Φ( θ, M τ,µ,α θ ) a . s . The second implication is trivial. (cid:3)
Let us show the following result, which will be crucial in the sequel.7 roposition 2.4
Fix τ ∈ T , µ ∈ L ([0 , , F τ ) . Then ( Y, Z ) ∈ S × H is a solution ofthe BSDE (2.7) - (2.8) if and only if ( Y, Z ) satisfies (2.7) and there exists α ∈ V τ,µ suchthat Y ν ≥ ess sup θ ∈T ν E gν,θ [Φ( θ, M τ,µ,αθ )] a.s. for all ν ∈ T τ . Proof.
Let (
Y, Z ) be a supersolution of BSDE (2.7)-(2.8). Then by Lemma 2.3, thereexists ˜ α ∈ V τ,µ such that for all θ ∈ T τ we have Ψ( θ, Y θ ) ≥ M τ,µ, ˜ αθ . We now define θ ˜ α :=inf { s ≥ τ, M τ,µ, ˜ αs = 0 } . Let us introduce the control ¯ α := ˜ α [0 ,θ ˜ α ] , which clearly belongs to V τ,µ . Let us fix ν ∈ T τ . One can remark that for all θ ∈ T ν we have Ψ( θ, Y θ ) ≥ M τ,µ, ¯ αθ a.s.The monotonocity of the map Φ and the above inequality imply that: Y θ ≥ Φ( θ, M τ,µ, ¯ αθ ) a . s . By the comparison theorem for BSDEs, we get that for all θ ∈ T ν , we have Y ν ≥ E gν,θ [Φ( θ, M τ,µ, ¯ αθ ]a.s. Now, by arbitrariness of θ ∈ T ν we finally obtain: Y ν ≥ ess sup θ ∈T ν E gν,θ [Φ( θ, M τ,µ, ¯ αθ ] a . s . (2.12)Let us show the converse implication. For all ν ∈ T τ , we have Y ν ≥ Φ( ν, M µ, ¯ αν ) a.s. Hencewe get Ψ( ν, Y ν ) ≥ M τ,µ, ¯ αν a.s. This implies that ( Y, Z ) satisfies (2.7) and (2.8). (cid:3)
Using the above results, we show in the following proposition how to relate the lower boundof the family Θ( τ, µ ) to the value of a stochastic control/optimal stopping game. To thisaim, we define the value function Y ( τ, µ ) := ess inf α ∈ V τ,µ ess sup θ ∈T τ E gτ,θ [Φ( θ, M τ,µ,αθ )] . (2.13) Proposition 2.5
We have ess inf Θ( τ, µ ) = Y ( τ, µ ) a.s. Proof.
Let Y τ ∈ Θ( τ, µ ). By Proposition 2.4, we obtain that Y τ ≥ ess sup θ ∈T τ E gτ,θ [Φ( θ, M τ,µ,αθ )] a.s.,which clearly implies that Y τ ≥ ess inf α ∈ V τ,µ ess sup θ ∈T τ E gτ,θ [Φ( θ, M τ,µ,αθ ))] = Y ( τ, µ ) a . s . By arbitrariness of Y τ , we derive that ess inf Θ( τ, µ ) ≥ Y ( τ, µ ) a.s.Conversely, we have that ess sup θ ∈T τ E gτ,θ [Φ( θ, M τ,µ,αθ )] belongs to Θ( τ, µ ), which leads toess sup θ ∈T τ E gτ,θ [Φ( θ, M τ,µ,αθ )] ≥ ess inf Θ( τ, µ ) a . s . By taking the essential infimum on α ∈ V τ,µ , the result follows. (cid:3) In the sequel, we assume that the map Φ is continuous with respect to t and m .We now introduce the nonlinear operator R ef g,ξ defined through the solution of a nonlinearreflected BSDE with driver g and lower obstacle ξ .8 efinition 2.6 (The nonlinear operator R ef g,ξ ) Let g be a Lipschitz driver and ξ aRCLL process belonging to S . Let L ξ,T be the set of random variables ζ included in L ( F T ) such that ζ ≥ ξ T a.s. Then there exists a unique solution ( Y, Z, A ) ∈ S × H × S to thefollowing lower reflected BSDE Y t = Y T + Z Tt f ( s, Y s , Z s ) ds − Z Tt Z s dW s + A T − A t for all t ∈ [0 , T ] a . s .Y t ≥ ξ t a . s . , ≤ t ≤ TY T = ζ a . s . Z T ( Y s − − ξ s − ) dA s = 0 a . s . For t ∈ [0 , T ] , the non-linear operator R ef g,ξt,T : L ξ,T
7→ L ξ,t is defined as follows: R ef g,ξt,T [ ζ ] := Y t , where Y is the first component at time t of the solution of the above Reflected BSDE. Thisnotion can be extended to the case where the (deterministic) terminal time T is replaced bya general stopping time τ ∈ T and t is replaced by a stopping time S such that S ≤ τ a.s.Remark 2.7 Note that, due to the flow property of reflected BSDEs, the nonlinear operator R ef g,ξ is consistent . Using the characterization of the first composant of the solution of a nonlinear reflectedBSDE as the value of an optimal stopping with nonlinear BSDEs, we obtain that Y ( τ, µ )can be rewritten as follows: Y ( τ, µ ) = ess inf α ∈ V τ,µ R ef g, Φ α τ,T [Φ( T, M τ,µ,αT )] , where Φ α corresponds to the obstacle process Φ( t, M τ,µ,αt ). In this section, we focus our study on Y ( τ, µ ) which is the lower bound of the set Θ( τ, µ ).For ease of notations, we fix m ∈ [0 ,
1] and set ( M t := M ,m ,αt , V ατ := { α ′ ∈ V τ,M ατ : α ′ = α dt × d P on [[0 , τ ]] } , V := V ,m and Y α ( τ ) := Y ( τ, M ατ ) for α ∈ V , t ∈ [0 , T ] and τ ∈ T . (3.14) Let us first recall the definition of a T -admissible system. Definition 3.1
A family S = { S ( τ ) , τ ∈ T } is admissible (or a T -system) if for all τ, τ ′ ∈ T ( S ( τ ) ∈ L ( F τ ) ,S ( τ ) = S ( τ ′ ) a.s. on { τ = τ ′ } . (3.15)9 emma 3.2 (Admissibility of the family ( Y α ( τ )) τ ∈T ) The family ( Y α ( τ )) τ ∈T is a square-integrable admissible family. Proof.
For each S ∈ T , Y α ( S ) is an F S -measurable square-integrable random variable,due to the definitions of the conditional g -expectation and of the essential supremum andessential infimum. Let S and S ′ be two stopping times in T . We set B := { S = S ′ } .We show that Y α ( S ) = Y α ( S ′ ) a.s. on B . Set θ B := θ B + T B c . We clearly have θ B ∈ T S ′ and moreover θ B = θ a.s. on B , for all θ ∈ T S . We also fix α ′ ∈ V αS ′ and set α ′ B := α [[0 ,S ]] + α ′ ]] S,T ]] B . Clearly α ′ B ∈ V αS and α ′ B = α ′ on ]] S ′ , T ]] on B . By using thefact that S = S ′ on B , as well as several properties of the g -expectation, we obtain B E gS,θ [Φ( θ, M α ′ B θ )] = B E gS ′ ,θ [Φ( θ, M α ′ B θ )] = E g B S ′ ,θ [ B Φ( θ, M α ′ B θ )] = E g B S ′ ,θ [ B Φ( θ B , M α ′ θ B )]= B E gS ′ ,θ [Φ( θ B , M α ′ θ B )] ≤ B ess sup θ ∈T S ′ E gS ′ ,θ [Φ( θ, M α ′ θ )] a . s . (3.16)By taking the essential supremum on θ ∈ T S and then the essential infimum on α ′ ∈ V αS ′ ,we get Y α ( S ) ≤ Y α ( S ′ ) a.s. By interchanging the roles of S and S ′ , the converse inequalityfollows by the same arguments. (cid:3) We now prove the existence of an optimizing sequence.
Lemma 3.3
Fix τ ∈ T , θ ∈ T τ , m ∈ L ([0 , , F τ ) and α ∈ V τ,µ . Then there exists a se-quence ( α ′ n ) ⊂ V θ,ατ,µ := { α ′ ∈ V τ,µ , α ′ [0 ,θ ) = α [0 ,θ ) } such that lim n →∞ ↓ R ef g, Φ α ′ n θ,T [Φ( T, M τ,m,α ′ n T )] = Y ( θ, M τ,m,α θ ) a.s. Proof. In order to prove the result, we have to show that the family { J ( α ′ ) := R ef g, Φ α ′ θ,T [Φ( T, M τ,m,α ′ T )] ,α ′ ∈ V θ,ατ,µ } is direct downward. Fix α ′ , α ′ ∈ V θ,ατ,µ and set˜ α ′ := α [0 ,θ ) + [ θ,T ] ( α ′ A + α ′ A c ) , where A := { J ( α ′ ) ≤ J ( α ′ ) } ∈ F θ , which implies that ˜ α ′ ∈ V θ,ατ,µ and, since A ∈ F θ , J ( ˜ α ′ ) = R ef g, Φ ˜ α ′ θ,τ [Φ( T, M τ,m,α ′ T ) A + Φ( T, M τ,m,α ′ T ) A c ]= A R ef g, Φ α ′ θ,τ [Φ( T, M τ,m,α ′ T )] + A c R ef g, Φ α ′ θ,τ [Φ( T, M τ,m,α ′ T )]= min( J ( α ′ ) , J ( α ′ )) . This gives the desired result. (cid:3)
Let us now introduce the notion of R ef g,ξ -submartingale system (resp. a R ef g,ξ -martingalesystem). Definition 3.4
An admissible family ( X ( τ ) , τ ∈ T ) is said to be a R ef g,ξ - submartingalefamily (resp. a R ef g,ξ - martingale family ) if for each τ ∈ T , X ( τ ) ∈ L ξ,τ and if, for all τ, σ ∈ T such that σ ∈ T τ a.s., X ( τ ) ≤ R ef g,ξτ,σ [ X ( σ )] a . s ., (resp .X ( τ ) = R ef g,ξτ,σ [ X ( σ )] ) a . s .
10e now proceed to show that for each α ∈ V , the family ( Y α ( τ ) , τ ∈ T ) is a R ef g, Φ α -submartingale family. This a direct consequence of the following dynamic programmingprinciple. Theorem 3.5 (
Dynamic programming principle ) The value family satisfies the fol-lowing dynamic programming principle: for all ( τ , τ , α ) ∈ T × T × V such that τ ≤ τ ,we have: Y α ( τ ) = ess inf α ′ ∈ V ατ R ef g, Φ α ′ τ ,τ [ Y α ′ ( τ )] a . s . (3.17) Proof.
Let us first show that Y α ( τ ) ≥ ess inf α ′ ∈ V ατ ess sup θ ∈T τ E gτ ,τ ∧ θ h Y α ′ ( τ ) θ ≥ τ + Φ( θ, M α ′ θ ) θ<τ i a . s . Fix α ′ ∈ V ατ . By the flow property for Reflected BSDEs we obtain: R ef g, Φ α ′ τ ,T h Φ( T, M α ′ T ) i = R ef g, Φ α ′ τ ,τ (cid:20) R ef g, Φ α ′ τ ,T [Φ( T, M α ′ T )] (cid:21) a . s . By the comparison theorem for Reflected BSDEs, we get: R ef g, Φ α ′ τ ,T h Φ( T, M α ′ T ) i ≥ R ef g, Φ α ′ τ ,τ h Y α ′ ( τ ) i a . s . By arbitrariness of α ′ ∈ V ατ , we finally obtain: Y α ( τ ) ≥ ess inf α ′ ∈ V ατ R ef g, Φ α ′ τ ,τ h Y α ′ ( τ ) i a . s . Conversely, we prove that Y α ( τ ) ≤ ess inf α ′ ∈ V ατ ess sup θ ∈T τ E gτ ,τ ∧ θ h Y α ′ ( τ ) θ ≥ τ + Φ( θ, M α ′ θ ) θ<τ i . (3.18)Let α n ∈ V α ′ τ such that: Y α ′ ( τ ) = lim n →∞ R ef g, Φ αn τ ,T (cid:2) Φ( T, M α n T ) (cid:3) a . s . The continuity of the reflected BSDEs with respect to its terminal condition gives: R ef g, Φ α ′ τ ,τ h Y α ′ ( τ ) i = lim n →∞ R ef g, Φ α ′ τ ,τ h R ef g, Φ αn τ ,T (cid:2) Φ( T, M α n T ) (cid:3)i a . s . We set: ˜ α ns := α s s<τ + α ns s ≥ τ . The two above relations and the consistency of the operator R ef g, Φ α ′ finally give: R ef g, Φ α ′ τ ,τ h Y α ′ ( τ ) i = lim n →∞ R ef g, Φ ˜ αn τ ,T (cid:2) Φ( T, M ˜ α n T ) (cid:3) ≥ Y α ( τ ) a . s . Now, by arbitrariness of α ′ ∈ V ατ , the result follows. (cid:3) .2 Aggregation results and E g -Mertens decomposition of R ef g,ξ -submartingales We now aim at proving that for each α ∈ V , the family ( Y α ( τ ) , τ ∈ T ) can be aggregatedby an optional process, that is it exists an optional process ( Y αt ) such that, for all stoppingtime τ ∈ T , it holds Y α ( τ ) = Y ατ a.s. The existence of such a process is in general a delicatequestion and, so far, it has only be addressed in the case of E g -(super)submartingales. Wethus show that this result can be extended to the case of R ef g, Φ -submartingales, with anoperator R ef g, Φ induced by the first composant of the solution of a lower reflected BSDE.
Theorem 3.6 (
Aggregation of the value family by an optional process ) For any α ∈ V , there exists an optional process ( Y αt ) which aggregates the family ( Y α ( τ ) , τ ∈ T ) , thatis Y α ( τ ) = Y ατ a.s., for all τ ∈ T . Proof.
Fix α ∈ V . Let ( τ n ) n ∈ N be a nondecreasing sequence of stopping times such that τ n ↓ τ a.s. The definition of Y α implies that Y α ( τ ) ≤ R ef g, Φ α τ,τ n [ Y α ( τ n )] a . s ., for all n ∈ N . (3.19)The nondecreasingness of the sequence ( τ n ) n together with the consistency of the operator R ef g, Φ α yield R ef g, Φ α τ,τ n [ Y α ( τ n )] = R ef g, Φ α τ,τ n +1 h R ef g, Φ α τ n +1 ,τ n [ Y α ( τ n )] i ≥ R ef g, Φ α τ,τ n +1 [ Y α ( τ n +1 )] a . s ., where the last inequality follows by (3.19).This implies that the sequence (cid:0) R ef g, Φ α τ,τ n [ Y α ( τ n )] (cid:1) n ∈ N is nondecreasing and thus it con-verges almost surely. Moreover, Y α ( τ ) ≤ lim n →∞ ↓ R ef g, Φ α τ,τ n [ Y α ( τ n )] a . s . (3.20)By Lebesgue’s theorem we have E [ Y α ( τ )] ≤ lim n →∞ ↓ E [ R ef g, Φ α τ,τ n [ Y α ( τ n )]] (3.21)Now, since lim sup n →∞ Y α ( τ n ) ≥ Φ( τ, M ατ ) a.s., we can apply the Fatou lemma for ReflectedBSDEs (see Proposition 3.13 in [12]). We therefore obtain E [ Y α ( τ )] ≤ E [lim sup n →∞ R ef g, Φ α τ,τ n [ Y α ( τ n )]] ≤ E [ R ef g, Φ α τ,τ [lim sup n →∞ Y α ( τ n )]] = E [lim sup n →∞ Y α ( τ n )] . (3.22)This implies that the family ( −Y α ( τ n )) n ∈ N satisfies E [ −Y α ( τ )] ≥ E [lim inf n →∞ ( −Y α ( τ n ))] . (3.23)Since the family ( −Y α ( τ ) , τ ∈ T ) is uniformly integrable, Theorem 12 in [8] gives theexistence of an optional process ( Y αt ) such that Y α ( τ ) = Y ατ a.s. for all τ ∈ T . Moreover,by the same Theorem, this process is right lower semicontinuous. (cid:3) Let us introduce the notion of strong R ef g,ξ - submartingale process.12 efinition 3.7 ( Strong R ef g,ξ - submartingale process) An optional process ( Y t ) sat-isfying Y σ ≥ ξ σ a.s. for all σ ∈ T and such that E [ess sup τ ∈T Y τ ] < ∞ is said to be astrong R ef g,ξ -submartingale if Y S ≤ R ef g,ξS,τ [ Y τ ] a.s. on S ≤ τ , for all S, τ ∈ T . We now show a E g -Mertens decomposition of r.l.s.c. R ef g,ξ -submartingales in the caseof a r.u.s.c. obstacle, which represents, to the best of our knowledge, a new result in theliterature. Moreover, our proof is simple, being based on some recent results on the theoryof optimal stopping with g -conditional expectations. Theorem 3.8 ( E g -Mertens decomposition of R ef g,ξ -submartingales) Let ( Y t ) be aright lower semicontinuous process such that E [ess sup τ ∈T ( Y τ ) ] < ∞ and ( ξ t ) a right uppersemicontinuous process such that E [ess sup τ ∈T ( ξ τ ) ] < ∞ . The process ( Y t ) is a strong R ef g,ξ -submartingale if and only if there exists two nondecreasing right-continuous predictableprocesses A, K ∈ S such that A = 0 and K = 0 , two nondecreasing right-continuousadapted purely discontinuous processes C, C ′ in S with C − = 0 and C ′ − = 0 and aprocess Z ∈ H such that a.s. for all t ∈ [0 , T ] , Y t = Y T + Z Tt g ( s, Y s , Z s ) ds + A T − A t + C T − − C t − − K T + K t − C ′ T − + C ′ t − , (3.24) Y t ≥ ξ t a . s ., ≤ t ≤ T. Z T ( Y s − − ξ s − ) dA s = 0 a . s . ; ( Y τ − ξ τ )( C τ − C τ − ) = 0 a . s . for all τ ∈ T ; dA t ⊥ dK t ; dC t ⊥ dC ′ t . (3.25) Proof.
Fix S ∈ T . Since ( Y t ) is a strong R ef g,ξ -submartingale, we derive that for each τ ∈ T S , we have Y S ≤ R ef g,ξS,τ [ Y τ ] a.s. By definition of the operator R ef g,ξ , thus we have Y S ≤ ess sup S ′ ∈T S E gS,S ′ ∧ τ ( Y τ S ′ ≥ τ + ξ S ′ S ′ <τ ) . By arbitrariness of τ ∈ T , hence we get Y S ≤ ess inf τ ∈T S ess sup S ′ ∈T S E gS,S ′ ∧ τ ( Y τ S ′ ≥ τ + ξ S ′ S ′ <τ ) a . s . (3.26)Now, one can remark that we have Y S = ess sup S ′ ∈T S E gS,S ∧ S ′ ( Y S S ′ ≥ S + ξ S ′ S>S ′ ) a . s . As S ∈ T S , we deduce: Y S ≥ ess inf τ ∈T S ess sup S ′ ∈T S E gS,τ ∧ S ′ ( Y τ S ′ ≥ τ + ξ S ′ τ>S ′ ) a . s . (3.27)The inequalities (3.26) and (3.27) allow to conclude that Y S = ess inf τ ∈T S ess sup S ′ ∈T S E gS,S ∧ S ′ ( Y τ S ′ ≥ τ + ξ S ′ τ>S ′ ) a . s . From the caracterization theorem of the solution of a DRBSDE (associated with twoobstacles supposed to be r.l.s.c., resp. r.u.s.c.) as the value function of a Generalized Dynkin13ame (that is, ¯ Y S = ess inf τ ∈T S ess sup σ ∈T S E gS,τ ∧ σ [ ξ τ τ<σ + ζ σ σ ≤ τ ], where ¯ Y is the first composantof the solution of the DRBSDE with driver g and obstacles ( ξ t ) and ( ζ t ), see Theorem 4.5in [16]), we derive that the process ( Y t ) coincides with the solution of the doubly reflectedBSDE associated with obstacles ( Y t ) and ( ξ t ). The result follows.Let us now show the converse implication.The reflected BSDE (3.24) can be seen as a reflected BSDE associated to the generalizeddriver f ( t, ω, y, z ) dt − dK t − dC ′ t − .Fix τ ∈ T S . Using the flow property for reflected BSDEs and their representation asthe value function of an optimal stopping problem, we get Y S = ess sup S ′ ∈T S E g − dK − dC ′ S,S ′ ∧ τ [ Y τ τ ≤ S ′ + ξ S ′ S ′ <τ ] a . s . (3.28)Using the comparison theorem for BSDEs with generalized driver , we deduce that Y S ≤ ess sup S ′ ∈T S E gS,S ′ ∧ τ [ Y τ τ ≤ S ′ + ξ S ′ S ′ <τ ] a . s ., (3.29)which implies that Y S ≤ R ef g,ξS,τ [ Y τ ] a . s ., for all τ ∈ T S . (cid:3) We now show the existence of a RCLL process which aggregates the value family ( Y α ) . Theorem 3.9 (
Existence of a RCLL aggregator process ) For any α ∈ V , there ex-ists a RCLL process ( Y αt ) which aggregates the family ( Y α ( τ ) , τ ∈ T ) , that is Y α ( τ ) = Y ατ a.s., for all τ ∈ T . Proof.
Fix α ∈ V . By Theorem 3.6, we get the existence of an optional process ( Y αt )that aggregates the family ( Y α ( τ ) , τ ∈ T ) and satisfies E [ess sup τ ∈T ( Y ατ ) ] < ∞ . Recall that,by Theorem 3 .
5, the process ( Y αt ) is a R ef g, Φ α -submartingale. We can thus use Theorem3 .
8, which shows that ( Y αt ) admits a E g -Mertens decomposition, giving the existence of itsleft and right limits.We thus define the process:( Y αt ) + := lim s ∈ ( t,T ] ↓ t Y αs , t ∈ [0 , T ] . (3.30)In order to show that the process Y α is indistinguishable of a RCLL process, we have toprove that ( Y α ) + τ = Y ατ a . s ., for all τ ∈ T . (3.31)Let us introduce ( τ n ) n ∈ N , a decreasing sequence of stopping times with values in [0 , T ] suchthat τ n ↓ τ a.s. as n → + ∞ . By the definition of the process ( Y α ) + , we have( Y α ) + τ = lim n →∞ Y ατ n a . s . (3.32)14he inequality ( Y α ) + τ ≥ ess inf α ′ ∈ V ατ R ef g, Φ α ′ τ,θ h Φ( θ, M α ′ θ ) i = Y ατ is clear by (3.20) and thecontinuity of the reflected BSDEs with respect to terminal time and terminal condition.It remains to show that ( Y α ) + τ ≤ ess inf α ′ ∈ V ατ R ef g, Φ α ′ τ,T h Φ( T, M α ′ T ) i = Y ατ a . s . Fix α ′ ∈ V ατ and set λ n := (cid:18) M ατ n M α ′ τ n ∧ − M ατ n − M α ′ τ n (cid:19) { M ατn / ∈{ , }} ∈ [0 , . We set α ′ n := α [0 ,τ n ) + λ n α ′ [ τ n ,T ] . This implies that α ′ n belongs to V ατ n .Now, relation (3.32) together with the F τ -measurability of lim n →∞ Y ατ n and the continuityof BSDEs with respect to the terminal time and terminal condition give:( Y α ) + τ ≤ E gτ,τ [ lim n →∞ Y ατ n ] = lim n →∞ E gτ,τ n [ Y ατ n ] a . s . (3.33)By the optimal stopping theory, there exists an optimal stopping time ˆ θ n ∈ T τ n for theoptimal stopping problem ess sup θ ∈T τn E gτ n ,θ h Φ( θ, M α ′ n θ ) i . We thus derive E gτ,τ n [ Y ατ n ] ≤ E gτ,τ n h ess sup θ ∈T τn E gτ n ,θ h(cid:0) θ, M α ′ n θ ) (cid:3)i = E gτ,τ n h E gτ n , ˆ θ n h Φ(ˆ θ n , M α ′ n ˆ θ n ) ii = E gτ, ˆ θ n h Φ(ˆ θ n , M α ′ n ˆ θ n ) i a . s ., where the first inequality follows by admissibility of the control α ′ n . Furthermore, we get E gτ, ˆ θ n h Φ(ˆ θ n , M α ′ n ˆ θ n ) i = E gτ, ˆ θ n h Φ(ˆ θ n , M α ′ n ˆ θ n ) i − E gτ, ˆ θ n h Φ(ˆ θ n , M α ′ ˆ θ n ) i + E gτ, ˆ θ n h Φ(ˆ θ n , M α ′ ˆ θ n ) i . (3.34)Since ˆ θ n ∈ T τ n ⊂ T τ , we have E gτ, ˆ θ n h Φ(ˆ θ n , M α ′ n ˆ θ n ) i ≤ E gτ, ˆ θ n h Φ(ˆ θ n , M α ′ n ˆ θ n ) i − E gτ, ˆ θ n h Φ(ˆ θ n , M α ′ ˆ θ n ) i + ess sup θ ∈T τ E gτ,θ h Φ( θ, M α ′ θ ) i a . s . (3.35)Now, by using the a priori estimates with BSDEs we have: E (cid:20)(cid:12)(cid:12)(cid:12) E gτ, ˆ θ n h Φ(ˆ θ n , M α ′ n ˆ θ n ) i − E gτ, ˆ θ n h Φ(ˆ θ n , M α ′ ˆ θ n ) i(cid:12)(cid:12)(cid:12) (cid:21) ≤ C E (cid:20)(cid:16) Φ(ˆ θ n , M α ′ n ˆ θ n ) − Φ(ˆ θ n , M α ′ ˆ θ n ) (cid:17) (cid:21) (3.36) ≤ C E " sup ≤ t ≤ T (cid:16) Φ( t, M α ′ n t ) − Φ( t, M α ′ t ) (cid:17) . The convergence M α ′ n T → M α ′ T when n → ∞ , together with Doob inequality, the uniformcontinuity of Φ and Lebesgue’s Theorem implies that E " sup ≤ t ≤ T (cid:16) Φ( t, M α ′ n t ) − Φ( t, M α ′ t ) (cid:17) → n → ∞ . (3.37)Using (3.34) , (3.35) , (3.36) , (3.37), taking the limit in n and then the essential infimum on α ′ ∈ V ατ , the result follows. (cid:3) .3 A Backward SDE representation of the value process In this subsection, we provide a Backward SDE representation of the value process ( Y αt ),for each α ∈ V . In order to do this, we first establish a Doob-Meyer decomposition of thevalue process ( Y αt ). Theorem 3.10 (Doob-Meyer decomposition of the value process)
For each α ∈ V ,the process ( Y αt ) admits the following Doob-Meyer decomposition: there exists Z α ∈ H andtwo RCLL predictable processes A α ∈ K and K α ∈ K with A α = 0 and K α = 0 such that Y αt = Φ( T, M αT ) + Z Tt g ( s, Y αs , Z αs ) ds + A αT − A αt − K αT + K αt , (3.38) Y αt ≥ Φ( t, M αt ) a . s ., ≤ t ≤ T. Z T ( Y αs − − Φ( s − , M αs − ) d A αs = 0 a . s .d A αt ⊥ d K αt . (3.39)Proof. By Theorems 3 . .
5, we obtain that ( Y αt ) is a r.l.s.c. R ef g, Φ α -submartingale.We can thus apply Theorem (3.8) and obtain the existence of the processes ( Z α , A α , K α , C α , C ′ α ) ∈ H × ( K ) such that (3.24) holds. Due to this equation, we have ∆ C αt − ∆ C ′ αt = − ( Y αt + − Y αt ). Since by the previous Theorem the process ( Y αt ) is right-continuous, theprocess C = 0. The result follows. (cid:3) We now show the following Backward SDE representation of the value process.
Theorem 3.1 (Representation of the value process)
There exists a family ( Z α , K α , A α ) α ∈ V ⊂ H × K × K such that, for all α ∈ V ,we have Y αt = Φ( T, M αT ) + R Tt g ( s, Y αs , Z αs ) ds − R Tt Z αs dW s + K αt − K αT − A αt + A αT , ≤ t ≤ T ; Y αt ≥ Φ( t, M αt ) a . s ., ≤ t ≤ T ; R T (cid:0) Y αs − − Φ( s, M αs − ) (cid:1) d A αs = 0 a.s. ; d A α ⊥ d K α ;ess inf α ′ ∈ V ατ E [ R Tτ M τ,α ′ s d ( A α ′ s − A α ′ s + K α ′ s )] = 0 a . s ., for all τ ∈ T ( Y α , Z α , K α , A α ) [[0 ,τ ]] = ( Y ¯ α , Z ¯ α , K ¯ α , A ¯ α ) [[0 ,τ ]] , ∀ τ ∈ T , α ∈ V ¯ ατ where for each τ ∈ T and α ′ ∈ V ατ , the process ( M τ,α ′ t ) t ≥ τ represents the linearization pro-cess associated with ( Y α ′ t ) and ( Y α ′ t ) satisfying M τ,α ′ τ = 1 a.s., with ( Y α ′ t , Y α ′ t , Y α ′ t ) the solu-tion of the reflected BSDE with driver g and obstacle Φ( t, M α ′ t ) . Moreover, ( Y α , Z α , A α , K α ) α ∈ V is the unique family satisfying the above BSDE. Proof.
First note that for ( α, τ ) ∈ V × T , we have V α ′ · = V α · on [0 , τ ] for α ′ ∈ V ατ . Thedefinition of Y implies that Y α [0 ,τ ] = Y α ′ [0 ,τ ] for α ′ ∈ V ατ . Fix τ ∈ T and α ∈ V . ByTheorem 3.10, we get the existence of ( Z α , K α , A α ) such that Y αt = Φ( T, M αT ) + R Tt g ( s, Y αs , Z αs ) ds + A αT − A αt − K αT + K αt − R Tt Z αs dW s , Y αt ≥ Φ( t, M αt ) a . s . ≤ t ≤ T R T ( Y αs − − Φ( s − , M αs − )) d A αs = 0; d A αs ⊥ d K αs . (3.40)16y the uniqueness of the representation of a semimartingale, we derive that ( Y α , Z α , K α , A α ) [[0 ,τ ]] =( Y ¯ α , Z ¯ α , K ¯ α , A ¯ α ) [[0 ,τ ]] , ∀ τ ∈ T , α ∈ V ¯ ατ . It remains to show the minimality condition sat-isfied by the process A α ′ − A α ′ − K α ′ .To do so, let us first consider an arbitrary control ¯ α ∈ V ατ and ( Y ¯ α , Z ¯ α , A ¯ α ) the solutionof the following reflected BSDE: Y ¯ αt = Φ( T, M ¯ αT ) + R Tt g ( s, Y ¯ αs , Z ¯ αs ) ds − R Tt Z ¯ αs dW s + A ¯ αT − A ¯ αt ,Y ¯ αt ≥ Φ( t, M ¯ αt ) a . s . ≤ t ≤ T R T ( Y ¯ αs − − Φ( s − , M ¯ αs − )) dA ¯ αs = 0 . We now define the linearization process M τ, ¯ α such that M ¯ ατ = 1 and M ¯ αt = exp (cid:18)R tτ β s dW s + R tτ ( λ s − β s ds (cid:19) , where λ s := g ( s, Y ¯ αs , Z ¯ αs ) − g ( s, Y ¯ αs , Z ¯ αs ) Y ¯ αs − Y ¯ αs {Y ¯ αs − Y ¯ αs =0 } ; β s := g ( s, Y ¯ αs , Z ¯ αs ) − g ( s, Y ¯ αs , Z ¯ αs ) |Z ¯ αs − Z ¯ αs | ( Z ¯ αs − Z ¯ αs ) {Z ¯ αs − Z ¯ αs =0 } . Using a classical linearization procedure, we obtain: Y ¯ ατ − Y ¯ ατ = E τ [ Z Tτ M τ, ¯ αs ( dA ¯ αs − d A ¯ αs + d K ¯ αs )] a . s . (3.41)We take now the ess inf on ¯ α ∈ V ατ and using the definition of the value function Y ¯ α , theminimality condition follows.We now show the uniqueness of the family. Let ( ˜ Y α , ˜ Z α , ˜ K α , ˜ A α ) be a solution of(3.40). Notice that, by using the comparison theorem between BSDEs with generalizeddriver and the characterization of the solution of a reflected BSDE as the solution of anoptimal stopping problem, we deduce that Y αt = ess inf α ′ ∈ V αt ess sup θ ∈T t E gt,θ [Φ( θ, M α ′ θ )] ≥ ess sup θ ∈T t E g − dKt,θ [Φ( θ, M α ′ θ )] = ˜ Y αt a.s. (3.42)By using the same linearization procedure, we obtain Y α ′ τ − ˜ Y α ′ τ = E [ Z Tτ M τ,α ′ s d ( A α ′ s − ˜ A α ′ s + ˜ K α ′ s )] a . s . (3.43)The minimality condition implies that ˜ Y ατ = ess inf α ′ ∈ V ατ Y α ′ τ a.s. Hence, the result follows. (cid:3) Remark 3.2
Note that since in general the process A α − A α − K α is not non-decreasing,we cannot reduce to a formulation only involving A α , A α and K α , as in the case of non-reflected BSDEs with weak terminal condition. We point out that in the case when Φ = −∞ and thus there is no reflection, the processes A α and A α become 0 for all α ∈ V . Hencethe minimality condition is indeed equivalent to ess inf α ′ ∈V ατ E τ h K α ′ T − K α ′ τ i = 0 a . s . (3.44)17 BSDEs with weak reflections and a related game problem
In this section, we study a related game problem. We show that, given a threshold pro-cess ( M αt ), the minimal initial process Y α corresponds to the value of an optimal stoppingproblem. More precisely, we provide some conditions under which one can interchange theinf-sup and obtain the existence of a saddle point. This problem is in general non trivial,and the additional complexity in our case is due to the presence of the control α in theobstacle Φ( t, M αt ).Let S ∈ T and α ∈ V . Define the first value function at time S as Y α ( S ) := ess inf α ′ ∈ V αS ess sup τ ∈T S E gS,τ [Φ( τ, M α ′ τ )] . (4.45)and the second value function at time S as Y α ( S ) := ess sup τ ∈T S ess inf α ′ ∈ V αS E gS,τ [Φ( τ, M α ′ τ )] . (4.46)By definition, we say that there exists a value function at time S for the game problemif Y α ( S ) = Y α ( S ) a.s.We recall the definition of a S − saddle point . Definition 4.1 (S-saddle point)
Let S ∈ T . A pair ( τ ∗ S , α ∗ S ) ∈ T S × V is called a S -saddle point if(i) Y α ( S ) = Y α ( S ) a.s.(ii) The essential infimum in (4.45) is attained at α ∗ S (iii) The essential supremum in (4.46) is attained at τ ∗ S . Let us now give the main result of this section.
Theorem 4.2
1. Assume that g ( t, ω, y, z ) ≥ , for all ( t, ω, y, z ) ∈ [0 , T ] × Ω × R × R d and suppose that Φ is increasing with respect to t and convex with respect to m . Then thegame problem admits a value function, that is Y α ( S ) = Y α ( S ) a . s ., for all S ∈ T . (4.47)
2. Assume that g ( t, ω, y, z ) ≤ , for all ( t, ω, y, z ) ∈ [0 , T ] × Ω × R × R d and suppose that Φ is decreasing with respect to t and concave with respect to m . Then the game problemadmits a value function, that is Y α ( S ) = Y α ( S ) a . s ., for all S ∈ T . (4.48)
3. Under the additional assumption that g is convex with respect to ( y, z ) , there existsa S -saddle point for the game problem (4.48) in the sense of Definition 4.1. roof.
1. Fix S ∈ T . First note thatess sup θ ∈T S ess inf α ′ ∈ V αS E gS,θ h Φ( θ, M α ′ θ )] i ≤ ess inf α ′ ∈ V αS ess sup θ ∈T S E gS,θ h Φ( θ, M α ′ θ ) i a . s . It remains to show the converse inequality.Fix θ ∈ T S and α ′ ∈ V αS . By the flow property for nonlinear BSDEs, we get E gS,T [Φ( T, M α ′ T )] = E gS,θ [ E gθ,T [Φ( T, M α ′ T )]] a . s . Applying the comparison theorem for BSDEs and using the assumption on the driver g , wederive E gS,θ h E gθ,T [Φ( T, M α ′ T )] i ≥ E gS,θ h E [Φ( T, M α ′ T ) |F θ ] i a . s . (4.49)The above relation, together with the properties of the map Φ and the conditional Jenseninequality implies that E gS,θ h E [Φ( T, M α ′ T ) |F θ ] i ≥ E gS,θ h E [Φ( θ, M α ′ T ) |F θ ] i ≥ E gS,θ h Φ( θ, E [ M α ′ T |F θ ]) i a . s . (4.50)The martingale property of M α ′ implies that E gS,θ h Φ( θ, E [ M α ′ T |F θ ]) i = E gS,θ h Φ( θ, M α ′ θ ) i a . s . (4.51)Combining (4.50) and (4.51), we get E gS,T h Φ( T, M α ′ T )] i ≥ E gS,θ h Φ( θ, M α ′ θ ) i a . s . By taking first the essential suprema on θ ∈ T S and then the essential infima on α ′ ∈ V αS ,it follows that ess inf α ′ ∈ V αS E gS,T [Φ( T, M α ′ T )] ≥ ess inf α ′ ∈ V αS ess sup θ ∈T S E gS,θ h Φ( θ, M α ′ θ ) i a . s . We clearly have ess inf α ′ ∈ V αS E gS,T [Φ( T, M α ′ T )] ≤ ess sup θ ∈T S ess inf α ′ ∈ V αS E gS,θ [Φ( θ, M α ′ θ )] a . s . The last two inequalities allow to conclude thatess sup θ ∈T S ess inf α ′ ∈ V αS E gS,θ h Φ( θ, M α ′ θ )] i ≥ ess inf α ∈ V αS ess sup θ ∈T S E gS,θ h Φ( θ, M α ′ θ ) i a . s . The result follows. (cid:3)
Remark 4.3
We emphasize that the above results still hold under different assumptionson the map Φ . Indeed, in the case of a positive driver g , one could consider the function Φ of the form Φ( t, ω, m ) = m + h ( S t ) , with S a submartingale process and h a convexfunction. In the case of a negative driver g , the proof still works for a function Φ of theform Φ( t, ω, m ) = m + h ( S t ) , with S a supermartingale process and h a concave function.
19. For the proof of this point, we mainly use the same ideas as for the previous proof. Forsake of clarity, we give it below. Fix S ∈ T . Notice thatess sup θ ∈T S ess inf α ′ ∈ V αS E gS,θ h Φ( θ, M α ′ θ ) i ≤ ess inf α ′ ∈ V αS ess sup θ ∈T S E gS,θ h Φ( θ, M α ′ θ ) i a . s . Let us show the converse inequality.Fix ϑ ∈ T S and α ′ ∈ V αS . By the martingale property of M α ′ , we deduce E gS,S [Φ( S, M α ′ S )] = E gS,S [Φ( S, E [ M α ′ ϑ |F S ])] a . s . (4.52)Using (4.52), the properties of the function Φ and the conditional Jensen inequality, wederive E gS,S [Φ( S, M α ′ S )] ≥ E gS,S [ E [Φ( ϑ, M α ′ ϑ ) |F S ]] a . s . The assumption on the driver g and the comparison theorem for BSDEs lead to E gS,S [ E [Φ( ϑ, M α ′ ϑ ) |F S ]] ≥ E gS,ϑ [Φ( ϑ, M αϑ )] a . s . By arbitrariness of ϑ ∈ T S , we have E gS,S [Φ( S, M α ′ S )] ≥ ess sup ϑ ∈T S E gS,ϑ [Φ( ϑ, M α ′ ϑ )] a . s . Since the above inequality holds for any α ′ ∈ V αS , we obtainess inf α ′ ∈ V αS E gS,S [Φ( S, M α ′ S )] ≥ ess inf α ′ ∈ V αS ess sup ϑ ∈T S E gS,ϑ [Φ( ϑ, M α ′ ϑ )] a . s . (4.53)Since S ∈ T S we deduce thatess inf α ′ ∈ V αS E gS,S [Φ( S, M α ′ S )] ≤ ess sup ϑ ∈T S ess inf α ′ ∈ V αS E gS,ϑ [Φ( ϑ, M α ′ ϑ )] a . s . (4.54)From the last two inequalities we deduceess sup ϑ ∈T S ess inf α ′ ∈ V αS E gS,ϑ h Φ( ϑ, M α ′ ϑ )] i ≥ ess inf α ′ ∈ V αS E gS,S [Φ( S, M α ′ S )] ≥ ess inf α ′ ∈ V αS ess sup ϑ ∈T S E gS,ϑ [Φ( ϑ, M αϑ )] a . s . (4.55)The result follows.3. We now show the existence of a S -saddle point, under the additional assumptionthat g is convex with respect to ( y, z ), that is Assumption 4.3 holds. Assumption 4.3
For all ( λ, m , m , t, y , y , z , z ) ∈ [0 , × [0 , × [0 , T ] × R × ( R d ) , g ( t, λy + (1 − λ ) y , λz + (1 − λ ) z ) ≤ λg ( t, y , z ) + (1 − λ ) g ( t, y , z ) a . s .
20e start by proving the existence of an optimal control α ∗ S for Problem 4.45. By Lemma3.3 , there exists a sequence of controls ( α n ) n belonging to V αS such that Y α ( S ) = lim n →∞ ↓ ess sup θ ∈T S E gS,θ [Φ( θ, M α n θ )] a . s . (4.56)As the sequence ( M α n T ) n is bounded in [0 , λ ni ) i ≥ n with P i ≥ n λ ni = 1, such that only a finite number of λ ni do not vanish, foreach n , and such that the sequence of convex combinations ( ˜ M nT ) n given by˜ M nT := X i ≥ n λ ni M α i T (4.57)converges a.s. to some ¯ M T . By dominated convergence, the convergence holds in L , inparticular E [ ¯ M T ] = m o and the martingale representation theorem gives the existence of acontrol ¯ α such that ¯ M T = M m , ¯ αT . Due to the fact that ( ˜ M nT ) and ¯ M T are martingales, weobtain that, for all θ ∈ T S , ˜ M nθ = P i ≥ n λ ni M α i θ a.s.Moreover, since Φ and g are convex, we have X i ≥ n λ ni E gτ,θ [Φ( θ, M α i θ )] ≥ E gτ,θ [Φ( θ, ˜ M nθ )] a . s . (4.58)We thus obtain that Y n ( S ) := X i ≥ n λ ni ess sup θ ∈T S E gS,θ [Φ( θ, M α i θ )] ≥ ess sup θ ∈T S X i ≥ n λ ni E gS,θ [Φ( θ, M α i θ )] ≥ ess sup θ ∈T S E gS,θ [Φ( θ, ˜ M nθ )] a . s . (4.59)Then (4.56) implies that Y n ( S ) → Y α ( S ) a . s . Let us now show thatess sup θ ∈T S E gS,θ [Φ( θ, ˜ M nθ )] → ess sup θ ∈T S E gS,θ [Φ( θ, ¯ M θ )] a . s . (4.60)The a priori estimates on BSDEs give: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ess sup θ ∈T S E gS,θ [Φ( θ, ˜ M nθ )] − ess sup θ ∈T S E gS,θ [Φ( θ, ˆ M θ )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ess sup θ ∈T S (cid:12)(cid:12)(cid:12) E gS,θ [Φ( θ, ˜ M nθ )] − E gS,θ [Φ( θ, ˆ M θ )] (cid:12)(cid:12)(cid:12) ≤ C ess sup θ ∈T S E S (cid:20)(cid:16) Φ( θ, ˜ M nθ ) − Φ( θ, ¯ M θ ) (cid:17) (cid:21) ≤ C E S " sup ≤ t ≤ T (cid:16) Φ( t, ˜ M nt ) − Φ( t, ¯ M t ) (cid:17) a . s ., with C a constant depending on T and the Lipschitz constant of the driver g .The Doob maximal inequality together with the uniform continuity of Φ with respectto t and m imply the convergence to 0, up to a subsequence, of the RHS term of the aboveinequality. Hence, we obtain (4.60). From (4.59) and (4.60) we derive that ¯ α is an optimalcontrol. (cid:3)
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