A dynamic dual representation of the buyer's price of American options in a nonlinear incomplete market
aa r X i v : . [ q -f i n . M F ] J un A dynamic dual representation of the buyer’s price ofAmerican options in a nonlinear incomplete market Roxana Dumitrescu ∗ First Version: 8 January 2019This Version: 1 June 2019
Abstract
In this paper we study the problem of nonlinear pricing of an American option witha right-continuous left-limited (RCLL) payoff process in an incomplete market with de-fault, from the buyer’s point of view. We show that the buyer’s price process can berepresented as the value of a stochastic control/optimal stopping game problem withnonlinear expectations, which corresponds to the maximal subsolution of a constrainedreflected Backward Stochastic Differential Equation (BSDE). We then deduce a nonlin-ear optional decomposition of the buyer’s price process. To the best of our knowledge,no dynamic dual representation (resp. no optional decomposition) of the buyer’s priceprocess can be found in the literature, even in the case of a linear incomplete marketand brownian filtration. Finally, we prove the ”infimum” and the ”supremum” in thedefinition of the stochastic game problem can be interchanged. Our method relies onnew tools, as simultaneous nonlinear Doob-Meyer decompositions of processes whichhave a Y ν -submartingale property for each admissible control ν . Keywords:
American options, buyer’s price, incomplete markets, nonlinear pricing, re-flected BSDEs with constraints, nonlinear optional decomposition
AMS MSC 2010:
Primary 60G40; 93E20; 60H30, Secondary 60G07; 47N10.
The aim of this paper is to study the problem of nonlinear pricing of an American option withRCLL payoff process ( ξ t ) in an incomplete market with default, from the buyer’s point of view .The financial market consists of one riskless asset and one risky asset, whose dynamics are ∗ Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, United Kingdom,email: [email protected] N defined by N t = ϑ ≤ t , where ϑ represents a defaulttime. The nonlinearity is incorporated in the wealth dynamics and allows to take into accountthe market imperfections. Moreover, the market is considered to be incomplete , in the sensethat it is not always possible to replicate the payoff of an European option by a controlledportfolio.The case of a nonlinear complete market has already been addressed in e.g. [20], [23], [35]within different frameworks. As shown in these papers, the seller’s price (resp. the buyer’sprice ) can be written in terms of an optimal stopping problem with nonlinear expectations,which is further related to the solution of a given reflected BSDE.The option pricing under incompleteness has been studied by many authors in the case of linear markets, using different techniques (see e.g. [1], [9], [25], [26], [28],[30], [32], [31]). Thestochastic control approach consists in embedding the initial market into an auxiliary familyof markets { M ν , ν ∈ D } (with D the set of admissible controls). The seller’s price (whichcorresponds to the minimal initial capital which allows the seller to be super-hedged) canbe expressed in terms of a mixed stochastic control/optimal stopping problem, and shownto admit an optional decomposition. The simultaneous Doob-Meyer decompositions, validunder a whole family of probability measures { Q ν , ν ∈ D } , play an important role in theanalysis (see e.g. [9], [32], [31], [36]). On its turn, the buyer’s price (defined as the supremumof initial prices which allow the buyer to select an exercise time τ and a portfolio strategy ϕ so that he/she is superhedged) can be represented as a stochastic control/optimal stoppinggame problem. The roles of the buyer and of the seller are asymmetric in the context ofAmerican options, and this asymmetry reflects itself in the definitions of the prices and inthe mathematical treatment of the control problems.In this paper, we show that the buyer’s price process in our nonlinear incomplete marketcan be characterized through the value family ¯ Y ( S ) := essinf ν ∈ D ess sup τ ∈T S E νS,τ ( ξ τ ) , (1.1)with T S the set of stopping times greater than S and E ν the nonlinear conditional expec-tation associated with a given driver f ν . Using tools from the control theory, we obtain a dynamic dual representation of the buyer’s price process in terms of the maximal subsolu-tion of a constrained reflected backward stochastic differential equation. From this dynamiccharacterization, we easily deduce a nonlinear optional decomposition. To the best of ourknowledge, no dynamic dual representation (and no optional decomposition) of the buyer’sprice process can be found in the previous literature, even in the case of linear markets andbrownian filtration, and this result is the main contribution of the paper. A key ingredientof our approach is represented by the simultaneous nonlinear Doob-Meyer decompositionsof the value process which aggregates the value family given by (1.1), which is shown to bea Y ν -submartingale for all ν ∈ D , where the nonlinear operator Y ν is defined through theunique solution of a reflected BSDE with obstacle process ( ξ t ) and driver f ν . Our methodseems to be completely new for the study of game problems written in the form (1.1). Using2he dynamic dual representation of the buyer’s price process , we also show that the ”in-fimum” and the ”supremum” in (1.1) can be interchanged. We would like to emphasizethat, due to the control/optimal stopping game aspect of the problem, the proofs are quiteinvolved and require fine techniques of the general theory of stochastic processes.The paper is organized as follows: in Section 2, we introduce the financial market model,as well as some notation and assumptions. In Section 3.1, we give some first properties ofthe value family given by (1.1) and in particular show that ( ¯ Y ( S )) is the greatest familysatisfying the Y ν -submartingale property for each ν ∈ D . In Section 3.2., we provide adetailed analysis of Y g -submartingale families/processes (with g a general nonlinear driver).Section 3.3 is devoted to the simultaneous nonlinear Doob-Meyer decompositions of processeswhich have the Y ν -submartingale property for each ν , leading to the representation of suchprocesses as the maximal subsolution of a constrained reflected BSDE. A nonlinear optionaldecomposition is deduced. In Section 3.4. we apply these results to the family value ¯ Y ( S ).In Section 3.5., we prove the dynamic dual representation of the buyer’s price process interms of the maximal subsolution of a constrained reflected BSDE. Finally, we show that( ¯ Y ( S )) corresponds to the buyer’s price process and that the ”infimum” and ”supremum”in (1.1) can be interchanged. We consider a financial market M that consists of one risk-free asset whose price process S = ( S t ) ≤ t ≤ T satisfies dS t = S t r t dt (2.1)and one risky asset with price process S = ( S t ) ≤ t ≤ T which evolves according to the equation dS t = S t − (cid:18) µ t dt + σ t dW t + β t dM t (cid:19) . (2.2)Here, W is a one-dimensional standard Brownian motion and M represents the compen-sated martingale associated with a jump process N given by N t = ϑ ≤ t for any t ∈ [0 , T ],where ϑ is a random variable which modelizes a default time. The processes W and N aredefined on a complete probability space (Ω , G , P ) and we shall denote by G := { G t , t ≥ } the P - augmentation of the filtration generated by W and N . We assume that the default ϑ can appear at any time, that is P ( ϑ ≥ t ) > t ≥
0. Moreover, we suppose that W is a G -Brownian motion. Let P be the G -predictable σ -algebra.We denote by (Λ t ) the predictable compensator of the nondecreasing process ( N t ). Notethat (Λ t ∧ ϑ ) then corresponds to the predictable compensator of ( N t ∧ ϑ ) = ( N t ). By uniquenessof the predictable compensator, we get that Λ t ∧ ϑ = Λ t , t ≥ , called the intensity process, such that Λ t = R t λ s ds , t ≥
0. Since Λ t ∧ ϑ = Λ t , λ vanishesafter ϑ . The compensated martingale M satisfies M t = N t − Z t λ s ds. The coefficients of M , that is, the processes r t , µ t , σ t and β t are supposed to be pre-dictable (that is P -measurable), satisfying σ > β ϑ > −
1, and such that σ , µ, λ, σ − , β are bounded. All processes encountered throughout the paper will be defined on the fixed,finite horizon [0 , T ]. Moreover, the following sets will be used: • S is the set of G -optional processes ϕ such that E [ ess sup τ ∈T | ϕ τ | ] < + ∞ . • A is the set of real-valued non decreasing RCLL predictable processes A with A = 0and E ( A T ) < ∞ . • A o is the set of real-valued non decreasing RCLL optional processes A with A = 0and E ( A T ) < ∞ . • C is the set of real-valued purely discontinuous non decreasing RCLL optional pro-cesses C with C = 0 and E ( C T ) < ∞ . • H is the set of G -predictable processes Z such that k Z k H := E h R T | Z t | dt i < ∞ . • H λ := L (Ω × [0 , T ] , P , λ t dt ), equipped with the scalar product h U, V i λ := E h R T U t V t λ t dt i ,for all U, V in H λ . For each U ∈ H λ , we set k U k λ := E h R T | U t | λ t dt i < ∞ . We can suppose, without loss of generality, that for each U in H λ = L (Ω × [0 , T ] , P , λ t dt ), U (or its representant still denoted by U ) vanishes after ϑ .Moreover, T is the set of stopping times τ such that τ ∈ [0 , T ] a.s. and for each S in T , T S is the set of stopping times τ such that S ≤ τ ≤ T a.s.We now give the definition of a λ - admissible driver . Definition 2.1 ( Driver, λ - admissible driver ) . A function g is said to be a driver if g : Ω × [0 , T ] × R → R ; ( ω, t, y, z, k ) g ( ω, t, y, z, k ) which is P ⊗ B ( R ) − measurable, andsuch that g ( ., , , ∈ H .A driver g is called a λ - admissible driver if moreover there exists a constant C ≥ suchthat dP ⊗ dt -a.s. , for each ( y, z, k ) , ( y , z , k ) , ( y , z , k ) , | g ( ω, t, y, z , k ) − g ( ω, t, y, z , k ) | ≤ C ( | y − y | + | z − z | + √ λ t | k − k | ) . (2.3) The positive real C is called the λ - constant associated with driver g . Note that condition (2.3) implies that for each t > ϑ , since λ t = 0, g does not dependon k . In other terms, for each ( y, z, k ), we have: g ( t, y, z, k ) = g ( t, y, z, t > ϑ dP ⊗ dt -a.s.4 ealth process. We consider an investor, endowed with an initial wealth equal to x , whocan invest his wealth in the two assets of the market. At each time t , he chooses the amount ϕ t of wealth invested in the risky asset.For an initial wealth x ∈ R and a portfolio strategy ϕ ∈ H , we denote by V x,ϕt (orsimply V t ) the value of the associated portfolio (also called wealth ), which is supposed tosatisfy the following dynamics: − dV t = f ( t, V t , ϕ t σ t ) dt − ϕ t σ t dW t − ϕ t β t dM t , (2.4)with V = x , where f is a nonlinear λ -admissible driver independent on k , which modelizesthe imperfections in the market and which satisfies f ( t, ,
0) = 0. Note that in the classicalcase ( linear market), the driver f is given by f ( t, ω, y, z ) = − yr t ( ω ) − zθ t , with θ t = µ t − r t σ t (see e.g. [20]).Using a change of variable which associates to ϕ ∈ H another process Z ∈ H givenby Z = ϕσ , one can write (2.4) as follows: − dV t = f ( t, V t , Z t ) dt − Z t dW t − Z t σ − t β t dM t . (2.5)Note that the market is incomplete , as it is not possible for all ζ ∈ L to find ( V, Z ) ∈ S × H satisfying (2.5) with V T = ζ . Dynamic dual representation of the buyer’s priceprocess of an American option
Let g be a λ - admissible driver and let ζ ∈ L ( G T ). By Proposition 2 in [18], for each T ′ ∈ [0 , T ] and η ∈ L ( G T ′ ) there exists a unique solution ( X ( T ′ , η ) , Z ( T ′ , η ) , K ( T ′ , η )) in S × H × H ν (simply denoted by ( X , Z , K )) of the following backward SDE: − d X t = g ( t, X t , Z t , K t ) dt − Z t dW t − K t dM t ; X T = ζ . (3.1)As it is already well known, one can define an associated nonlinear operator (called g - expectation ) as follows: E g · ,T ′ ( η ) := X · . In order to ensure the monotonicity of the operator E g ( · ), the driver g should satisfy the following assumption. Assumption 3.1.
Assume that there exists a map γ : Ω × [0 , T ] × R R ; ( ω, t, y, z, k , k ) γ y,z,k ,k t ( ω ) P ⊗ B ( R ) -measurable, satisfying dP ⊗ dt -a.e., for each ( y, z, k , k ) ∈ R , | γ y,z,k ,k t p λ t | ≤ C and γ y,z,k ,k t ≥ − and g ( t, y, z, k ) − g ( t, y, z, k ) ≥ γ y,z,k ,k t ( k − k ) λ t (where C is a positive constant).
5e now address the problem of pricing and hedging the American option from the buyer’spoint of view. We define the superhedging price for the buyer of the American option withRCLL payoff process ξ · belonging to S as the maximal initial capital which allows thebuyer to find a superhedging strategy for the claim, that is v := sup { x ∈ R : ∃ ( ϕ, τ ) ∈ B ( x ) } , (3.2)where B ( x ) = { ( ϕ, τ ) ∈ H × T such that V − x,ϕτ + ξ τ ≥ . s . } . Now, we aim at providing a dual representation of the buyer’s superhedging price interms of a stochastic control/optimal stopping game , which will be later on characterizedas the maximal supersolution of a constrained reflected BSDE. To this purpose, we definethe driver ¯ f ( t, ω, y, z ) := − f ( t, ω, − y, − z ), which is clearly λ - admissible and denote by ¯ E the associated nonlinear conditional expectation. Let D be the set of bounded predictableprocesses ν such that ν t > − t ∈ [0 , T ] λ t dP ⊗ dt -a.s.Fix ν ∈ D . We denote by E ¯ f ν or ¯ E ν the nonlinear conditional expectation associatedwith the Lipschitz driver ¯ f ν ( t, y, z, k ) := ¯ f ( t, y, z ) + ν t λ t ( k − β t ( σ t ) − z ).For each S ∈ T , we define the G S -measurable random variable ¯ Y ( S ) as follows:¯ Y ( S ) := essinf ν ∈ D ess sup τ ∈T S ¯ E νS,τ ( ξ τ ) . Note that for each S ∈ T , τ ∈ T S and ν ∈ D , E νS,τ ( ξ τ ) depends on the control ν onlythrough the values of ν on the interval [ S, ν ]. For each S ∈ D , define D S the set of boundedpredictable processes ν defined on [ S, T ] such that ν t > − λ t dP ⊗ dt . Therefore, we have¯ Y ( S ) := essinf ν ∈ D S ess sup τ ∈T S ¯ E νS,τ ( ξ τ ) a . s . (3.3)In order to ensure some integrability properties of the above value family, we introducethe following assumption: Assumption 3.2.
There exists x ∈ R and ϕ ∈ H such that | ξ t | ≤ V x,ϕt , ≤ t ≤ T a.s. Under the above assumption, we can show that E [ ess sup τ ∈T ¯ Y ( τ )] < ∞ . Indeed, as ν ≡ D , we have ¯ Y ( S ) ≤ ess sup τ ∈T S E S,τ ( ξ τ ) = ¯ Y S a.s., where ( ¯ Y t ) is the first coordinateof the solution of the reflected BSDE associated with driver ¯ f and lower obstacle ( ξ t ). Now,since | ξ t | ≤ V x,ϕt , 0 ≤ t ≤ T a.s., we get that for all S ∈ T , τ ∈ T S and ν ∈ D S ,¯ E νS,τ ( ξ τ ) = − E νS,τ ( − ξ τ ) ≥ − E νS,τ ( | ξ τ | ) ≥ − E νS,τ ( V x,ϕτ ) = − V x,ϕS a.s., where the last equalityfollows by the E ν -martingale property of V x,ϕ for all ν ∈ D S . Hence, taking the essentialsupremum over τ ∈ T S and then the essential infimum over ν ∈ D S in this inequality, weobtain ¯ Y ( S ) ≥ − V x,ϕS a.s. Since ¯ Y ∈ S and V x,ϕ ∈ S , it follows that E [ ess sup S ∈T ¯ Y ( S ) ] < + ∞ . (3.4)6 .1 First properties of the value family ( ¯ Y ( S )) For each driver g , we denote by Y g the nonlinear operator (semigroup) associated with thereflected BSDE with lower obstacle ( ξ t ) and driver g , which is the analogous of the operator E g , induced by the non-reflected BSDE with driver g . Definition 3.3 ( Nonlinear operator Y g ). Let g be a λ - admissible driver. For each τ ∈ T and each ζ ∈ L ( F τ ) such that ζ ≥ ξ τ a.s., we define Y g · ,τ ( ζ ) := Y · , where Y · corresponds tothe first componant of the solution of the reflected BSDE associated with terminal time τ ,driver g and lower obstacle ( ξ t t<τ + ζ t ≥ τ ) . Recall that, by the flow property for reflected BSDEs, for each driver g , the operator Y g is consistent (or, equivalently, satisfies a semigroup property) with respect to terminalcondition ζ . Under Assumption 3.1, by the comparison theorem for reflected BSDEs withRCLL obstacle (see Th. 4.4. in [40]), we get that Y g is monotonous with respect to theterminal condition.Using the characterization of the solution of a reflected BSDE with RCLL lower obstaclein terms of an optimal stopping problem with g -expectations (see Th. 3.3. in [40]), we canrewrite (3.3) as follows ¯ Y ( S ) = essinf ν ∈ D S Y νS,T ( ξ T ) , (3.5)where for simplicity we denote by Y ν the operator Y ¯ f ν associated with driver ¯ f ν .Using standard arguments (see e.g. Lemma 3.2. in [13]), one can show that the family( ¯ Y ( S ) , S ∈ T ) is admissible . Moreover, we give below a result concerning the existence ofan optimizing sequence. Proposition 3.4.
Let S ∈ T . There exists a sequence of controls ( ν n ) n ∈ N with ( ν n ) in D S ,for all n , such that the sequence ( Y ν n S,T ( ξ T )) n ∈ N is non increasing and satisfies: ¯ Y ( S ) = lim n →∞ ↓ Y ν n S,T ( ξ T ) a.s. (3.6)Proof. It is enough to show that for each S ∈ T , the family { Y νS,T ( ξ T ) , ν ∈ D S } is directeddownward. Indeed, let ν, ν ′ ∈ D S . Set A = { Y ν ′ S,T ( ξ T ) ≥ Y νS,T ( ξ T ) } . We have A ∈ F S . Set˜ ν = ν A + ν ′ A c . Then ˜ ν ∈ D S . We have Y ˜ νS,T ( ξ T ) A = Y f ˜ ν A S,T ( ξ T A ) = Y f ν A S,T ( ξ T A ) = Y νS,T ( ξ T ) A a.s. and similarly on A c . It follows that Y ˜ νS,T ( ξ T ) = Y νS,T ( ξ T ) A + Y ν ′ S,T ( ξ T ) A c = Y νS,T ( ξ T ) ∧ Y ν ′ S,T ( ξ T ). (cid:3) We now recall the definition of an Y g - submartingale family (resp. an Y g - martingalefamily ) for a given λ -admissible driver. This notion is first introduced in [13]. Definition 3.5.
An admissible family ( X ( S ) , S ∈ T ) is said to be an Y g - submartingalefamily (resp. an Y g - martingale family ) if E [ ess sup τ ∈T X ( τ )] < ∞ , if for each S ∈ T , X ( S ) ≥ ξ S and if, for all S, S ′ ∈ T such that S ≥ S ′ a.s., Y gS ′ ,S ( X ( S )) ≥ X ( S ′ ) a.s., (resp . Y gS ′ ,S ( X ( S )) = X ( S ′ ) a.s.).
7e also recall the definition of a strong Y g - submartingale process (resp. strong Y g - martingale process ) (see [13]). Definition 3.6.
An optional process X ∈ S is said to be a strong Y g - submartingaleprocess (resp. a strong Y g - martingale process ) if for each S ∈ T , X S ≥ ξ S and if, for all S, S ′ ∈ T such that S ≥ S ′ a.s., Y gS ′ ,S ( X S ) ≥ X S ′ a.s., (resp . Y gS ′ ,S ( X S ) = X S ′ a.s.). We now give the following characterization of the family ( ¯ Y ( S )). Proposition 3.7.
The family ( ¯ Y ( S )) is the greatest family such that for each ν ∈ D , it isan Y ν -submartingale family equal to ξ T at terminal time T . Proof.We first show that ( ¯ Y ( S )) is a Y ν -submartingale family, for all ν ∈ D . Let θ ′ ∈ T and θ ∈ T θ ′ . By Proposition 3.4, there exists ( ν n ) n ∈ N such that equality (3.6) holds with S = θ .First, notice that ¯ Y ( θ ) ≥ ξ θ a.s for all θ ∈ T . By the continuity property of reflected BSDEswith respect to terminal condition, Y νθ ′ ,θ ( ¯ Y ( θ )) = lim n →∞ Y νθ ′ ,θ ( Y ν n θ,T ( ξ T )) a.s. For each n , weset ˜ ν nt := ν t [ θ ′ ,θ ] ( t ) + ν nt [ θ,T ] ( t ). Note that ˜ ν n ∈ D θ ′ and that ¯ f ˜ ν n = ¯ f ν [ θ ′ ,θ ] + ¯ f ν n [ θ,T ] . Wethus obtain Y νθ ′ ,θ ( Y ν n θ,T ( ξ T )) = Y ˜ ν n θ ′ ,θ ( Y ˜ ν n S,T ( ξ T )) = Y ˜ ν n θ ′ ,T ( ξ T ) a . s ., where the last equality follows from the consistency property of the operator Y ˜ ν n . We thusget that Y νθ ′ ,θ ( ¯ Y ( θ )) = lim n →∞ Y ˜ ν n θ ′ ,T ( ξ T ) ≥ ¯ Y ( θ ′ ) a.s. , where the last equality follows from thedefinition of ¯ Y ( θ ′ ). We now show the second assertion. Let ( Y ′ ( S ) , S ∈ T ) be an admissiblefamily such that for each ν ∈ D , it is an Y ν -submartingale family such that Y ′ ( T ) = ξ T a.s. Let ν ∈ D . By the properties of Y ′ , for all θ ∈ T , Y ′ ( θ ) ≤ Y νθ,T ( Y ′ ( T )) = Y νθ,T ( ξ T ) a.s.Taking the essential infimum over ν ∈ D , we derive Y ′ ( θ ) ≤ ¯ Y ( θ ) a.s. (cid:3) Y g - submartingale families/processes We give here some properties of Y g - submartingale families/processes in the case of a RCLLpayoff process ( ξ t ). We first provide an aggregation result, which has been first establishedin a more specific setting in [13]. Lemma 3.8 ( Aggregation of a Y g -submartingale family by a right-l.s.c. process ) . Let ( X ( S ) , S ∈ T ) be an Y g -submartingale family. Then, there exists a right-l.s.c. optionalprocess ( X t ) belonging to S which aggregates the family ( X ( S ) , S ∈ T ) , that is such that X ( S ) = X S a.s. for all S ∈ T . Moreover, the process ( X t ) is a strong Y g -submartingale,that is for each S ∈ T , X S ∈ L , X S ≥ ξ S and for all S, S ′ ∈ T such that S ≥ S ′ a.s., Y gS ′ ,S ( X S ) ≥ X S ′ a.s. τ n ) n ∈ N be a nondecreasing sequence of stopping times such that τ n ↓ τ a.s.The definition of Y g implies that X ( τ ) ≤ Y gτ,τ n ( X ( τ n )) a . s ., for all n ∈ N . (3.7)Since the sequence ( τ n ) n is nondecreasing and the operator Y g is consistent, we derive that Y gτ,τ n ( X ( τ n )) = Y gτ,τ n +1 ( Y gτ n +1 ,τ n ( X ( τ n ))) ≥ Y gτ,τ n +1 ( X ( τ n +1 )) a . s ., where the last inequality follows by (3.7). This implies that the sequence Y gτ,τ n ( X ( τ n )) n ∈ N isnondecreasing and thus it converges almost surely. Moreover, X ( τ ) ≤ lim n →∞ ↓ Y gτ,τ n ( X ( τ n )) a . s . (3.8)Since lim sup n →∞ X ( τ n ) ≥ ξ τ a.s., one can use the Fatou lemma for Reflected BSDEs (seeProposition 3.13 in [15]). We thus get X ( τ ) ≤ lim sup n →∞ Y gτ,τ n ( X ( τ n )) ≤ Y gτ,τ (lim sup n →∞ X ( τ n )) = lim sup n →∞ X ( τ n ) . (3.9)By Lemma 5 in [11], we conclude that the family ( X ( S ) , S ∈ T ) is right lower semicontin-uous. It follows from Theorem 4 in [11] that there exists a right-l.s.c. optional process ( X t )which aggregates the family ( X ( S ) , S ∈ T ), which is clearly a strong Y g -submartingale. (cid:3) Remark 3.9.
The above proposition implies that any strong Y g -submartingale process isright lower semicontinuous. We will now show that, if a process ( X t ) is a strong Y g -submartingale, then the process( X t + ) is a Y g -submartingale as well. This is an analagous result of the one given in the caseof classical linear expectations (see e.g. [34]). Lemma 3.10.
Suppose that ( ξ t ) is a strong semimartingale. If ( X t ) t ∈ [0 ,T ] is a strong Y g -submartingale, then the process of right-limits ( X t + ) t ∈ [0 ,T ] (where, by convention, X T + := X T )is a strong Y g -submartingale. Proof. Since ( X t ) is a strong Y g -submartingale, X t ≥ ξ t , 0 ≤ t ≤ T a.s. Moreover,since ( ξ t ) is a strong semimartingale, the strong Y g -submartingale ( X t ) has right limits (seeRemark 4.2) and X t ≥ ξ t , 0 ≤ t ≤ T a.s.We have to show that the process ( X t + ) is a strong Y g -submartingale. Let us first showthat ( X t + ) is greater than ( ξ t ). Since ( X t ) is a strong Y g -submartingale, by Remark 3.9, itfollows that ( X t ) is right-l.s.c., which implies that for each θ ∈ T , we have X θ + ≥ X θ a.s.Since X θ ≥ ξ θ a.s., we derive that X θ + ≥ ξ θ a.s.Let S, θ ∈ T with S ≤ θ a.s. There exist two nondecreasing sequences of stoppingtimes ( S n ) and ( θ n ) such that for each n , S n ≤ θ n a.s. , S n > S a.s. on { S < T } , θ n > a.s. on { θ < T } and S n → S a.s. (resp. θ n → θ ) when n → ∞ . Since ( X t ) is astrong Y g -submartingale, using the consistency and the monotonicity properties of Y g , weget Y gS,θ n ( X θ n ) = Y gS,S n ( Y gS n ,θ n ( X θ n )) ≥ Y gS,S n ( X S n ) a . s . Since ( ξ t ) is RCLL , the continuityproperty with respect to terminal time and terminal condition of reflected BSDEs holds.Hence, letting n tend to + ∞ in the above inequality, we obtain Y gS,θ ( X θ + ) ≥ Y gS,S ( X S + ) = X S + a . s . We thus conclude that the process ( X t + ) is a strong Y g -submartingale. (cid:3) Y ν -submartingales for all ν ∈ D In this subsection, we show that a RCLL process which is a Y ν -submartingale for all ν ∈ D admits a dynamic characterization via constrained reflected BSDEs. To this purpose, wefirst prove that a process ( X t ) which is a strong Y ν -submartingale for all ν ∈ D admits aRCLL version. Proposition 3.11.
Suppose that ( ξ t ) is a strong semimartingale. Let ( X t ) t ∈ [0 ,T ] be an op-tional process. Suppose that ( X t ) is the largest strong Y ν -submartingale for all ν ∈ D suchthat X T = ξ T a.s. Then, ( X t ) admits a RCLL version (still denoted by ( X t ) ). Proof.Since ( X t ) is a strong Y ν -submartingale for all ν ∈ D , it follows by Lemma 3.10 that( X t + ) is a strong Y ν -submartingale for all ν ∈ D . By the maximality property of ( X t ), itfollows that X t ≥ X t + , 0 ≤ t ≤ T a.s. On the other hand, as ( X t ) is right-l.s.c. (cf. Remark3.9), we have X t + ≥ X t , 0 ≤ t ≤ T a.s. We conclude that X t = X t + , 0 ≤ t ≤ T a.s. (cid:3) We recall here the following definition from [17].
Definition 3.12.
Let A = ( A t ) ≤ t ≤ T and A ′ = ( A ′ t ) ≤ t ≤ T belonging to A . The measures dA t ⊥ dA ′ t are said to be mutually singular and we write dA t ⊥ dA ′ t if there exists D ∈ P such that E [ Z T D c dA t ] = E [ Z T D c dA ′ t ] = 0 . Similarly, one can define mutually singular random measures associated with non-decreasingRCLL optional processes.We now prove a constrained reflected BSDE characterization of a RCLL process whichis a Y ν -submartingale for all ν ∈ D . Proposition 3.13.
Suppose that ( ξ t ) is a strong semimartingale. Let ( X t ) ∈ S be a RCLLstrong Y ν -submartingale for all ν ∈ D . There exists an unique process ( Z, K, A, A ′ ) ∈ H × H ν × ( A ) such that − dX t = ¯ f ( t, X t , Z t ) dt − Z t dW t − K t dM t − dA ′ t + dA t , (3.10)10 ith dA t ⊥ dA ′ t and Z T ( Y s − − ξ s − ) dA s = 0 a . s . (3.11) and { Y t − >ξ t − } ( K t − β t σ − t Z t ) λ t ≥ , t ∈ [0 , T ] , dt ⊗ dP a . s . (3.12) and { Y t − >ξ t − } (cid:18) dA ′ t − ( K t − β t σ − t Z t ) λ t dt (cid:19) ≥ , t ∈ [0 , T ] , a . s . (3.13)Proof.First note that, by the Y -Mertens decomposition of the strong Y - submartingale( X t ) (proved in [13] and recalled in Appendix, see Th.4.1), there exists an unique process( Z, K, A, A ′ , C, C ′ ) ∈ H × H ν × ( A ) × ( C ) such that − dX t = ¯ f ( t, X t , Z t ) dt − Z t dW t − K t dM t − dA ′ t + dA t − dC ′ t − + dC t − ; Z T ( X s − − ξ s − ) dA s = 0 a . s . ; dA t ⊥ dA ′ t ; dC t ⊥ dC ′ t . Since the process ( X t ) is assumed to be RCLL and ∆ C τ = ( X τ + − X τ ) − ( resp. ∆ C ′ τ =( X τ + − X τ ) + ), we deduce that C = C ′ = 0.Fix ν ∈ D . Since ( X t ) is a RCLL strong Y ν -submartingale in S and using similararguments as above, there exists an unique process ( Z ν , K ν , A ν , A ′ ν ) ∈ H × H ν × ( A ) such that − dX t = (cid:0) ¯ f ( t, X t , Z νt ) + ( K νt − β t σ − t Z νt ) ν t λ t (cid:1) dt − Z νt dW t − K νt dM t − dA ′ νt + dA νt ; Z T ( X s − − ξ s − ) dA νs = 0 a . s . ; dA νt ⊥ dA ′ νt . The uniqueness of the decompositions of a semimartingale and of a martingale lead to Z t = Z νt dt ⊗ dP -a.s. and K t = K νt dP ⊗ dt -a.s. This implies that ¯ f ( t, X t , Z t ) = ¯ f ( t, X t , Z νt ) dt ⊗ dP -a.s. Then, using the uniqueness of the finite variation part of the decomposition ofthe semimartingale ( X t ), we derive that dA νt − dA ′ νt = dA t − dA ′ t − ( K t − β t σ − t Z t ) ν t λ t dt. (3.14)Since by the Skorohod conditions dA νt = dA t = 0 on { X t − > ξ t − } , we derive that dA ′ νt = dA ′ t + ( K t − β t σ − t Z t ) ν t λ t dt on { X t − > ξ t − } . (3.15)11e now show that this implies that { X t − >ξ t − } ( K t − β t σ − t Z t ) λ t ≥ dt ⊗ dP a.s. Letus define the set B := { ( K t − β t σ − t Z t ) λ t < , X t − > ξ t − } . Suppose by contradiction that P ( B ) >
0. For each n ∈ N , set ν n := n B , which belongs to D . From relation (3.15),we get for n sufficiently large, E [ R T { X t − >ξ t − } dA ′ ν n t ] = E [ R T { X t − >ξ t − } dA ′ t + n R T ( K t − β t σ − t Z t ) λ t B dt ] <
0. This leads to a contradiction, which implies that { X t − >ξ t − } ( K t − β t σ − t Z t ) λ t ≥ dt ⊗ dP a.s. We now show that (3.13) holds. Assume by contradictionthat there exists ε > u, v ∈ [0 , T ] with u < v and D ∈ G T with P ( D ) > R vu X t − >ξ t − (cid:18) dA ′ t − ( K t − β t σ − t Z t ) λ t dt (cid:19) ≤ − ε a.s. on D . Considering the sequence of controls ν n ≡ − n (which are clearly admissible) and using (3.15), we get − n R vu X t − >ξ t − ( K t − βσ − t Z t ) λ t dt ≤ − ε on D . Letting n tend to infinity, we get a contradiction and thus concludethat (3.13) holds. (cid:3) Using the previous proposition, we can provide a nonlinear optional decomposition of Y ν -submartingales, for all ν ∈ D . Theorem 3.14 (Nonlinear optional decomposition) . Let ( X t ) be a RCLL process belongingto S . Suppose that it is an Y ν -strong submartingale for each ν ∈ D . Then, there exists Z ∈ H and k , k ′ ∈ A o such that − dX t = ¯ f ( t, X t , Z t ) dt − Z t σ − t ( σ t dW t + β t dM t ) + dk t − dk ′ t ; (3.16) dk t ⊥ dk ′ t ; Z T ( X s − − ξ s − ) dk s = 0 a . s . (3.17) Moreover, this decomposition is unique.
Proof. By Proposition 3.13, there exists an unique process (
Z, K, A, A ′ ) ∈ H × H ν × ( A ) such that (3.10), (3.11), (3.12), (3.13) hold.By classical results, the finite variational optional RCLL process f t := A t − A ′ t − R t ( K s − β s σ − s Z s ) dM s can be uniquely decomposed as l · = k · − k ′· , where ( k t ) and ( k ′ t ) are twoprocesses in A with k = k ′ = 0 and E [ k ′ T ] < ∞ (resp. E [ k T ] < ∞ ). Recall that dk t ⊥ dk ′ t .Using classical notation of Measure Theory, we can write: dk t = (cid:0) dA t − dA ′ t − ( K t − β t σ − t Z t ) dM t (cid:1) + and dk ′ t = (cid:0) dA t − dA ′ t − ( K t − β t σ − t Z t ) dM t (cid:1) − . Since dM t = dN t − λ t dt , we have dk t = (cid:18) dA t − ( K t − β t σ − t Z t ) dN t − ( dA ′ t − ( K t − β t σ − t Z t ) λ t dt ) (cid:19) + . (3.18)12sing the constraints (3.11), (3.12), (3.13), we derive that { X t − >ξ t − } dk t = 0. Hence, theSkorohod condition (3.17) hold. By (3.10) and using the definition of l , we derive thatequation (3.16) is satisfied.Let us show that this decomposition is unique. By equation (3.16), we have∆ X ϑ = Z ϑ σ − ϑ β ϑ − ∆ k ϑ + ∆ k ′ ϑ . (3.19)Set dB t := dk t − ∆ k ϑ , dB ′ t := dk ′ t − ∆ k ′ ϑ and dX ′ t = dX t − ∆ X ϑ . Note that the non decreasingprocesses B and B ′ have only predictable jumps, which implies that B, B ′ ∈ A . Moreover, dB t ⊥ dB ′ t . By (3.16), using dN t = dM t + λ t dt , we get − dX ′ t = f ( t, X t , Z t ) dt − Z t dW t + Z t σ − t β t λ t dt + dB t − dB ′ t . (3.20)By uniqueness of the semimartingale and martingale decompositions, we derive the unique-ness of the processes Z , B and B ′ . By (3.19), we obtain ∆ k ′ ϑ − ∆ k ϑ = ∆ X ϑ − Z ϑ σ − ϑ β ϑ . Sincemoreover dk ⊥ dk ′ , we finally derive the uniqueness of dk t = dB t + ∆ k ϑ and dk ′ t = d ′ t + ∆ k ′ ϑ . (cid:3) ( ¯ Y t ) Using the results given in the previous sections, we will obtain an infinitesimal characteriza-tion of the value process ( ¯ Y t ). We first introduce the following definition. Definition 3.15.
A process ( X t ) is called a subsolution of the reflected BSDE driven by themartingale m St := W t + R t σ − s β s dM s associated with driver ¯ f and obstacle ξ if there exists aprocess ( Z, k, k ′ ) ∈ H × ( A o ) such that − dX t = ¯ f ( t, X t , Z t ) dt − Z t dm St + dk t − dk ′ t , (3.21) with X t ≥ ξ t , t ∈ [0 , T ] a . s . ; X T = ξ T a . s . and Z T ( X s − − ξ s − ) dk s = 0 , dk t ⊥ dk ′ t . (3.22) Remark 3.16.
Since M has only a totally inaccessible jump, for each predictable τ ∈ T ,we have ∆ k τ = (∆ X τ ) − . Since k satisfies the Skorokhod condition (3.22) , we get ∆ k τ = { Y τ − = ξ τ − } ( ξ τ − − X τ ) + ≤ { X τ − = ξ τ − } ( ξ τ − X τ ) + a.s. where the (last) inequality follows fromthe left u.s.c. property of ξ . Since ξ ≤ X , we derive ∆ k τ ≤ a.s. , which implies that ∆ k τ = 0 a.s. We note also that X can jump (on the left) at totally inaccessible stoppingtimes; these jumps of X come from the jumps of the stochastic integral with respect to M in (3.10) .
13e now show that the value process ( ¯ Y t ) is a maximal subsolution of the reflected BSDEgiven in the above definition. Theorem 3.17.
The process ( ¯ Y t ) is the maximal subsolution of the reflected BSDE (3.21) ,that is, if ( Y t ) is a subsolution of (3.21) , then Y t ≤ ¯ Y t , t ∈ [0 , T ] a . s . Proof.By Proposition 3.7 and Proposition 3.14, we derive that ( ¯ Y t ) is a subsolution of thereflected BSDE (3.21). By Proposition 3.7, we also derive that ( ¯ Y t ) is the greatest processwhich is a strong Y ν -submartingale, for all ν ∈ D .It remains to prove that ( ¯ Y t ) is the maximal subsolution of the reflected BSDE (3.21).Assume that ( Y, Z, K, k, k ′ ) be a subsolution of the same reflected BSDE (cf. (3.21)). Let ν ∈ D . Note that we have − dY t = ¯ f ν ( t, Y t , Z t , Z t σ − t β t ) dt − Z t dm St + dk t − dk ′ t , (3.23)with Y · ≥ ξ · , Y T = ξ T and the Skorohod condition (3.22). This implies that ( Y, Z, Zσ − β, k )is the solution of the reflected BSDE associated with generalized driver ¯ f ν ( · ) dt − dk ′ t andobstacle ( ξ t ). Using the (generalized) comparison theorem for reflected BSDEs, we have thatfor all S, S ′ ∈ T with S ≥ S ′ a.s., Y νS ′ ,S ( Y S ) ≥ Y S ′ a.s. since Y ν · ,S ( Y S ) is the solution of thereflected BSDE associated with driver ¯ f ν , obstacle ( ξ t ) and terminal condition Y S . Hence,( Y t ) is a strong Y ν - submartingale for each ν ∈ D . Moreover, Y T = ξ T a.s. Hence, byProposition 3.7, we get Y t ≤ ¯ Y t , 0 ≤ t ≤ T a.s. (cid:3) Dynamic dual representation of the buyer’s price process
Taking advantage of the previous theorem, we are now able to provide a dynamic dualrepresentation of the buyer’s price process of an American option in a nonlinear incompletemarket. We first consider the simpler case when ξ is left-u.s.c. and, for simplicity, firstprovide the dual representation of the price at time 0, which will be extended in Theorem3.19 to any stopping time. Theorem 3.18 ( Buyer’s superhedging price and super-hedge) . Let ( ξ t ) be a left-u.s.c. alongstopping times strong semimartingale. The buyer’s price v of the American option is givenby v = inf ν ∈ D sup τ ∈T ¯ E ν ,τ ( ξ τ ) . (3.24) Moreover, v = ¯ Y , where ¯ Y is the maximal subsolution of the constrained reflected BSDE (3.21) . Let ( ¯ Z, ¯ k, ¯ k ′ ) be the associated processes which appear in the representation (3.21) .The risky assets strategy ¯ ϕ := − σ − ¯ Z and the stopping time ¯ τ := inf { t ≥ Y t = ξ t } is asuperhedging strategy for the buyer, that is, (¯ τ , ¯ ϕ ) ∈ B ( v ) . ν ∈ D sup τ ∈T ¯ E ν ,τ ( ξ τ ) = ¯ Y . Therefore, it is sufficient to showthat v = ¯ Y and (¯ τ , ¯ ϕ ) ∈ B ( ¯ Y ). Let S be the set of initial capitals which allow the buyerto be “super-hedged”, that is S = { x ∈ R : ∃ ( τ, ϕ ) ∈ B ( x ) } . Remark that v = sup S .Let us first show that ¯ Y ≤ v . To this aim, we prove that(¯ τ , ¯ ϕ ) ∈ B ( ¯ Y ) . (3.25)We consider the portfolio associated with the initial capital − ¯ Y and the strategy ¯ ϕ = − σ − ¯ Z . By (2.4), the value of the portfolio process ( V − ¯ Y , ¯ ϕt ) satisfies the following forwarddifferential equation: V − ¯ Y , ¯ ϕt = − ¯ Y − Z t f ( s, V − ¯ Y , ¯ ϕs , − ¯ Z s ) ds − Z t ¯ Z s dm Ss , ≤ t ≤ T. (3.26)Moreover, since ¯ Y is the solution of the reflected BSDE (3.21), it satisfies:¯ Y t = ¯ Y − Z t ¯ f ( s, ¯ Y s , ¯ Z s ) ds + Z t ¯ Z s dm Ss − ¯ k t + ¯ k ′ t , ≤ t ≤ T. (3.27)We have ¯ k = ¯ k c + ¯ k d , where ¯ k c (resp. ¯ k d ) is the continuous (resp. discontinuous) part of¯ k . We first show that ¯ h c = 0 on [0 , ¯ τ ]. Now, by definition of ¯ τ , we have that almost surelyon [0 , ¯ τ [, ¯ Y t > ξ t . By the Skorokhod condition (3.22), we get that the process ¯ k c is equalto 0 on [0 , ¯ τ [. The continuity of ¯ k c implies that ¯ k c = 0 a.s. on [0 , ¯ τ ]. Under the left upper-semicontinuity assumption on the process ( ξ t ), by Remark 3.16 we derive that ∆ k τ = 0 a.s.for all predictable stopping time τ ∈ T . It remains to show that ∆ k ϑ = 0 a.s. on { ϑ = ¯ τ } .For each n ∈ N , we define τ n := inf { t ≥ Y t ≤ ξ t + n } . Note that ( τ n ) n is anon-decreasing sequence of stopping times, which satisfies lim n →∞ τ n = ¯ τ a.s. Since ϑ isa totally inaccessible stopping time, we get that for a.e. ω such that ¯ τ ( ω ) = ϑ ( ω ), thereexists n ( ω ) such that for all n ≥ n ( ω ) we have τ n ( ω ) = ϑ ( ω ). Let us consider such an ω . By definition of τ n ( ω ), we get that ¯ Y τ n ( ω ) − ≥ ξ τ n ( ω ) − + n , from which we derive that¯ Y ϑ ( ω ) − ( ω ) ≥ ξ ϑ ( ω ) − ( ω ) + n > ξ ϑ ( ω ) − ( ω ) . By the Skorokhod condition (3.22), we get that∆ k ϑ ( ω ) ( ω ) = 0 . We thus conclude that ¯ k ¯ τ = 0 a.s.By multiplying by ( −
1) the equation (3.27) and using the definition of the driver ¯ f , wederive that the ( − ¯ Y t ) satisfies the following equation: − ¯ Y t = − ¯ Y − Z t f ( s, − ¯ Y s , − ¯ Z s ) ds − Z t ¯ Z s dm Ss − ¯ k ′ t , ≤ t ≤ ¯ τ , a . s . (3.28)Therefore, by the comparison result for forward differential equations, we get V − ¯ Y , ¯ ϕt ≥ − ¯ Y t ,0 ≤ t ≤ ¯ τ a.s. By definition of the stopping time ¯ τ , and the right continuity of the processes( ¯ Y t ) and ( ξ t ), we derive that ¯ Y ¯ τ = ξ ¯ τ a.s. We thus conclude that V − ¯ Y , ¯ ϕ ¯ τ ≥ − ξ ¯ τ a . s ., whichimplies that (¯ τ , ¯ ϕ ) ∈ B ( ¯ Y ) and thus ¯ Y ≤ v .We now prove the converse inequality. Let x ∈ S . By definition of S , there exists( τ, ϕ ) ∈ B ( x ) such that V − x,ϕt ≥ − ξ τ a.s. Let ν ∈ D . By taking the E ν -evaluation in15he above inequality, using the monotonicity of E ν and the E ν -martingale property of thewealth process V − x,ϕ , we derive that − x = E ν ,τ ( V − x,ϕτ ) ≥ E ν ,τ ( − ξ τ ) = − ¯ E ν ,τ ( ξ τ ) . We thusget x ≤ ¯ E ν ,τ ( ξ τ ), which implies x ≤ sup τ ∈T ¯ E ν ,τ ( ξ τ ) . By arbitrariness of ν ∈ D , we get x ≤ inf ν ∈ D sup τ ∈T ¯ E ν ,τ ( ξ τ ) = ¯ Y , which holds for any x ∈ S . By taking the supremum over x ∈ S , we get v ≤ ¯ Y . Itfollows that v = ¯ Y . By (3.25), we get (¯ τ , ¯ ϕ ) ∈ B ( v ), which completes the proof. (cid:3) We now define the buyer’s price of the American option at each stopping time S ∈ T .We define for each initial wealth X ∈ L ( G S ), a super-hedge against the American optionfrom the buyer’s point of view as a portfolio strategy ϕ ∈ H and a stopping time τ ∈ T S such that V S, − X,ϕτ + ξ τ ≥ V S, − X,ϕ represents the wealth process associated withinitial time S and initial condition X . The buyer’s price at time S is defined by the randomvariable v ( S ) = ess sup { X ∈ L ( G S ) , ∃ ( ϕ, τ ) ∈ B S ( X ) } , with B S ( X ) the set of all super-hedges associated with initial time S and initial wealth X . By Theorem 3.17 and using similar arguments as in Theorem 3.18, one can show thefollowing result. Theorem 3.19 ( Buyer’s price process and dynamic dual representation) . Let ( ξ t ) be a left-u.s.c. along stopping times strong semimartingale. For each time S ∈ T , the buyer’s price v ( S ) at time S of the American option satisfies v ( S ) = essinf ν ∈ D S ess sup τ ∈T S ¯ E νS,τ ( ξ τ ) = ¯ Y S a . s ., where ¯ Y is the maximal subsolution of the constrained reflected BSDE (3.21) . Let ( ¯ Z, ¯ k, ¯ k ′ ) bethe associated processes which appear in the representation (3.21) . The risky assets strategy ¯ ϕ := − σ − ¯ Z and the stopping time ¯ τ S := inf { t ≥ S : ¯ Y t = ξ t } is a superhedging strategy forthe buyer, that is (¯ τ S , ¯ ϕ ) ∈ B S ( v ( S )) . Let us now address the general case when ξ is only RCLL. Again, for simplicity, we willprovide the results for the initial time 0, which can be easily extended to any time/stoppingtime S ∈ T as in the case of a left upper-semicontinuous payoff process ( ξ t ).We introduce the definition of an ε - super-hedge for the buyer . Definition 3.20.
For each initial price x and for each ε > , an ε -super-hedge for the buyerof an American option is a pair ( x, ϕ ) of a stopping time τ ∈ T and a risky-assets strategy ϕ ∈ H such that V − z,ϕτ + ξ τ ≥ − ε a . s . Theorem 3.21 ( Buyer’s superhedging price and super-hedge) . Let ( ξ t ) be a RCLL strongsemimartingale. The buyer’s price v of the American option is satisfies v = inf ν ∈ D sup τ ∈T ¯ E ν ,τ ( ξ τ ) = ¯ Y , (3.29)16 here ¯ Y is the maximal subsolution of the constrained reflected BSDE (3.21) . Let ( ¯ Z, ¯ k, ¯ k ′ ) be the associated processes which appear in the representation (3.21) .Consider the risky assets strategy ¯ ϕ := − σ − ¯ Z and for each ε > , define ¯ τ ε := inf { t ≥ Y t ≤ ξ t + ε } . (3.30) The pair ( ¯ ϕ, ¯ τ ε ) is a an ε -superhedging strategy for the buyer (associated with the initial price v ). Proof. By the same arguments as in the previous proof, we derive that the equation(3.28) holds on [0 , ¯ τ ε ]. Hence, by the comparison theorem for forward equations, we get V − ¯ Y , ¯ ϕ ¯ τ ε ≥ − ¯ Y ¯ τ ε a.s. Moreover, using the same arguments as in the previous proof, weobtain that ¯ Y ¯ τ ε ≤ ξ ¯ τ ε + ε a . s . (3.31)We thus conclude that V − ¯ Y , ¯ ϕ ¯ τ ε ≥ − ¯ Y ¯ τ ε ≥ − ξ ¯ τ ε − ε a . s ., which implies that the pair(¯ τ ε , ¯ ϕ ) is an ε -super-hedge for the buyer associated with the initial price ¯ Y . We now provethat ¯ Y = v . By Theorem 3.17, we have inf ν ∈ D sup τ ∈T ¯ E ν ,τ ( ξ τ ) = ¯ Y . Using this property andthe same arguments as in the proof of the second part of the previous theorem (which donot require the continuity of the process ¯ k ), we derive that v ≤ ¯ Y .We now show the converse inequality ¯ Y ≤ v . Fix ε > Y ′ , Z ′ , K ′ ) be thesolution of the BSDE associated with generalized driver ¯ f − d ¯ k ′ , terminal time τ ε and terminalcondition ξ τ ε ∧ ¯ Y τ ε . By (3.31) and a priori estimates on BSDEs with jumps, we derive that¯ Y ≤ Y ′ + Cε , with C a constant dependending on ¯ f . Now, by the assumption Y ′ τ ε = ξ τ ε ∧ ¯ Y τ ε , we deduce that Y ′ τ ε ≤ ξ τ ε . Moreover, one can easily remark that − V Y ′ ,ϕ ≤ Y ′ . Therefore, wededuce that − V Y ′ ,ϕτ ε ≤ ξ τ ε . This implies that ( τ ε , ϕ ′ ) is a super-hedging strategy associatedwith the initial price Y ′ . Since the price Y ′ allows the buyer to be super-hedged, we derivethat ¯ Y − Cε ≤ Y ′ ≤ v for each ε >
0. Hence, v ≥ ¯ Y − Cε , for all ε >
0. We thusconclude that v ≥ ¯ Y . (cid:3) We now show that the operations of ”infimum” and ”supremum” in the dual representa-tion (3.29) of the buyer’s superhedging price v can be interchanged. We prove this resultin the case of a left upper semicontinuous payoff process; the proof in the general case of aRCLL process follows exactly the same steps, by replacing the optimal stopping time by an ε -optimal stopping time. Proposition 3.22 (Interchange inf-sup) . Assume that the process ( ξ t ) is left upper semi-continuous along stopping times. The buyer’s superhedging price of the American optionsatisfies: v = inf ν ∈ D sup τ ∈T ¯ E ν ,τ ( ξ τ ) = sup τ ∈T inf ν ∈ D ¯ E ν ,τ ( ξ τ ) . v = inf ν ∈ D sup τ ∈T ¯ E ν ,τ ( ξ τ ) = ¯ Y , where the pro-cess ( ¯ Y t ) is the maximal subsolution of the constrained reflected BSDE (3.10) and ( ¯ Z, ¯ A, ¯ A ′ )the associated processes. We define: Y := sup τ ∈T inf ν ∈ D ¯ E ν ,τ ( ξ τ ) . We have to show that ¯ Y = Y . We clearly have ¯ Y ≥ Y . It thus remains to prove that¯ Y ≤ Y .Let ¯ τ := inf { t ≥ Y t = ξ t } . The right continuity of the processes ( ξ t ) and ( ¯ Y t ) yieldsthe equality ¯ Y ¯ τ = ξ ¯ τ a.s. Since ( ¯ Y , ¯ Z, ¯ A, ¯ A ′ ) is a subsolution of the constrained reflectedBSDE (3.10), we have¯ Y = ¯ Y ¯ τ + Z ¯ τ ¯ f ( s, ¯ Y s , ¯ Z s ) ds − Z ¯ τ ¯ Z s dW s − Z ¯ τ ¯ K s dM s + A ¯ τ − A ′ ¯ τ . (3.32)Note that ¯ Y · > ξ · on [0 , ¯ τ [. Hence, the process ( A t ) is constant equal to 0 on [0 , ¯ τ [ a.s. Now,the left upper semicontinuity assumption on the process ( ξ t ) ensures that the process ( A t )is continuous (see Th. 3.7 in [17]). It follows that A ¯ τ = 0 a.s.Moroever, by the constraints satisfied by A ′ , ˜ Z and ˜ K on the set { ¯ Y t − > ξ t − } , and since ν > −
1, we derive that¯ f ( s, ¯ Y s , ¯ Z s ) ds − d ¯ A ′ s ≤ ¯ f ( s, ¯ Y s , ¯ Z s ) ds − ( ¯ K s − β s σ − s ¯ Z s ) λ s ds ≤ ¯ f ( s, ¯ Y s , ¯ Z s ) ds +( ¯ K s − β s σ − s ¯ Z s ) λ s ν s ds, ≤ t ≤ ¯ τ a.s. By the comparison theorem for BSDEs (with generalized driver) on [0 , ¯ τ ], weget ¯ Y ≤ ¯ E ν , ¯ τ ( ¯ Y ¯ τ ) = ¯ E ν , ¯ τ ( ξ ¯ τ ) . By arbitrariness of ν ∈ D , we derive¯ Y ≤ inf ν ∈ D ¯ E ν , ¯ τ ( ξ ¯ τ ) ≤ sup τ ∈T inf ν ∈ D ¯ E ν ,τ ( ξ τ ) = Y . We thus conclude that ¯ Y = Y , which completes the proof. (cid:3) We recall here the Y g -Mertens decomposition of Y g -submartingales proved in [13] (see Th.3.9). Theorem 4.1 ( Y g -Mertens decomposition of Y g -submartingales) . Let ( ξ t ) be an optionalstrong semimartingale right upper semicontinuous belonging to S and let ( Y t ) be an optionalprocess in S . Then ( Y t ) is a Y g -submartingale if and only if there exist two non decreasingright continuous and predictable processes A, A ′ in A , two non decreasing adapted right ontinuous and purely discontinuous processes C and C ′ in C and ( Z, K ) ∈ H × H ν suchthat − dY s = g ( s, Y s , Z s , K s ) ds − Z s dW s − K t dM t − dA ′ s − dC ′ s − + dA s + dC s − ; (4.1) Y t ≥ ξ t , t ∈ [0 , T ] a . s . ; 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 ⊥ dA ′ t ; dC t ⊥ dC ′ t . Moreover, this decomposition is unique.
Remark 4.2.
Using the above decomposition, we deduce that a Y g -submartingale admitsleft and right limits. Note also that by Remark 3.9, a strong Y g -submartingale process isright-l.s.c. , which gives that Y t + ≥ Y t for all t ∈ [0 , T ] a.s. References [1] Ansel, J.-P., Stricker, C., Couverture des actifs contingents,
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