Optimal stopping for dynamic risk measures with jumps and obstacle problems
aa r X i v : . [ m a t h . O C ] J un Optimal stopping for dynamic risk measures with jumpsand obstacle problems
Roxana DUMITRESCU ∗ Marie-Claire Quenez † Agn`es SULEM ‡ October 1, 2018
Abstract
We study the optimal stopping problem for a monotonous dynamic risk measureinduced by a BSDE with jumps in the Markovian case. We show that the valuefunction is a viscosity solution of an obstacle problem for a partial integro-differentialvariational inequality, and we provide an uniqueness result for this obstacle problem.
Key-words:
Dynamic risk-measures, optimal stopping, obstacle problem, reflectedbackward stochastic differential equations with jumps,viscosity solution, comparison princi-ple, partial integro-differential variational inequality ∗ CEREMADE, Universit´e Paris 9 Dauphine, CREST and INRIA Paris-Rocquencourt, email: [email protected] † LPMA, Universit´e Paris 7 Denis Diderot, Boite courrier 7012, 75251 Paris Cedex 05, France, email: [email protected] ‡ INRIA Paris-Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, Le Chesnay Cedex, 78153,France, and Universit´e Paris-Est, F-77455 Marne-la-Vall´ee, France, email: [email protected] Introduction
In the last years, there has been several studies on dynamic risk measures and their linkswith nonlinear backward stochastic differential equations (BSDEs). We recall that non-linear BSDEs have been introduced in [12] in a Brownian framework, in order to providea probabilistic representation of semilinear parabolic partial-differential equations. BSDEswith jumps and their links with partial integro-differential equations are studied in [2]. Acomparison theorem is established in [16] and generalized in [14], where properties of dy-namic risk measures induced by BSDEs with jumps are also provided. An optimal stoppingproblem for such risk measures is addressed in [15], and the value function is characterizedas the solution of a reflected BSDE with jumps and RCLL obstacle process.In the present paper, we focus on the optimal stopping problem for dynamic risk measuresinduced by BSDEs with jumps in a Markovian framework. In this case the driver of theBSDE depends on a given state process X , which can represent, for example, an index or astock price. This process will be assumed to be driven by a Brownian motion and a Poissonrandom measure.Our main contribution consists in establishing the link between the value function of ouroptimal stopping problem and parabolic partial integro-differential variational inequalities(PIDVIs). We prove that the minimal risk measure, which corresponds to the solution of areflected BSDE with jumps, is a viscosity solution of a PIDVI. This provides an existenceresult for the obstacle problem under relatively weak assumptions. In the Brownian case,this result was obtained in [8] by using a penalization method via non-reflected BSDEs. Notethat this method could also be adapted to our case with jumps, but would involve heavycomputations in order to prove the convergence of the solutions of the penalized BSDEs tothe solution of the reflected BSDE. It would also require some convergence results of theviscosity solutions theory in the integro-differential case. We provide here instead a directand shorter proof.Furthermore, under some additional assumptions, we prove a comparison theorem in theclass of bounded continuous functions, relying on a non-local version of Jensen-Ishii Lemma(see [3]), from which the uniqueness of the viscosity solution follows. We point out that ourproblem is not covered by the study in [3], since we are dealing with nonlinear BSDEs, andthis leads to a more complex integro-differential operator in the associated PDE.The paper is organized as follows: In Section 2 we give the formulation of our optimalstopping problem. In Section 3, we prove that the value function is a solution of an obstacleproblem for a PIDVI in the viscosity sense. In Section 4, we establish an uniqueness result.In the Appendix, we prove some estimates, from which we derive that the value function iscontinuous and has polynomial growth and provide some complementary results.2 Optimal Stopping Problem for Dynamic Risk Mea-sures with Jumps in the Markovian Case
Let (Ω , F , P ) be a probability space. Let W be a one-dimensional Brownian motion and N ( dt, du ) be a Poisson random measure with compensator ν ( du ) dt such that ν is a σ -finitemeasure on R ∗ equipped with its Borel field B ( R ∗ ) , and satisfies R R ∗ (1 ∧ e ) ν ( de ) < ∞ .Let ˜ N ( dt, du ) be its compensated process. Let IF = {F t , t ≥ } be the natural filtrationassociated with W and N .We consider a state process X which may be interpreted as an index, an interest rateprocess, an economic factor, an indicator of the market or the value of a portfolio, which hasan influence on the risk measure and the position. For each initial time t ∈ [0 , T ] and eachcondition x ∈ R , let X t,x be the solution of the following stochastic differential equation(SDE): X t,xs = x + Z st b ( X t,xr ) dr + Z st σ ( X t,xr ) dW r + Z st Z R ∗ β ( X t,xr − , e ) ˜ N ( dr, de ) , (2.1) { SDE } where b, σ : R → R are Lipschitz continuous, and β : R × R ∗ → R is a measurable functionsuch that for some non negative real C , and for all e ∈ R | β ( x, e ) | ≤ C (1 ∧ | e | ) , x ∈ R | β ( x, e ) − β ( x ′ , e ) | ≤ C | x − x ′ | (1 ∧ | e | ) , x, x ′ ∈ R . We introduce a dynamic risk measure ρ induced by a BSDE with jumps. For this, weconsider two functions γ and f satisfying the following assumption: Assumption H • γ : R × R ∗ → R is B ( R ) ⊗ B ( R ∗ )-measurable, | γ ( x, e ) − γ ( x ′ , e ) | < C | x − x ′ | (1 ∧ | e | ) , x, x ′ ∈ R , e ∈ R ∗ − ≤ γ ( x, e ) ≤ C (1 ∧ | e | ), e ∈ R ∗ • f : [0 , T ] × R × L ν → R is continuous in t uniformly with respect to x, y, z, k , andcontinuous in x uniformly with respect to y, z, k .(i) | f ( t, x, , , | ≤ C (1 + x p ) , ∀ x ∈ R (ii) | f ( t, x, y, z, k ) − f ( t, x ′ , y ′ , z ′ , k ′ ) | ≤ C ( | y − y ′ | + | z − z ′ | + k k − k ′ k L ν ), ∀ t ∈ [0 , T ], y, y ′ , z, z ′ ∈ R , k, k ′ ∈ L ν (iii) f ( t, x, y, z, k ) − f ( t, x, y, z, k ) ≥ < γ ( x, · ) , k − k > ν , ∀ t, x, y, z, k , k . Here, L ν denotes the set of Borelian functions ℓ : R ∗ → R such that k ℓ k ν := R R ∗ | ℓ ( u ) | ν ( du ) < + ∞ . It is a Hilbert space equipped with the scalar product h δ, ℓ i ν := R R ∗ δ ( e ) ℓ ( e ) ν ( de ) forall δ, ℓ ∈ L ν × L ν .
3e also introduce the set H (resp. H ν ) of predictable processes ( π t ) (resp. ( l t ( · )))such that E R T π s ds< ∞ (resp. E R T k l s k L ν ds< ∞ ); the set S of real-valued RCLL adaptedprocesses ( ϕ s ) with E [sup s ϕ s ] < ∞ , and the set L ( F T ) of F T -measurable and square-integrable random variables.Let ( t, x ) be a fixed intial condition. For each maturity S in [ t, T ] and each position ζ in L ( F S ), the associated risk measure at time s ∈ [ t, S ] is defined by ρ t,xs ( ζ , S ) := −E t,xs,S ( ζ ) , t ≤ s ≤ S, (2.2)where E t,x · ,S ( ζ ) denotes the f -conditional expectation, starting at ( t, x ), defined as the solutionin S of the BSDE with Lipschitz driver f ( s, X t,xs , y, z, k ), terminal condition ζ and terminaltime S , that is the solution ( E t,xs ) of − d E s = f ( s, X t,xs , E s , π s , l s ( · )) ds − π s dW s − Z R ∗ l s ( u ) ˜ N ( dt, du ) ; E S = ζ , (2.3) { } where ( π s ), ( l s ) are the associated processes, which belong to H and H ν respectively.The functional ρ : ( ζ , S ) → ρ · ( ζ , S ) defines then a dynamic risk measure induced by theBSDE with driver f (see [14]). Assumption H implies that the driver f ( s, X t,xs , y, z, k )satisfies Assumption 3.1 in [15], which ensures the monotonocity property of ρ with respectto ζ . More precisely, for each maturity S and for each positions ζ , ζ ∈ L ( F S ), with ζ ≤ ζ a.s., we have ρ t,xs ( ζ , S ) ≥ ρ t,xs ( ζ , S ) a.s.We now formulate our optimal stopping problem for dynamic risk measures. For each( t, x ) ∈ [0 , T ] × R , we consider a dynamic financial position given by the process ( ξ t,xs , t ≤ s ≤ T ),defined via the state process ( X t,xs ) and two functions g and h such that • g ∈ C ( R ) with at most polynomial growth at infinity, • h : [0 , T ] × R → R is continuous in t , x and there exist p ∈ N and a real constant C ,such that | h ( t, x ) | ≤ C (1 + | x | p ) , ∀ t ∈ [0 , T ] , x ∈ R , (2.4) { } • h ( T, x ) ≤ g ( x ) , ∀ x ∈ R . For each initial condition ( t, x ) ∈ [0 , T ] × R , the dynamic position is then defined by: ( ξ t,xs := h ( s, X t,xs ) , s < Tξ t,xT := g ( X t,xT ) . Let t ∈ [0 , T ] be the initial time and let x ∈ R be the initial condition. The minimal riskmeasure at time t is given by:ess inf τ ∈T t ρ t,xt ( ξ t,xτ , τ ) = − ess sup τ ∈T t E t,xt,τ ( ξ t,xτ ) . (2.5) { vs } Here T t denotes the set of stopping times with values in [ t, T ].4y Th. 3.2 in [15], the minimal risk measure is characterized via the solution Y t,x in S of the following reflected BSDE (RBSDE) associated with driver f and obstacle ξ : Y t,xs = g ( X t,xT ) + Z Ts f ( r, X t,xr , Y t,xr , Z t,xr , K t,xr ( · )) dr + A t,xT − A t,xs − Z Ts Z t,xr dW r − Z Ts Z R ∗ K t,x ( r, e ) ˜ N ( dr, de ) Y t,xs ≥ ξ t,xs , ≤ s ≤ T a.s. A t,x is a nondecreasing, continuous predictable process in S with A t,xt = 0 and such that Z Tt ( Y t,xs − ξ t,xs ) dA t,xs = 0 a.s. , (2.6) { markRBSDE } with Z t,x , K t,x ∈ H (resp. H ν ). Note that by the assumptions made on h and g , the obstacle( ξ ,t,xs ) s ≥ t is continuous except at the inaccessible jump times of the Poisson measure, andat time T with ∆ ξ t,xT ≤ A t,x by Th. 2.6 in [15].Moreover, Th. 3.2 in [15] ensures that Y t,xt = ess sup τ ∈T t E t,xt,τ ( ξ t,xτ ) a.s. (2.7)The SDE (2.1) and the RBSDE (2.6) can be solved with respect to the translated Brownianmotion ( W s − W t ) s ≥ t . Hence Y t,xt is constant for each t, x . We can thus define a deterministicfunction u called value function of our optimal stopping problem by setting for each t, xu ( t, x ) := Y t,xt . (2.8) { DEF } By Lemma A. A. u is continuous and has atmost polynomial growth.The continuity of u implies that Y t,xs = u ( s, X t,xs ), t ≤ s ≤ T a.s.Moreover, the stopping time τ ∗ ,t,x (also denoted by τ ∗ ), defined by τ ∗ := inf { s ≥ t, Y t,xs = ξ t,xs } = inf { s ≥ t, u ( s, X t,xs ) = ¯ h ( s, X t,xs ) } is an optimal stopping time for (2.5) (see Th. 3.6 in [15]). Here, the function ¯ h is definedby ¯ h ( t, x ) := h ( t, x ) t We consider the following related obstacle problem for a parabolic PIDE: min( u ( t, x ) − h ( t, x ) , − ∂u∂t ( t, x ) − Lu ( t, x ) − f ( t, x, u ( t, x ) , ( σ ∂u∂x )( t, x ) , Bu ( t, x )) = 0 , ( t, x ) ∈ [0 , T [ × R u ( T, x ) = g ( x ) , x ∈ R (3.1) { } where L := A + K,Aφ ( t, x ) := 12 σ ( x ) ∂ φ∂x ( t, x ) + b ( x ) ∂φ∂x ( t, x ) ,Kφ ( t, x ) := Z R ∗ (cid:18) φ ( t, x + β ( x, e )) − φ ( t, x ) − ∂φ∂x ( t, x ) β ( x, e ) (cid:19) ν ( de ) , (3.2) { defK3 } Bφ ( t, x )( · ) := φ ( t, x + β ( x, · )) − φ ( t, x ) ∈ L ν . The operator B and K are well defined for φ ∈ C , ([0 , T ] × R ). Indeed, since β is bounded,we have | φ ( t, x + β ( x, e )) − φ ( t, x ) | ≤ C | β ( x, e ) | and | φ ( t, x + β ( x, e )) − φ ( t, x ) − ∂φ∂x ( t, x ) β ( x, e ) | ≤ Cβ ( x, e ) . We prove below that the value function u defined by (2.8) is a viscosity solution of theabove obstacle problem. Definition 3.1. • A continuous function u is said to be a viscosity subsolution of (3.1)iff u ( T, x ) ≤ g ( x ) , x ∈ R , and iff for any point ( t , x ) ∈ (0 , T ) × R and for any φ ∈ C , ([0 , T ] × R ) such that φ ( t , x ) = u ( t , x ) and φ − u attains its minimum at ( t , x ), wehave min( u ( t , x ) − h ( t , x ) , − ∂φ∂t ( t , x ) − Lφ ( t , x ) − f ( t , x , u ( t , x ) , ( σ ∂φ∂x )( t , x ) , Bφ ( t , x )) ≤ . In other words, if u ( t , x ) > h ( t , x ), then − ∂φ∂t ( t , x ) − Lφ ( t , x ) − f ( t , x , u ( t , x ) , ( σ ∂φ∂x )( t , x ) , Bφ ( t , x )) ≤ . • A continuous function u is said to be a viscosity supersolution of (3.1) iff u ( T, x ) ≥ g ( x ) , x ∈ R , and iff for any point ( t , x ) ∈ (0 , T ) × R and for any φ ∈ C , ([0 , T ] × R ) suchthat φ ( t , x ) = u ( t , x ) and φ − u attains its maximum at ( t , x ), we havemin( u ( t , x ) − h ( t , x ) , − ∂φ∂t ( t , x ) − Lφ ( t , x ) − f ( t , x , u ( t , x ) , ( σ ∂φ∂x )( t , x ) , Bφ ( t , x )) ≥ . 6n other words, we have both u ( t , x ) ≥ h ( t , x ), and − ∂φ∂t ( t , x ) − Lφ ( t , x ) − f ( t , x , u ( t , x ) , ( σ ∂φ∂x )( t , x ) , Bφ ( t , x )) ≥ . Theorem 3.2. The function u , defined by (2.8) , is a viscosity solution (i.e. both a viscositysub- and supersolution) of the obstacle problem (3.1) .Proof. • We first prove that u is a subsolution of (3.1).Let ( t , x ) ∈ (0 , T ) × R and φ ∈ C , ([0 , T ] × R ) be such that φ ( t , x ) = u ( t , x ) and φ ( t, x ) ≥ u ( t, x ), ∀ ( t, x ) ∈ [0 , T ] × R . Suppose by contradictionthat u ( t , x ) > h ( t , x ) and that − ∂φ∂t ( t , x ) − Lφ ( t , x ) − f ( t , x , φ ( t , x ) , ( σ ∂φ∂x )( t , x ) , Bφ ( t , x )) > . By continuity of Kφ (which can be shown using Lebesgue’s theorem) and that of Bφ :[0 , T ] × R → L ν , we can suppose that there exists ε > η ε > ∀ ( t, x ) suchthat t ≤ t ≤ t + η ε < T and | x − x | ≤ η ε , we have: u ( t, x ) ≥ h ( t, x ) + ε and − ∂φ∂t ( t, x ) − Lφ ( t, x ) − f ( t, x, φ ( t, x ) , ( σ ∂φ∂x )( t, x ) , Bφ ( t, x )) ≥ ε. (3.3) { } Note that Y t ,x s = Y s,X t ,x s s = u ( s, X t ,x s ) a.s. because X t ,x is a Markov process and u iscontinuous. We define the stopping time θ as: θ := ( t + η ε ) ∧ inf { s ≥ t , | X t ,x s − x | > η ε } . (3.4) { } By definition of the stopping time θ , u ( s, X t ,x s ) ≥ h ( s, X t ,x s ) + ε > h ( s, X t ,x s ) , t ≤ s < θ a.s.This means that for a.e. ω the process ( Y t ,x s ( ω ) , s ∈ [ t , θ ( ω )[) stays strictly above thebarrier. It follows that for a.e. ω , the function s → A cs ( ω ) is constant on [ t , θ ( ω )]. Inother words, Y t ,x s = E t ,x s,θ ( Y θ ), t ≤ s ≤ θ a.s , that is ( Y t ,x s , s ∈ [ t , θ ]) is the solutionof the classical BSDE associated with driver f , terminal time θ and terminal value Y t ,x θ .Applying Itˆo’s lemma to φ ( t, X t ,x t ), we get: φ ( t, X t ,x t ) = φ ( θ, X t ,x θ ) − Z θt ψ ( s, X t ,x s ) ds − Z θt ( σ ∂φ∂x )( s, X t ,x s ) dW s − Z θt Z R ∗ Bφ ( s, X t ,x s − ) ˜ N ( ds, de ) (3.5) { } where ψ ( s, x ) := ∂φ∂s ( s, x ) + Lφ ( s, x ) . Note that ( φ ( s, X t ,x s ) , ( σ ∂φ∂x )( s, X t ,x s ) , Bφ ( s, X t ,x s − ); s ∈ [ t , θ ]) is the solution of the BSDEassociated to terminal time θ , terminal value φ ( θ, X t ,x θ ) and driver process − ψ ( s, X t ,x s ).7y (3.3) and the definition of the stopping time θ , we have a.s. that for each s ∈ [ t , θ ]: − ∂φ∂t ( s, X t ,x s ) − Lφ ( s, X t ,x s ) − f (cid:18) s, X t ,x s , φ ( s, X t ,x s ) , ( σ ∂φ∂x )( s, X t ,x s ) , Bφ ( s, X t ,x s ) (cid:19) ≥ ε. (3.6) { } Using the definition of the function ψ , (3.6) can be rewritten: for all s ∈ [ t , θ ], − ψ ( s, X t ,x s ) − f (cid:18) s, X t ,x s , φ ( s, X t ,x s ) , ( σ ∂φ∂x )( s, X t ,x s ) , Bφ ( s, X t ,x s ) (cid:19) ≥ ε. This gives a relation between the drivers − ψ ( s, X t ,x s ) and f ( s, X t ,x s , · ) of the two BSDEs.Also, φ ( θ, X t ,x θ ) ≥ u ( θ, X t ,x θ ) = Y t ,x θ a.s.Consequently, the extended comparison result for BSDEs with jumps given in the Appendix(see Proposition A.7) implies that: φ ( t , x ) = φ ( t , X t ,x t ) > Y t ,x t = u ( t , x ) , which leads to a contradiction. • We now prove that u is a viscosity supersolution of (3.1).Let ( t , x ) ∈ (0 , T ) × R and φ ∈ C , ([0 , T ] × R ) be such that φ ( t , x ) = u ( t , x ) and φ ( t, x ) ≤ u ( t, x ), ∀ ( t, x ) ∈ [0 , T ] × R . Since the solution ( Y t ,x s )stays above the obstacle, we have: u ( t , x ) ≥ h ( t , x ) . We must prove that: − ∂φ∂t ( t , x ) − Lφ ( t , x ) − f (cid:18) t , x , φ ( t , x ) , ( σ ∂φ∂x )( t , x ) , Bφ ( t , x ) (cid:19) ≥ . Suppose by contradiction that: − ∂φ∂t ( t , x ) − Lφ ( t , x ) − f (cid:18) t , x , φ ( t , x ) , ( σ ∂φ∂x )( t , x ) , Bφ ( t , x ) (cid:19) < . By continuity, we can suppose that there exists ε > η ε > t, x )such that t ≤ t ≤ t + η ε < T and | x − x | ≤ η ε , we have: − ∂φ∂t ( t, x ) − Lφ ( t, x ) − f (cid:18) t, x, φ ( t, x ) , ( σ ∂φ∂x )( t, x ) , Bφ ( t, x ) (cid:19) ≤ − ε. (3.7) { } We define the stopping time θ as: θ := ( t + η ε ) ∧ inf { s ≥ t / | X t ,x s − x | > η ε } . φ ( s, X t ,x s ), we get that( φ ( s, X t ,x s ) , ( σ ∂φ∂x )( s, X t ,x s ) , Bφ ( s, X t ,x s − ); s ∈ [ t , θ ]) is the solution of the BSDE associ-ated with terminal value φ ( θ, X t ,x θ ) and driver − ψ ( s, X t ,x s ).The process ( Y t ,x , s ∈ [ t , θ ]) is the solution of the classical BSDE associated withterminal condition Y t ,x θ = u ( θ, X t ,x θ ) and generalized driver f ( s, X t ,x s , y, z, q ) ds + dA t ,x s . By (3.7) and the definition of the stopping time θ , we have :( − ∂φ∂t ( s, X t ,x s ) − Lφ ( s, X t ,x s ) − f ( s, X t ,x s , φ ( s, X t ,x s ) , ( σ ∂φ∂x )( s, X t ,x s ) , Bφ ( s, X t ,x s ))) ds − dA t ,x s ≤ − ε ds, t ≤ s ≤ θ a.s.or, equivalently, − ψ ( s, X t ,x s ) ds ≤ ( f ( s, X t ,x s , φ ( s, X t ,x s ) , ( σ ∂φ∂x )( s, X t ,x s ) , Bφ ( s, X t ,x s ))) ds + dA t ,x s − ε ds, t ≤ s ≤ θ a.s.This gives a relation between the drivers of the two BSDEs.Also, φ ( θ, X t ,x θ ) ≤ u ( θ, X t ,x θ ) = Y t ,x θ a.s. Consequently, Proposition A.7 in the Appendiximplies that: φ ( t , x ) = φ ( t , X t ,x t ) < Y t ,x t = u ( t , x ) , which leads to a contradiction. We provide a uniqueness result for (3.1) in the particular case when for each φ ∈ C , ([0 , T ] × R ), Bφ is a map valued in R instead of L ν . More precisely, Bφ ( t, x ) := Z R ∗ ( φ ( t, x + β ( x, e )) − φ ( t, x )) γ ( x, e ) ν ( de ) , (4.1) { defB } which is well defined since | φ ( t, x + β ( x, e )) − φ ( t, x ) | ≤ C | β ( x, e ) | . We suppose that Assumption H holds and we make the additional assumptions: Assumption H : . f ( s, X t,xs ( ω ) , y, z, k ) := f (cid:18) s, X t,xs ( ω ) , y, z, Z R ∗ k ( e ) γ ( X t,xs ( ω ) , e ) ν ( de ) (cid:19) s ≥ t , where f : [0 , T ] × R → R is continuous in t uniformly with respect to x, y, z, k , continuousin x uniformly with respect to y, z, k , and satisfies:9i) | f ( t, x, , , | ≤ C, for all t ∈ [0 , T ] , x ∈ R . (ii) | f ( t, x, y, z, k ) − f ( t, x ′ , y ′ , z ′ , k ′ ) | ≤ C ( | y − y ′ | + | z − z ′ | + | k − k ′ | ), for all t ∈ [0 , T ], y, y ′ , z, z ′ , k, k ′ ∈ R .(iii) k f ( t, x, y, z, k ) is non-decreasing, for all t ∈ [0 , T ], x, y, z ∈ R .2. For each R > 0, there exists a continuous function m R : R + → R + such that m R (0) = 0and | f ( t, x, v, p, q ) − f ( t, y, v, p, q ) | ≤ m R ( | x − y | (1 + | p | )) , for all t ∈ [0 , T ], | x | , | y | ≤ R, | v | ≤ R, p, q ∈ R . | γ ( x, e ) − γ ( y, e ) | ≤ C | x − y | (1 ∧ e ) and 0 ≤ γ ( x, e ) ≤ C (1 ∧ | e | ), for all x, y ∈ R , e ∈ R ∗ . 4. There exists r > t ∈ [0 , T ], x, u, v, p, l ∈ R : f ( t, x, v, p, l ) − f ( t, x, u, p, l ) ≥ r ( u − v ) when u ≥ v. | h ( t, x ) | + | g ( x ) | ≤ C , for all t ∈ [0 , T ], x ∈ R .To simplify notation, f is denoted by f in the sequel.We state below a comparison theorem, which uses results of three lemmas. The proofsof these lemmas are given in Subsection 4.1. Theorem 4.1 (Comparison principle) . Under the above hypotheses, if U is a viscosity sub-solution and V is a viscosity supersolution of the obstacle problem (3.1) in the class ofcontinuous bounded functions, then U ( t, x ) ≤ V ( t, x ) , for each ( t, x ) ∈ [0 , T ] × R .Proof. Set M := sup [0 ,T ] × R ( U − V ) . It is sufficient to prove that M ≤ 0. For each ε, η > 0, we introduce the function: ψ ε,η ( t, s, x, y ) := U ( t, x ) − V ( s, y ) − ( x − y ) ε − ( t − s ) ε − η ( x + y ) , for t, s, x, y in [0 , T ] × R . Let M ε,η := max [0 ,T ] × R ψ ε,η . This supremum is reached at some point ( t ε,η , s ε,η , x ε,η , y ε,η ) . Using that ψ ε,η ( t ε,η , s ε,η , x ε,η , y ε,η ) ≥ ψ ε,η (0 , , , U ( t ε,η , x ε,η ) − V ( s ε,η , y ε,η ) − ( t ε,η − s ε,η ) ε − ( x ε,η − y ε,η ) ε − η (( x ε,η ) + ( y ε,η ) ) ≥ U (0 , − V (0 , , (4.2) { } or, equivalently,( t ε,η − s ε,η ) ε + ( x ε,η − y ε,η ) ε + η (( x ε,η ) + ( y ε,η ) ) ≤ k U k ∞ + k V k ∞ − U (0 , − V (0 , . (4.3) { } C such that: | x ε,η − y ε,η | + | t ε,η − s ε,η | ≤ Cε (4.4) { } | x ε,η | ≤ Cη , | y ε,η | ≤ Cη . (4.5) { } Extracting a subsequence if necessary, we may suppose that for each η the sequences ( t ε,η ) ε and ( s ε,η ) ε converge to a common limit t η when ε tends to 0, and from (4.4) and (4.5) we mayalso suppose, extracting again, that for each η , the sequences ( x ε,η ) ε and ( y ε,η ) ε converge toa common limit x η . Lemma 4.2. We have: lim ε → ( x ε,η − y ε,η ) ε = 0 ; lim ε → ( t ε,η − s ε,η ) ε = 0lim η → lim ε → M ε,η = M. We now introduce the functions:Ψ ( t, x ) := V ( s ε,η , y ε,η ) + ( x − y ε,η ) ε + ( t − s ε,η ) ε + η ( x + ( y ε,η ) )Ψ ( s, y ) := U ( t ε,η , x ε,η ) − ( x ε,η − y ) ε − ( t ε,η − s ) ε − η (( x ε,η ) + y ) . As ( t, x ) → ( U − Ψ )( t, x ) reaches its maximum at ( t ε,η , x ε,η ) and U is a subsolution wehave two cases: • t ε,η = T and then U ( t ε,η , x ε,η ) ≤ g ( x ε,η ), • t ε,η = T and thenmin (cid:18) U ( t ε,η , x ε,η ) − h ( t ε,η , x ε,η ) , ∂ Ψ ∂t ( t ε,η , x ε,η ) − L Ψ ( t ε,η , x ε,η ) −− f (cid:18) t ε,η , x ε,η , U ( t ε,η , x ε,η ) , ( σ ∂ Ψ ∂x )( t ε,η , x ε,η ) , B Ψ ( t ε,η , x ε,η ) (cid:19)(cid:19) ≤ . (4.6) { } As ( s, y ) → (Ψ − V )( s, y ) reaches its maximum at ( s ε,η , y ε,η ) and V is a supersolution wehave the two following cases: • s ε,η = T and then V ( s ε,η , y ε,η ) ≥ g ( y ε,η ), • s ε,η = T and thenmin( V ( s ε,η , y ε,η ) − h ( s ε,η , y ε,η ) ,∂ Ψ ∂t ( s ε,η , y ε,η ) − L Ψ ( s ε,η , y ε,η ) − f ( s ε,η , y ε,η , V ( s ε,η , y ε,η ) , (4.7) { } ( σ ∂ Ψ ∂x )( s ε,η , y ε,η ) , B Ψ ( s ε,η , y ε,η )) ≥ . 11e now prove that M ≤ . Three cases are possible. There exists a subsequence of ( t η ) such that t η = T for all η (of this subsequence).As U is continuous, for all η and for ε small enough U ( t ε,η , x ε,η ) ≤ U ( t η , x η ) + η ≤ g ( x η ) + η, and as V is continuous, for all η and for ε small enough V ( s ε,η , y ε,η ) ≥ V ( t η , x η ) − η ≥ g ( x η ) − η. Hence U ( t ε,η , x ε,η ) − V ( s ε,η , y ε,η ) ≤ η and M ε,η = U ( t ε,η , x ε,η ) − V ( s ε,η , y ε,η ) − ( x ε,η − y ε,η ) ε − ( t ε,η − s ε,η ) ε − η (( x ε,η ) + ( y ε,η ) ) ≤ U ( t ε,η , x ε,η ) − V ( s ε,η , y ε,η ) ≤ η. Letting ε → η → M ≤ There exists a subsequence such that t η = T , and for all η belonging to thissubsequence, there exists a subsequence of ( x ε,η ) η such that U ( t ε,η , x ε,η ) − h ( t ε,η , x ε,η ) ≤ . As from (4.7) one has V ( s ε,η , y ε,η ) − h ( s ε,η , y ε,η ) ≥ , it comes that M ε,η ≤ U ( t ε,η , x ε,η ) − V ( s ε,η , y ε,η ) ≤ h ( t ε,η , x ε,η ) − h ( s ε,η , y ε,η ) . Letting ε → η → 0, using the equality lim η → lim ε → M ε,η = M (see Lemma 4.2),we derive that M ≤ Last case: We are left with the case when, for a subsequence of η , we have t η = T andfor all η belonging to this subsequence there exists a subsequence of ( x ε,η ) ε such that: U ( t ε,η , x ε,η ) − h ( t ε,η , x ε,η ) > . Set ϕ ( t, s, x, y ) := ( x − y ) ε + ( t − s ) ε + η ( x + y ) . (4.8) { } The maximum of the function ψ ε,η ( t, s, x, y ) := U ( t, x ) − V ( s, y ) − ϕ ( t, s, x, y ) is reached atthe point ( t ε,η , s ε,η , x ε,η , y ε,η ). We apply the non-local version of Jensen Ishii’s lemma [3] andwe obtain that there exist:( a, p, X ) ∈ P , + U ( t ε,η , x ε,η ) , ( b, q, Y ) ∈ P , − V ( s ε,η , y ε,η )12uch that p = p + 2 η x ε,η ; q = p − η y ε,η ; p = x ε,η − y ε,η ) ε a = b = t ε,η − s ε,η ) ε X − Y ! ≤ ε − − ! + 2 η ! . Here, P , + (resp. P , − ) is the set of superjets (resp. subjets) defined in [3] (see Definition3). Since ( t ε,η , s ε,η , x ε,η , y ε,η ) is a global maximum of ψ ε,η ,we have: ψ ε,η ( t ε,η , s ε,η , x ε,η + β ( x ε,η , e ) , y ε,η + β ( y ε,η , e )) ≤ ψ ε,η ( t ε,η , s ε,η , x ε,η , y ε,η ) ⇔ U ( t ε,η , x ε,η + β ( x ε,η , e )) − V ( s ε,η , y ε,η + β ( y ε,η , e )) − ( x ε,η + β ( x ε,η , e ) − y ε,η − β ( y ε,η , e )) ε − ( t ε,η − s ε,η ) ε − η (( x ε,η + β ( x ε,η , e )) + ( y ε,η + β ( y ε,η , e )) ) ≤ U ( t ε,η , x ε,η ) − V ( s ε,η , y ε,η ) − ( x ε,η − y ε,η ) ε − ( t ε,η − s ε,η ) ε − η (( x ε,η ) + ( y ε,η ) ) . Consequently, we get: U ( t ε,η , x ε,η + β ( x ε,η , e )) − U ( t ε,η , x ε,η ) ≤ V ( s ε,η , y ε,η + β ( y ε,η , e )) − V ( s ε,η , y ε,η ) + ( β ( x ε,η , e ) − β ( y ε,η , e )) ε + p ( β ( x ε,η , e ) − β ( y ε,η , e ))+ η ( β ( x ε,η , e ) + 2 x ε,η β ( x ε,η , e ) + 2 y ε,η β ( y ε,η , e ) + β ( y ε,η , e )) . (4.9) { } Let us fix δ > B δ = B (0 , δ ). We introduce the operators K δ ,˜ K δ , B δ , ˜ B δ corresponding to the operators K and B defined in (3.2) and (4.1), but integratingon B δ or R \B δ (also denoted by B cδ ) only.They are defined respectively for all φ ∈ C , , Φ ∈ C by K δ [ t, x, φ ] := Z B δ (cid:18) φ ( t, x + β ( x, e )) − φ ( t, x ) − ∂φ∂x ( t, x ) β ( x, e ) (cid:19) ν ( de ) (4.10) { estim1 } ˜ K δ [ t, x, π, Φ] := Z B cδ (cid:18) Φ( t, x + β ( x, e )) − Φ( t, x ) − πβ ( x, e ) (cid:19) ν ( de ) . (4.11) { estim2 } B δ [ t, x, φ ] := Z B δ (cid:18) φ ( t, x + β ( x, e )) − φ ( t, x ) (cid:19) γ ( x, e ) ν ( de ) (4.12) { estim5 } ˜ B δ [ t, x, Φ] := Z B cδ (cid:18) Φ( t, x + β ( x, e )) − Φ( t, x ) (cid:19) γ ( x, e ) ν ( de ) (4.13) { estim6 } Here C denotes the set of bounded continuous functions.13ote that the operators K δ , ˜ K δ , B δ and ˜ B δ satisfy the hypotheses (NLT) of [3] (see Sec-tion 2.2 in [3]). Hence we can use the alternative definition for sub-superviscosity solutionsin terms of sub-superjets (see Definition 4 in [3]). Since U is a subviscosity solution and V is superviscosity solution, we have: F ( t ε,η , x ε,η , U ( t ε,η , x ε,η ) , a, p, X, K δ [ t ε,η , x ε,η , ϕ x ]+ ˜ K δ [ t ε,η , x ε,η , p, U ] , B δ [ t ε,η , x ε,η , ϕ x ] + ˜ B δ [ t ε,η , x ε,η , U ]) ≤ F ( s ε,η , y ε,η , V ( s ε,η , y ε,η ) , a, q, Y, K δ [ s ε,η , y ε,η , − ϕ y ]+ ˜ K δ [ s ε,η , y ε,η , q, V ] , B δ [ s ε,η , y ε,η , − ϕ y ] + ˜ B δ [ s ε,η , y ε,η , V ]) ≥ { } where F ( t, x, u, a, p, X, l , l ) := − a − σ ( x ) X − b ( x ) p − l − f ( t, x, u, pσ ( x ) , l ) . (4.15) { } We denote by ϕ x the function ( t, x ) ϕ ( t, x, s ε,η , y ε,η ) and by ϕ y the function ( s, y ) ϕ ( t ε,η , x ε,η , s, y ). The two following lemmas hold. Lemma 4.3. Let l K := K δ [ t ε,η , x ε,η , ϕ x ] + ˜ K δ [ t ε,η , x ε,η , p, U ] l ′ K := K δ [ s ε,η , y ε,η , − ϕ y ] + ˜ K δ [ s ε,η , y ε,η , q, V ] . (4.16) { } We have l K ≤ l ′ K + O ( ( x ε,η − y ε,η ) ε ) + O ( η ) + ( 1 ε + η ) O ( δ ) . (4.17) { estim } Lemma 4.4. Let l B := B δ [ t ε,η , x ε,η , ϕ x ] + ˜ B δ [ t ε,η , x ε,η , U ] l ′ B := B δ [ s ε,η , y ε,η , − ϕ y ] + ˜ B δ [ s ε,η , y ε,η , V ] . (4.18) { } We have l B ≤ l ′ B + ( η + 1 ε ) O ( δ ) + O ( ( x ε,η − y ε,η ) ε ) + O ( | x ε,η − y ε,η | ) + O ( η ) . (4.19) { } We argue now by contradiction by assuming that M > . (4.20) { absurde } H ) . 4, we get0 < r M ≤ rM ε,η ≤ r ( U ( t ε,η , x ε,η ) − V ( s ε,η , y ε,η )) ≤ F ( s ε,η , y ε,η , U ( t ε,η , x ε,η ) , a, q, Y, l ′ K , l ′ B ) − F ( s ε,η , y ε,η , V ( s ε,η , y ε,η ) , a, q, Y, l ′ K , l ′ B )= F ( s ε,η , y ε,η , U ( t ε,η , x ε,η ) , a, q, Y, l ′ K , l ′ B ) − F ( s ε,η , y ε,η , U ( s ε,η , y ε,η ) , a, q, Y, l ′ K , l ′ B )+ F ( s ε,η , y ε,η , U ( s ε,η , y ε,η ) , a, q, Y, l ′ K , l ′ B ) − F ( s ε,η , y ε,η , U ( s ε,η , y ε,η ) , a, q, Y, l K , l B )+ F ( s ε,η , y ε,η , U ( s ε,η , y ε,η ) , a, q, Y, l K , l B ) − F ( t ε,η , x ε,η , U ( t ε,η , x ε,η ) , a, p, X, l K , l B )+ F ( t ε,η , x ε,η , U ( t ε,η , x ε,η ) , a, p, X, l K , l B ) − F ( s ε,η , y ε,η , V ( s ε,η , y ε,η ) , a, q, Y, l ′ K , l ′ B ) ≤ K | U ( t ε,η , x ε,η ) − U ( s ε,η , y ε,η ) | + F ( s ε,η , y ε,η , U ( s ε,η , y ε,η ) , a, q, Y, l K , l B ) − F ( t ε,η , x ε,η , U ( t ε,η , X ε,η ) , a, p, X, l K , l B )+ ( η + 1 ε ) O ( δ ) + O ( ( x ε,η − y ε,η ) ε ) + O ( | x ε,η − y ε,η | ) + O ( η ) . (4.21) { } We have used here the (nonlocal) ellipticity of F , the Lipschitz property of F , (4.14) andthe estimates proven in Lemma 4.3 and Lemma 4.4. From the hypothesis on b and σ , wehave: σ ( x ε,η ) X − σ ( y ε,η ) Y ≤ C ( x ε,η − y ε,η ) ε + O ( η ) ,b ( x ε,η ) p − b ( y ε,η ) q ≤ C | x ε,η − y ε,η | ε + O ( η ) . We thus obtain the inequality: F ( s ε,η , y ε,η , U ( s ε,η , y ε,η ) , a, q, Y, l K , l B ) − F ( t ε,η , x ε,η , U ( t ε,η , x ε,η ) , a, p, X, l K , l B ) ≤ C ( x ε,η − y ε,η ) ε + O ( η )+ f ( t ε,η , x ε,η , U ( t ε,η , x ε,η ) , ( p + 2 η ) σ ( x ε,η ) , l B ) − f ( s ε,η , y ε,η , U ( s ε,η , y ε,η ) , ( p − η ) σ ( y ε,η ) , l B ) ≤ f ( t ε,η , x ε,η , U ( t ε,η , x ε,η ) , ( p + 2 η ) σ ( x ε,η ) , l B ) − f ( s ε,η , x ε,η , U ( t ε,η , x ε,η ) , ( p + 2 η ) σ ( x ε,η ) , l B )+ m R ( | x ε,η − y ε,η | (1 + ( p + 2 η ) σ ( x ε,η )))+ K | U ( t ε,η , x ε,η ) − U ( s ε,η , y ε,η ) | + O ( ( x ε,η − y ε,η ) ε ) + O ( η ) . (4.22) { } < r M ≤ rM ε,η ≤ f ( t ε,η , x ε,η , U ( t ε,η , x ε,η ) , ( p + 2 η ) σ ( x ε,η ) , l B ) − f ( s ε,η , x ε,η , U ( t ε,η , x ε,η ) , ( p + 2 η ) σ ( x ε,η ) , l B )+ m R ( | x ε,η − y ε,η | (1 + ( p + 2 η ) σ ( x ε,η ))+ K | U ( t ε,η , x ε,η ) − U ( s ε,η , y ε,η ) | ++ O ( ( x ε,η − y ε,η ) ε ) + O ( | x ε,η − y ε,η | ) + ( η + 1 ε ) O ( δ ) + O ( η ) . (4.23) { } By Lemma 4.2, letting successively δ, ε and η tend to 0 in (4.23) we obtain that 0 < r M ≤ M > Corollary 4.5 (Uniqueness) . Under the additional Assumption ( H ) , the value function isthe unique solution of the obstacle problem (3 . in the class of bounded continuous functions. Proof of Lemma 4.2. For η > 0, we introduce the functions:˜ U η ( t, x ) = U ( t, x ) − η x and ˜ V η ( t, x ) = V ( t, x ) + η x . Set M η := sup [0 ,T ] × R ( ˜ U η − ˜ V η ) . The maximum M η is reached at some point (ˆ t η , ˆ x η ). From the form of ψ ε,η , we have thatfor fixed η , there exists a subsequence ( t ε,η , s ε,η , x εη , y ε,η ) ε which converges to some point( t η , s η , x η , y η ) when ε tends to 0.Since M ε,η is reached at ( t ε,η , s ε,η , x ε,η , y ε,η ), we have:( ˜ U η − ˜ V η )(ˆ t η , ˆ x η ) = ( U − V )(ˆ t η , ˆ x η ) − η ((ˆ x η ) + (ˆ y η ) ) ≤ M ε,η = U ( t ε,η , x ε,η ) − V ( s ε,η , y ε,η ) − ( t ε,η − s ε,η ) ε − ( x ε,η − y ε,η ) ε − η (( x ε,η ) + ( y ε,η ) ) . Setting l η := lim sup ε → ( x ε,η − y ε,η ) ε , l η := lim inf ε → ( x ε,η − y ε,η ) ε we get 0 ≤ l η ≤ l η ≤ ( ˜ U η − ˜ V η )( t η , x η ) − ( ˜ U η − ˜ V η )(ˆ t η , ˆ x η ) ≤ . (4.24) { lim1 } We derive that, up to a subsequence, lim ε → x ε,η − y ε,η ) ε = 0 andlim ε → M ε,η = M η . Similarly, we get lim ε → t ε,η − s ε,η ) ε = 0.16et us prove that lim η → M η = M . First, note that M η ≤ M, for all η .By definition of M , for all δ > t δ , x δ ) ∈ [0 , T ] × R such that M − δ ≤ ( U − V )( t δ , x δ ) . Consequently, we get M − η x δ − δ ≤ ( U − V )( t δ , x δ ) − η x δ = ( ˜ U η − ˜ V η )( t δ , x δ ) ≤ M η ≤ M. By letting η and then δ tend to 0, the result follows. Proof of Lemma 4.3. We have: K δ [ t ε,η , x ε,η , ϕ x ] = Z B δ ( 1 ε + η ) β ( x ε,η , e ) ν ( de ) (4.25) { op1 } K δ [ s ε,η , y ε,η , − ϕ y ] = Z B δ ( − ε − η ) β ( y ε,η , e ) ν ( de ) . (4.26) { op2 } Equations (4.25) and (4.26) imply: K δ [ t ε,η , x ε,η , ϕ x ] ≤ K δ [ s ε,η , y ε,η , − ϕ y ]+( 1 ε + η ) Z B δ β ( y ε,η , e ) ν ( de )+ ( 1 ε + η ) Z B δ β ( x ε,η , e ) ν ( de ) ≤ K δ [ s ε,η , y ε,η , − ϕ y ]+( 1 ε + η ) O ( δ ) . (4.27) { } Using inequality (4.9) and integrating on B cδ , we obtain:˜ K δ [ t ε,η , x ε,η , p, U ] = Z B cδ (cid:18) U ( t ε,η , x ε,η + β ( x ε,η , e )) − U ( t ε,η , x ε,η ) − ( p + 2 η x ε,η ) β ( x ε,η , e ) (cid:19) ν ( de ) ≤ Z B cδ (cid:18) V ( s ε,η , y ε,η + β ( y ε,η , e )) − V ( s ε,η , y ε,η ) − ( p − η y ε,η ) β ( y ε,η , e ) (cid:19) ν ( de ) + Z B cδ ( β ( x ε,η , e ) − β ( y ε,η , e )) ε ν ( de )+ η Z B cδ ( β ( x ε,η , e ) + β ( y ε,η , e )) ν ( de ) ≤ ˜ K δ [ s ε,η , y ε,η , q, V ] + O ( ( x ε,η − y ε,η ) ε ) + O ( η ) . Using (4.16) and (4.27), we derive (4.17), which ends the proof of Lemma 4.3. Proof of Lemma 4.4. From (4.12), we derive that: B δ [ t ε,η , x ε,η , ϕ x ] = Z B δ (cid:18) ( η + 1 ε ) β ( x ε,η , e ) + 2 β ( x ε,η , e ) ε ( x ε,η − y ε,η )+ 2 η x ε,η β ( x ε,η , e ) (cid:19) γ ( x ε,η , e ) ν ( de ) (4.28) { e1 } B δ [ s ε,η , y ε,η , − ϕ y ] = Z B δ (cid:18) ( − η − ε ) β ( y ε,η , e ) + 2 β ( y ε,η , e ) ε ( x ε,η − y ε,η ) − η y ε,η β ( y ε,η , e ) (cid:19) γ ( y ε,η , e ) ν ( de ) . (4.29) { e2 } (cid:18) ( η + 1 ε ) β ( x ε,η , e ) + 2 β ( x ε,η , e ) ε ( x ε,η − y ε,η ) + 2 η x ε,η β ( x ε,η , e ) (cid:19) γ ( x ε,η , e )= ( − η − ε ) β ( y ε,η , e ) γ ( y ε,η , e ) + 2 β ( y ε,η , e ) ε ( x ε,η − y ε,η ) γ ( y ε,η , e ) − η y ε,η β ( y ε,η , e ) γ ( y ε,η , e )+ ( η + 1 ε ) (cid:18) β ( y ε,η , e ) γ ( y ε,η , e ) + β ( x ε,η , e ) γ ( x ε,η , e ) (cid:19) + 2 ε ( x ε,η − y ε,η ) (cid:18) β ( x ε,η , e ) γ ( x ε,η , e ) − β ( y ε,η , e ) γ ( y ε,η , e ) (cid:19) + 2 η (cid:18) x ε,η β ( x ε,η , e ) γ ( x ε,η , e ) + y ε,η β ( y ε,η , e ) γ ( y ε,η , e ) (cid:19) . (4.30) { e3 } From (4.28), (4.29), (4.30) and using the hypothesis on β and γ , we get: B δ [ t ε,η , x ε,η , ϕ x ] ≤ B δ [ s ε,η , y ε,η , − ϕ y ] + ( η + 1 ε ) O ( δ ) + O ( ( x ε,η − y ε,η ) ε ) + O ( η ) . (4.31) { } We now estimate the operator ˜ B δ . Inequality (4.9) implies: (cid:18) U ( t ε,η , x ε,η + β ( x ε,η , e )) − U ( t ε,η , x ε,η ) (cid:19) γ ( x ε,η , e ) ≤ (cid:18) V ( s ε,η , y ε,η + β ( y ε,η , e )) − V ( s ε,η , y ε,η )+ | β ( x ε,η , e ) − β ( y ε,η , e ) | ε + p ( β ( x ε,η , e ) − β ( y ε,η , e ))+ η ( β ( x ε,η , e ) + 2 x ε,η β ( x ε,η , e ) + 2 y ε,η β ( y ε,η , e ) + β ( y ε,η , e ) (cid:19) γ ( x ε,η , e )= (cid:18) V ( s ε,η , y ε,η + β ( y ε,η , e )) − V ( s ε,η , y ε,η ) (cid:19) γ ( y ε,η , e )+ (cid:18) V ( s ε,η , y ε,η + β ( y ε,η , e )) − V ( s ε,η , y ε,η ) (cid:19)(cid:18) γ ( x ε,η , e ) − γ ( y ε,η , e ) (cid:19) + | β ( x ε,η , e ) − β ( y ε,η , e ) | ε γ ( x ε,η , e ) + p (cid:18) β ( x ε,η , e ) − β ( y ε,η , e ) (cid:19) γ ( x ε,η , e )+ η (cid:18) β ( x ε,η , e ) + 2 x ε,η β ( x ε,η , e ) + 2 y ε,η β ( y ε,η , e ) + β ( y ε,η , e ) (cid:19) γ ( x ε,η , e ) . Now, by (4.5), we have | x ε,η | ≤ Cη and | y ε,η | ≤ Cη . Hence, using the hypothesis on β, γ andintegrating on B cδ , we get˜ B δ [ t ε,η , x ε,η , U ] ≤ ˜ B δ [ s ε,η , y ε,η , V ] + O ( | x ε,η − y ε,η | ) + O ( ( x ε,η − y ε,η ) ε ) + O ( η ) . (4.32) { } Finally, from (4.31), (4.18) and (4.32), we derive inequality (4.19).18 Conclusions In this paper, we have studied the optimal stopping problem for a monotonous dynamic riskmeasure defined by a Markovian BSDE with jumps. We have proven that, under relativelyweak hypotheses, the value function is a viscosity solution of an obstacle problem for apartial integro-differential variational inequality. To obtain the uniqueness of the solutionunder appropriate conditions, we have proven a comparison theorem, based on the nonlocalversion of the Jensen Ishii Lemma, which extends some results established in [3] (Section5.1, Th.3) to the case of a nonlinear BSDE.The links given in this paper between optimal stopping problems for BSDEs and obstacleproblems for PDEs can be extended to a larger class of problems. Among them, we canmention generalized Dynkin games with nonlinear expectation (see [6]), and mixed optimalstopping/stochastic control problems (see [5]). However, the latter case requires to establisha dynamic programming principle, which does not follow from the flow property of reflectedBSDEs only, and needs rather sophisticated techniques. A Appendix A.1 Some Useful Estimates Let T > f : [0 , T ] × Ω × R × L ν → R ; ( t, ω, y, z, k ) f ( t, ω, y, z, k ) is said to be a Lipschitz driver if it is predictable, uniformly Lipchitz with respect to y, z, k and such that f ( t, , , ∈ H . Let ξ t , ξ t ∈ S . Let f , f be two admissible Lipschitz drivers with Lipchitz constant C .For i = 1 , 2, let E i be the f i -conditional expectation associated with driver f i , and let ( Y it )be the adapted process defined for each t ∈ [0 , T ], Y it := ess sup τ ∈T t E it,τ ( ξ iτ ) . (A.1) { procV } Proposition A.1. For s ∈ [0 , T ] , denote Y s = Y s − Y s , ξ s = ξ s − ξ s and f s = sup y,z,k | f ( s, y, z, k ) − f ( s, y, z, k ) | . Let η, β > be such that β ≥ η + 2 C and η ≤ C .Then for each t , we have: e βt Y t ≤ e βT ( E [sup s ≥ t ξ s |F t ] + η E [ Z Tt f s ds |F t ]) a.s. (A.2) { eqA.1 } Proof. For i = 1 , τ ∈ T , let ( X i,τ , π i,τs , l i,τs ) be the solution of the BSDEassociated with driver f i , terminal time τ and terminal condition ξ iτ . Set X τs = X ,τs − X ,τs .By a priori estimate on BSDEs (see Proposition A. e βt ( X τt ) ≤ e βT E [ ξ τ |F t ] + η E [ Z Tt e βs ( f ( s, X ,τs , π ,τs , l ,τs ) − f ( s, X ,τs , π ,τs , l ,τs )) ds |F t ] a.s. (A.3) { A.2 } e βt ( X τt ) ≤ e βT ( E [sup s ≥ t ξ s |F t ] + η E [ Z Tt f s ds |F t ]) . (A.4) { A.3 } Now, by definition of Y i , we have Y it = ess sup τ ≥ t X i,τt a.s. for i = 1 , 2. We thus get | Y t | ≤ ess sup τ ≥ t | X τt | a.s. The result follows.Let ξ t ∈ S . Let f be a Lipschitz driver with Lipschitz constant C > 0. Set Y t := ess sup τ ∈T t E t,τ ( ξ τ ) (A.5) { procV } where E is the f -conditional expectation associated with driver f . Proposition A.2. Let η, β > be such that β ≥ η + 2 C and η ≤ C . Then for each t , wehave: e βt Y t ≤ e βT ( E [sup s ≥ t ξ s |F t ] + η E [ Z Tt f ( s, , , ds |F t ]) a.s. (A.6) Proof. Let X τt be the solution of the BSDE associated with driver f , terminal time τ andterminal condition ξ τ . By applying inequality (A.3) with f = f , ξ = ξ , f = 0 and ξ = 0,we get: e βt ( X τt ) ≤ e βT E [ ξ τ |F t ] + η E [ Z Tt e βs ( f ( s, , , |F t ] . (A.7) { A.5 } The result follows. Remark A.3 . If the drivers satisfy Assumption 3.1 in [15], then Y (resp. Y i ) is the solutionof the RBSDE associated with driver f (resp. f i ) and obstacle ξ (resp. ξ i ). Hence theabove estimates provide some new estimates on RBSDEs. Note that η and β are universalconstants, i.e. they do not depend on T , ξ, ξ , ξ , f, f , f . This was not the case for theestimates given in the previous literature (see e.g. [8]). A.2 Some Properties of the Value Function u We prove below the continuity and polynomial growth of the function u defined by (2.8). Lemma A.4. The function u is continuous in ( t, x ) .Proof. It is sufficient to show that, when ( t n , x n ) → ( t, x ), | u ( t n , x n ) − u ( t, x ) | → h be the map defined by ¯ h ( t, x ) = h ( t, x ) for t < T and ¯ h ( T, x ) = g ( x ), so that, foreach ( t, x ), we have ξ t,xs = ¯ h ( s, X t,xs ), 0 ≤ s ≤ T a.s. By applying Proposition A.1 with X s = X t n ,x n s , X s = X t,xs , f ( s, ω, y, z, q ) := [ t,T ] ( s ) f ( s, X t,xs ( ω ) , y, z, q ) and f ( s, ω, y, z, q ) := [ t n ,T ] ( s ) f ( s, X t n ,x n s ( ω ) , y, z, q ), we obtain: | u ( t n , x n ) − u ( t, x ) | ≤ K C,T E [ sup ≤ s ≤ T | h ( s, X t n ,x n s ) − h ( s, X t,xs ) | + Z T ( f ns ) ] , K C,T := e (3 C +2 C ) T max(1 , C ) f ns ( ω ) := sup y,z,q | [ t,T ] f ( s, X t,xs ( ω ) , y, z, q ) − [ t n ,T ] f ( s, X t n ,x n s ( ω ) , y, z, q ) | . The continuity of u is then a consequence of the following convergences as n → ∞ : E ( sup ≤ s ≤ T | h ( s, X t,xs ) − h ( s, X t n s ( x n )) | ) → E [ Z T ( f ns ) ds ] → , which follow from the Lebesgue’s theorem, using the continuity assumptions and polynomialgrowth of f and h . Lemma A.5. The function u has at most polynomial growth at infinity.Proof. By applying Prop. A. u ( t, x ) ≤ K C,T ( E ( Z T f ( s, X t,xs , , , ds + sup ≤ s ≤ T h ( s, X t,xs ) ) . (A.8) { } Using now the hypothesis of polynomial growth on f, h, g and the standard estimate E [ sup ≤ s ≤ T | X t,xs | ] ≤ C ′ (1 + x ) , we derive that there exist ¯ C ∈ R and p ∈ N such that | u ( t, x ) | ≤ ¯ C (1 + x p ) , ∀ t ∈ [0 , T ], ∀ x ∈ R . Remark A.6 . By (A.8) , if ( t, x ) f ( t, x, , , h and g are bounded, then u is bounded. A.3 An Extension of the Comparison Result for BSDEs withJumps We provide here an extension of the comparison theorem for BSDEs given in [14] whichformally states that if two drivers f , f satisfy f ≥ f + ε , then the associated solutions X and X satisfy X > X . Proposition A.7. Let t ∈ [0 , T ] and let θ be a stopping time such that θ > t a.s.Let ξ and ξ ∈ L ( F θ ) . Let f be a driver. Let f be a Lipschitz driver. For i = 1 , , let ( X it , π it , l it ) be a solution in S × H × H ν of the BSDE − dX it = f i ( t, X it , π it , l it ) dt − π it dW t − Z R ∗ l it ( u ) ˜ N ( dt, du ); X iθ = ξ i . (A.9) { eq7 } Assume that there exists a bounded predictable process ( γ t ) such that dt ⊗ dP ⊗ ν ( de ) -a.s. γ t ( e ) ≥ − and | γ t ( e ) | ≤ C (1 ∧ | e | ) , and such that f ( t, X t , π t , l t ) − f ( t, X t , π t , l t ) ≥ h γ t , l t − l t i ν , t ≤ t ≤ θ, dt ⊗ dP a.s. (A.10) { autre } uppose also that ξ ≥ ξ a.s. f ( t, X t , π t , l t ) ≥ f ( t, X t , π t , l t ) + ε, t ≤ t ≤ θ, dt ⊗ dP a.s.where ε is a real constant. Then, X t − X t ≥ εα a . s . where α is a non negative F t -measurable r.v. which does not depend on ε , with P ( α > > .Proof. From inequality (4.22) in the proof of the Comparison Theorem in [14], we derivethat X t − X t ≥ e − CT E (cid:20)Z θt H t ,s ε ds |F t (cid:21) a . s . , where C is the Lipschitz constant of f , and ( H t ,s ) s ∈ [ t ,T ] is the square integrable non negativemartingale satisfying dH t ,s = H t ,s − (cid:20) β s dW s + Z R ∗ γ s ( u ) ˜ N ( ds, du ) (cid:21) ; H t ,t = 1 , ( β s ) being a predictable process bounded by C . We get X t − X t ≥ e − CT ε E [ H t ,θ ( θ − t ) |F t ] a . s . Since θ > t a.s. , we have H t ,θ ( θ − t ) ≥ P ( H t ,θ ( θ − t ) > > 0. Setting α := e − CT E [ H t ,θ ( θ − t ) |F t ], the result follows. References [1] Bally, V., Caballero, M.E., Fernandez, B., El-Karoui N.: Reflected BSDE’s, PDE’s andVariational Inequalities, INRIA Research report, (2002)[2] Barles, G., Buckdahn R., Pardoux, E.: Backward stochastic differential equations andintegral-partial differential equations, Stochastics and Stochastics Reports, (1-2), 57-83(1997)[3] Barles, G., Imbert, C.: Second-order elliptic integro-differential equations: viscositysolutions theory revisited, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, , 567–585 (2008)[4] Crandall, M., Ishii, H., Lions P-L.: User’s guide to viscosity solutions of second orderpartial differential equations, American Mathematical Society, , 1-67 (1992)[5] Dumitrescu, R., Quenez, M.-C., Sulem A.: Dynamic programming principle for mixedoptimal/stopping problems with f -conditional expectations, manuscript (2014)226] Dumitrescu, R., Quenez M.-C., Sulem A.: Double barrier reflected BSDEs with jumpsand generalized Dynkin games, arXiv:1310.2764 (2013)[7] Essaky, E.H., Reflected backward stochastic differential equation with jumps and RCLLobstacle, Bulletin des Sciences Math´ematiques, , 690–710 (2008)[8] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S., Quenez, M-C.: Reflected solutionsof backward SDE’s, and related obstacle problems for PDE’s, The Annals of Probability, , 702-737 (1997)[9] Hamad`ene, S., Ouknine, Y.: Reflected backward stochastic differential equation withjumps and random obstacle, Electronic Journal of Probability, , 1–20 (2003)[10] Hamad`ene S., Ouknine, Y.: Reflected backward SDEs with general jumps, Manuscript(2007)[11] Ouknine, Y.: Reflected backward stochastic differential equation with jumps, Stochasticand Stoch. Reports, , 111-125 (1998)[12] Pardoux E., Peng, S.: Backward Stochastic Differential equations and QuasilinearParabolic Partial Differential equations, Lect. Notes in CIS, , 200–217 (1992)[13] Peng, S.: Nonlinear expectations, nonlinear evaluations and risk measures, 165-253,Lecture Notes in Math., , Springer, Berlin (2004)[14] Quenez, M.-C., Sulem A., BSDEs with jumps, optimization and applications to dynamicrisk measures, Stoch. Proc. and Their Appl., , 3328–3357 (2013)[15] Quenez, M.-C., Sulem A.: Reflected BSDEs and robust optimal stopping for dynamicrisk measures with jumps, Stoch. Proc. and Their Appl., , 3031–3054 (2014)[16] Royer, M.: Backward stochastic differential equations with jumps and related non-linearexpectations, Stoch. Proc. and Their Appl., , 1358-1376 (2006)[17] Tang, S.H., Li X.: Necessary conditions for optimal control of stochastic systems withrandom jumps, SIAM J. Cont. and Optim.,32