Constrained BSDE and Viscosity Solutions of Variation Inequalities
aa r X i v : . [ m a t h . S G ] J u l Constrained BSDE and Viscosity Solutions of VariationInequalities ∗ Shige PENG a,c
Mingyu XU a,c † a School of Mathematics and System Science, Shandong University,250100, Jinan, China b Institute of Applied Mathematics, Academy of Mathematics and Systems Science,Chinese Academy of Sciences, Beijing, 100080, China. c Department of Financial Mathematics and Control science, School of Mathematical Science,Fudan University, Shanghai, 200433, China.
August 23, 2007
Abstract.
In this paper, we study the relation between the smallest g -supersolutionof constraint backward stochastic differential equation and viscosity solution of constraintsemilineare parabolic PDE, i.e. variation inequalities. And we get an existence result ofvariation inequalities via constraint BSDE, and prove a uniqueness result under certaincondition. Keywords:
Backward stochastic differential equation with a constraint, viscosity solution,variation inequality.
El Karoui, Kapoudjian, Pardoux, Peng and Quenez (1997) studied the problem of BSDE(backward stochastic differential equation) with reflection, which is, a standard BSDE withan additional continuous, increasing process in this equation to keep the solution abovea certain given continuous boundary process. This increasing process must be chosen incertain minimal way, i.e. an integral condition, called Skorohod reflecting condition (cf. [ ? ]),is satisfied. It was proved in this paper that the solution of the reflected BSDE associatedto a terminal condition ξ , a coefficient g and a lower reflecting obstacle L , is the smallestsupersolution of BSDE with same parameter ( ξ, g ), which dominates the given boundaryprocess L . Then in same paper, they give a probabilistic interpretation of viscosity solutionof variation inequality by the solution of reflected BSDEs. ∗ This work is supported by the National Basic Research Program of China (973 Program), No.2007CB814902 and No. 2007CB814906. † Corresponding author, Email: [email protected]
1n important application of the constrained BSDE is the pricing of contingent claimswith constraint of protfolios, i.e. portfolios of an asset is constrained in a given subset. Inthis case the solution ( y, z ) of the corresponding reflected BSDE must remain in this subset.This problem was studied by Karaztas and Kou (cf. [ ? ]), then by [4] and [2].The most general case of the constraint Φ, which is discribed by a Lipschitz continuousfunction, is first studied in [6]. Author proved that under the Lipschitz condition of coefficient g , the smallest supersolution of BSDE with coefficient g and constraint Φ ≥ g is a Lipschitz function and theconstraint Φ( ω, t, x, y, z ), t ∈ [0 , T ] is a Lipschitz continuous function. In this paper westudy the relation between the smallest g -supersolution and viscosity solution of constraintsemilineare parabolic PDE, i.e. variation inequalities. And we get an existence result ofvariation inequalities via constraint BSDE, and prove a uniqueness result under certaincondition. Let (Ω , F , P ) be a probability space, and B = ( B , B , · · · , B d ) T be a d -dimensional Brown-ian motion defined on [0 , ∞ ). We denote by {F t ; 0 ≤ t < ∞} the natural filtration generatedby this Brownian motion B : F t = σ {{ B s ; 0 ≤ s ≤ t } ∪ N } , where N is the collection of all P − null sets of F . The Euclidean norm of an element x ∈ R m is denoted by | x | . We also need the following notations for p ∈ [1 , ∞ ): • L p ( F t ; R m ) := { R m -valued F t –measurable random variables X s.t. E [ | X | p ] < ∞} ; • L p F (0 , t ; R m ) := { R m –valued and F t –progressively measurable processes ϕ defined on[0 , t ], s.t. E R t | ϕ s | p ds < ∞} ; • D p F (0 , t ; R m ) := { R m –valued and RCLL F t –progressively measurable processes ϕ de-fined on [0 , t ], s.t. E [sup ≤ s ≤ t | ϕ s | p ] < ∞} ; • A p F (0 , t ) := { increasing processes A in D p F (0 , t ; R ) with A (0) = 0 } .When m = 1, they are denoted by L p ( F t ), L p F (0 , t ) and D p F (0 , t ), respectively. We are mainlyinterested in the case p = 2. In this paper, we consider BSDE on the interval [0 , T ], with afixed T > dX t,xs = b ( s, X t,xs ) ds + σ ( s, X t,xs ) dW s , t ≤ s ≤ T, (1) X t,xt = x. where b : [0 , T ] × R d → R d , σ : [0 , T ] × R d → R d × d are continuous mappings, satisfying2i) b and σ are continuous in t (ii) | b ( t, x ) − b ( t, x ′ ) | + | σ ( t, x ) − σ ( t, x ′ ) | ≤ k ( | x − x | ) , dP × dt a.s.for some k >
0, and for all x , x ′ ∈ R d . And for each ( t, x ) ∈ [0 , T ] × R d , { X t,xs ; t ≤ s ≤ T } is denoted as the unique solution of SDE (1).Let g be a coefficient g ( t, x, y, z ) : [0 , T ] × R d × R × R d → R , which satisfies the followingassumptions: there exists a constant µ > p ∈ N such that, for each x in R d , y, y ′ in R and z, z ′ in R d , we have(i) | g ( t, x, , | ≤ µ (1 + | x | p )(ii) | g ( t, x, y, z ) − g ( t, x, y ′ , z ′ ) | ≤ µ ( | y − y ′ | + | z − z ′ | ) , dP × dt a.s. (2)Our BSDE with a constraint is − dY t,xs = g ( s, X t,xs , Y t,xs , Z t,xs s ) ds + dA t,xs − Z t,xs dW s , (3) Y t,xT = Ψ( X t,xT ) , with Φ( s, X t,xs , Y t,xs , Z t,xs ) ≥ , d P × dt -a.s..Here Ψ : R d → R , has at most polynomial growth at infinity. Φ : [0 , T ] × R d × R × R d → R ,which plays a role of constraint in this paper, satisfying: there exists a constant µ >
0, suchthat, for each x ∈ R d , y , y ′ ∈ R and z , z ′ ∈ R d , we have(i) | Φ( t, x, , | ≤ µ (1 + | x | p )(ii) | Φ( t, x, y, z ) − Φ( t, x, y ′ , z ′ ) | ≤ µ ( | y − y ′ | + | z − z ′ | ) , dP × dt a.s.(ii) y → Φ( t, x, · , z ) and z → Φ( t, x, y, · ) are continuous. (4)The constraint Φ is an equivalent form of the constraint we have discussed before, as [3], [6]and [9]. Definition 2.1.
The solution of (3) is ( Y t,xs , Z t,xs , A t,xs ) t ≤ s ≤ T defined as the smallest g –supersolution constrained by Φ ≥ , i.e. Y t,x ∈ D F ( t, T ) and there exist a predictable process Z t,x ∈ L F ( t, T ; R d ) and an increasing RCLL process A t,x ∈ A F ( t, T ) such that (3) is satisfiedand if there is another process Y t,x ′ ∈ D F ( t, T ) , with ( Z t,x ′ , A t,x ′ ) ∈ L F ( t, T ; R d ) × A F ( t, T ) ,satisfying (3), then we have Y t,x ′ s ≥ Y t,xs . The following theorem of the existence of the smallest solution was obtained in [6].
Theorem 2.1.
Suppose that ξ ∈ L ( F T ) , the function g satisfies (2) and the constraint Φ satisfies (4). We assume that (H) there is one g –supersolution y ′ ∈ D F (0 , T ) , constrainedby Φ ≥ : y ′ t = ξ + Z Tt g ( s, y ′ s , z ′ s ) ds + A ′ T − A ′ t − Z Tt z ′ s dB s , (5) A ′ ∈ A F (0 , T ) , Φ( t, y ′ t , z ′ t ) ≥ , dP × dt a.s.Then there exists the smallest g –supersolution y ∈ D F (0 , T ) constrained by Φ ≥ , with theterminal condition y T = ξ , i.e. there exists a triple ( y t , z t , A t ) ∈ D F ( t, T ) × L F ( t, T ; R d ) × A F ( t, T ) , such that y t = ξ + Z Tt g ( s, y s , z s ) ds + A T − A t − Z Tt z s dB s ,A ∈ A F (0 , T ) , Φ( t, y t , z t ) ≥ , dP × dt a.s. oreover, this smallest g –supersolution is the limit of a sequence of g n –solutions with g n = g + n Φ − , where the convergence is in the following sense: y nt ր y t , with lim n →∞ E [ | y nt − y t | ] = 0 , lim n →∞ E Z T | z t − z nt | p dt = 0 , p ∈ [1 , , (6) A nt : = Z t ( g + n Φ − )( s, y ns , z ns ) ds → A t weakly in L ( F t ) , (7) where z and A are the corresponding martingale part and increasing part of y , respectively. And we recall an interesting proposition proved in [9].
Proposition 2.1.
A process y ∈ D F (0 , T ) is the smallest g -supersolution on [0 , T ] constraintby Φ with y T = ξ , if and only if for all m ≥ , it is a ( g + m Φ) -supersolution on [0 , T ] with y T = ξ . In the following, we assume that (H) holds, and denote the smallest solution of (3) by( Y t,xs , Z t,xs , A t,xs ) t ≤ s ≤ T . Define u ( t, x ) := Y t,xt . The variation inequality we concerned ismin {− ∂ t u − F ( t, x, u, Du, D u ) , Φ( x, u, σ T ( x ) Du ) } = 0 , where F ( t, x, u, q, S ) := P ni,j =1 [ σσ T ] ij ( t, x ) S ij + h b ( t, x ) , q i + g ( t, x, u, σ T ( t, x ) q ). We studythis problem by the following penalization approach: for each α ≥ − dY t,x,αs = ( g + α Φ − )( s, X t,xs , Y t,x,αs , Z t,x,αs ) ds − Z t,x,αs dW s ,Y t,x,αT = Ψ( X t,xT ) . Define u α ( t, x ) := Y t,x,αt . (8)Then by theorem 2.1, we have u α ( t, x ) ր u ( t, x ) , ( t, x ) ∈ [0 , T ] × R n , as α → ∞ . (9)We introduce the following penalized PDE ∂ t u + F α ( t, x, u, Du, D u ) = 0 , ∀ α = 1 , , · · · , (10)where F α ( t, x, u, q, S ) := P ni,j =1 [ σσ T ] ij ( t, x ) S ij + h b ( t, x ) , q i + ( α Φ − + g )( t, x, u, σ T ( t, x ) q ).To introduce the definition of viscosity solution. First we need the notions of parabolicsuperjet and subjet. 4 efinition 3.1. For a function u ∈ LSC ([0 , T ] × R n ) (resp. USC ([0 , T ] × R n ) ), we definethe parabolic superjet (resp. parabolic subjet) of u at ( t, x ) by P , + u ( t, x ) (resp. P , − u ( t, x ) ),the set of triples ( p, q, X ) ∈ R × R d × S n , satisfying u ( s, y ) ≤ (resp. ≥ ) u ( t, x ) + p ( s − t ) + h q, y − s i + 12 h X ( y − x ) , y − x i + o (cid:0) | s − t | + | y − x | (cid:1) . Then we have
Definition 3.2.
A function u ∈ LSC ([0 , T ] × R n ) (resp. USC ([0 , T ] × R n ) ) is called aviscosity supersolution (resp. subsolution) of ∂ t u + F α = 0 if for each ( t, x ) ∈ (0 , T ) × R n ,for any ( p, q, X ) ∈ P , − u ( t, x ) (resp. ( p, q, X ) ∈ P , + u ( t, x ) ), we have p + F α ( t, x, u ( t, x ) , q, X ) ≤ , (resp. ≥ ). The following result can be found in [5].
Proposition 3.1.
We assume (2) and Ψ has at most polynomial growth at infinity. Thenfor each α = 1 , , · · · , the function u α ∈ C ([0 , T ] × R n ) defined by (8) is the viscosity solutionof ∂ t u α + F α = 0 . Now we return to the variation inequalitymin {− ∂ t u − F ( t, x, u, Du, D u ) , Φ( x, u, σ T ( x ) Du ) } = 0 . (11)The solution of this equation may be not continuous, so we need the definition of discon-tinuous viscosity solution. For a given locally bounded function v , we define its upper andlower semicontinuous envelope of v , denoted as v ∗ and v ∗ respectively, where v ∗ ( t, x ) = lim sup t ′ → t,x ′ → x v ( t ′ , x ′ ) , v ∗ ( t, x ) = lim inf t ′ → t,x ′ → x v ( t ′ , x ′ ) . Then
Definition 3.3. (i) A locally bounded function u is called a viscosity supersolution (11) iffor each ( t, x ) ∈ (0 , T ) × R n , for any ( p, q, X ) ∈ P , − u ∗ ( t, x ) , then we have min {− p − F ( t, x, u ∗ , q, X ) , Φ( x, u ∗ , σ T ( x ) q ) } ≥ , (12) i.e. we have both Φ( x, u ∗ , σ T ( x ) q ) ≥ and − p − F ( t, x, u ∗ , q, X ) ≥ . (ii) A locally bounded function u is called a viscosity subsolution of (11), if for each ( t, x ) ∈ (0 , T ) × R n , for any ( p, q, X ) ∈ P , + u ∗ ( t, x ) , then we have min {− p − F ( t, x, u ∗ , q, X ) , Φ( x, u ∗ , σ T ( x ) q ) } ≤ , (13) i.e. for ( t, x ) ∈ (0 , T ) × R n where Φ( x, u ∗ , σ T ( x ) q ) > , we have − p − F ( t, x, u ∗ , q, X ) ≤ . (iii) A locally bounded function u is called a viscosity solution of (11), if it is both viscositysuper- and subsolution.
5e recall the function u ( t, x ) is denoted by u ( t, x ) := Y t,xt , where ( Y t,xs , Z t,xs , A t,xs ) t ≤ s ≤ T isthe smallest solution of BSDE (3) constraint by Φ ≥
0. And such solution exists. Our firstreault is following.
Proposition 3.2.
For each α = 1 , , · · · , u ( t, x ) is a discontinuous viscosity supersolutionof ∂ t u + F α = 0 . Proof.
It is an application of Proposition 2.1 and the fact that a g -supersolution relate toviscosity supersolution. (cid:3) Then we have
Theorem 3.1.
The function u is a discontinuous viscosity solution of (11). Proof.
From the above discussion, we know that for each α = 1 , , · · · , u α , defined by u α ( t, x ) := Y t,x,αt , is a viscosity solution of ∂ t u α + F α = 0. And u α ( t, x ) ր u ( t, x ), so u ( t, x )is lower semicontinuous, i.e. u ( t, x ) = u ∗ ( t, x ).We now prove that u is a subsolution of (11). Let ( t, x ) be a point such that Φ( t, x, u ∗ ( t, x ) , σ T q ) >
0, and ( p, q, X ) ∈ P , + u ∗ ( t, x ).By Lemma 6.1 in [1], there exist sequences α j → ∞ , ( t j , x j ) → ( t, x ) , ( p j , q j , X j ) ∈ P , + u α j ( t, x ) , such that ( u α j ( t j , x j ) , p j , q j , X j ) → ( u ∗ ( t, x ) , p, q, X ) . While for each j , − p j − F α ( t j , x j , u n j ( t j , x j ) , q j , X j )= − p j − [ F ( t j , x j , u n j ( t j , x j ) , q j , X j ) + α Φ − ( t j , x j , u α j ( t j , x j ) , σ T q j )] ≤ . From the assumption that Φ( t, x, u ∗ ( t, x ) , σ T q ) >
0, continuity assumption of Φ and conver-gence of u α , it follows for j large enough, Φ − ( t j , x j , u α j ( t j , x j ) , σ T q j ) >
0. Hence taking limitin the above inequality, we get − p − F ( t, x, u ∗ ( t, x ) , q, X ) ≤ , we prove that u is viscosity subsolution of (11).Then we conclude by proving that u is a viscosity supersolution of (11). Let ( t, x ) ∈ [0 , T ] × R n , and ( p, q, X ) ∈ P , − u ∗ ( t, x ), by proposition 3.2, we know that u is a discontinuousviscosity supersolution of ∂ t u + F α = 0, for each α ≥
0, i.e. − p − F α ( t, x, u ∗ ( t, x ) , q, X )= − p − F ( t, x, u ∗ ( t, x ) , q, X ) − α Φ − ( t, x, u ∗ , σ T ( t, x ) q ) ≥ . By the arbitrary of α , we have − p − F ( t, x, u ∗ ( t, x ) , q, X ) ≥ − ( t, x, u ∗ , σ T ( t, x ) q ) = 0 , i.e. Φ( t, x, u ∗ ( t, x ) , σ T q ) ≥ (cid:3) Then we consider the uniqueness of the solution. First, we have a characterization prop-erty of u ( t, x ). 6 roposition 3.3. The function u ( t, x ) is the smallest viscosity supersolution of (11). Proof.
Consider another viscosity supersolution of (11) denoted by u ( t, x ). By thedefinition, we have for each ( t, x ) ∈ (0 , T ) × R n , for any ( p, q, X ) ∈ P , − u ∗ ( t, x ), thenΦ( x, u ∗ , σ T ( x ) q ) ≥ − p − F ( t, x, u ∗ , q, X ) ≥ . So for α = 1 , , · · ·− p − F ( t, x, u ∗ ( t, x ) , q, X ) − α Φ − ( t, x, u ∗ , σ T ( t, x ) q ) ≥ , which follows that u ∗ ( t, x ) is also a viscosity supersolution of (10). While u α ( t, x ) is aviscosity solution of (10), then u α ( t, x ) ≤ u ∗ ( t, x ) ≤ u ( t, x ) . By the limit property in (9), we have u ( t, x ) ≥ u ( t, x ). And the result follows.For the uniqueness of viscosity solution, we have following result. Theorem 3.2.
Under assumptions (2) and (4), we assume that for each
R > , there existsa function m R : R + → R + , such that m R (0) = 0 and | g ( t, x , y, p ) − g ( t, x , y, p ) | ≤ m R ( | x − x | (1 + | p | ) , for all t ∈ [0 , T ] , | x | , | x | ≤ R , | y | ≤ R , p ∈ R d . And Φ is strictly increasing in y for each ( t, x, z ) . Then the constraint PDE 11 has at most one locally bounded viscosity solution. Proof.
The proof is done by the same techniques in theorem 8.6 in [3], so we omit it. (cid:3)
Remark 3.1.
The constraint satisfies assumptions in this theorem, if Φ( t, x, y, z ) = y − h ( t, x ) , here h ( t, x ) may be a discontinuous function with certain integral condition. In factsuch constraint introduces a reflected BSDE with a discontinuous barrier h ( s, X t,xs ) , c.f. [7].Another example is solution y reflected on function of z , i.e. Φ( t, x, y, z ) = y − ϕ ( t, x ) , where ϕ is a Lipschitz function on z . References [1] Crandall, M., Ishii, H and Lions, P.-L. (1992) User’s guide to viscosity solutions of seconsorder partial differential equations, Bulletin of the American mathematical Society, Vol.27, Num. 1, 1-67.[2] J. Cvitanic, I. Karatzas, M. Soner, (1998) Backward stochastic differential equationswith constraints on the gain-process, The Annals of Probability, , No. 4, 1522–1551.[3] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M. C. (1997) Re-flected solutions of Backward SDE’s, and related obstacle problems for PDE’s, Annalsof Probability, Vol. 25, No. 2, 702-737. 74] El Karoui, N. and Quenez, M. C. (1995). Dynamic programming and pricing of contin-gent claims in an incomplete market. SIAM J. Control Optim. 33 29–66.[5] Peng, S., (1991). Probabilistic interpretation for system of quasilinear parabolic partialdifferential equations. Stochastics and stochastics reports, Vol. 37, pp 61-74.[6] Shige Peng. Monotonic limit theory of BSDE and nonlinear decomposition theorem ofDoob-Meyer’s type. Probab. Theory and Related Fields, (1999), 473–499.[7] S. Peng and M. Xu, (2005), Smallest g –Supermartingales and Related Reflected BSDEswith Single and Double L Barriers,