Reflected BSDEs and continuous solutions of parabolic obstacle problem for semilinear PDEs in divergence form
aa r X i v : . [ m a t h . P R ] D ec Reflected BSDEs and continuous solutions of parabolicobstacle problem for semilinear PDEs in divergence form
Tomasz KlimsiakNovember 8, 2018
Abstract
We consider the Cauchy problem for semilinear parabolic equation in divergenceform with obstacle. We show that under natural conditions on the right-hand sideof the equation and mild conditions on the obstacle a unique continuous solutionof the problem admits a stochastic representation in terms of reflected backwardstochastic differential equations. We derive also some regularity properties of so-lutions and prove useful approximation results.
In the present paper we are interested in stochastic representation of solutions of theCauchy problem for semilinear parabolic equation in divergence form with obstacle.Let a : Q T ≡ [0 , T ] × R d → R d ⊗ R d be a measurable, symmetric matrix valued functionsuch that λ | ξ | ≤ d X i,j =1 a ij ( t, x ) ξ i ξ j ≤ Λ | ξ | , ξ ∈ R d (1.1)for some 0 < λ ≤ Λ and let A t be a linear operator of the form A t = 12 d X i,j =1 ∂∂x i ( a ij ∂∂x j ) . (1.2)Roughly speaking the problem consist in finding u : Q T → R such that for given ϕ : R d → R , h : Q T → R , f : Q T × R × R d → R , (cid:26) min( u − h, − ∂u∂t − A t u − f u ) = 0 in Q T ,u ( T ) = ϕ on R d , (1.3)where f u = f ( · , · , u, σ ∇ u ) and σσ ∗ = a , i.e. u satisfies the prescribed terminal condition,takes values above a given obstacle h , satisfies inequality ∂u∂t + A t u ≤ − f u in Q T andequation ∂u∂t + A t u = − f u on the set { u > h } . T. Klimsiak: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University,[email protected]
Mathematics Subject Classification (2010):
Primary 60H30, 35K60; Secondary 35K85.
Key words or phrases:
Backward stochastic differential equation, Semilinear parabolic partial dif-ferential equation, Divergence form operator, Obstacle problem, Weak solution. ϕ, f, h and coefficients of A t , viscosity solutions (see [11]) orsolutions of variational inequalities associated with (1.3) are considered. In the lattercase one can consider weak solutions (see [4, 17, 19]) or strong solutions (see [4, 8, 9, 10]).In the present paper by a solution of (1.3) we understand a pair ( u, µ ) consistingof a measurable function u : Q T → R having some regularity properties and a Radonmeasure µ on Q T such that ∂u∂t + A t u = − f u − µ, u ( T ) = ϕ, u ≥ h, Z Q T ( u − h ) dµ = 0 (1.4)(see Section 2.2 for details). We adopt the above definition for three reasons. Firstly,it may be viewed as an analogue of the definition of the obstacle problem for ellipticequations (see [14, 16]). It is worth pointing out, however, that contrary to the case ofelliptic equations, it is not obvious how solution of a parabolic variational inequalityassociated with (1.3) is related to the solution in the sense of (1.4). Secondly, since inmany cases we are able to prove some additional information on µ , using (1.4) insteadof variational formulation gives more information on solutions of (1.3). Finally, defini-tion (1.4) is well suited with our main purpose which consists in providing stochasticrepresentation of solutions of the obstacle problem.In the case where A t is a non-divergent operator of the form A t = 12 d X i,j =1 a ij ∂ ∂x i ∂x j + d X i =1 b i ∂∂x i , problem (1.3) has been investigated carefully in [11] by using probabilistic methods.Let X s,x be a solution of the Itˆo equation dX s,xt = σ ( t, X s,xt ) dW t + b ( t, X s,xt ) dt, X s,xs = x ( σσ ∗ = a )associated with A t . In [11] it is proved, that under suitable assumptions on a, b andthe data ϕ, f, h , for each ( s, x ) ∈ Q T there exists a unique solution ( Y s,x , Z s,x , K s,x ) ofreflected backward stochastic differential equation with forward driving process X s,x ,terminal condition ϕ ( X s,xT ), coefficient f and obstacle h ( · , X s,x · ) (RBSDE( ϕ, f, h ) forshort), and moreover, u defined by the formula u ( s, x ) = Y s,xs , ( s, x ) ∈ Q T is a uniqueviscosity solution of (1.3) in the class of functions satisfying the polynomial growthcondition. In the present paper we give a representation similar to that proved in [11]for weak solutions of (1.4) with A t defined by (1.2).In the paper we assume that(H1) ϕ ∈ L loc ( R d ), h ∈ L loc ( Q T ),(H2) f : [0 , T ] × R d × R × R d → R is a measurable function satisfying the followingconditions:a) there is L > | f ( t, x, y , z ) − f ( t, x, y , z ) | ≤ L ( | y − y | + | z − z | )for all ( t, x ) ∈ [0 , T ] × R d , y , y ∈ R and z , z ∈ R d ,b) there exist M > g ∈ L loc ( Q T ) such that | f ( t, x, y, z ) | ≤ g ( t, x ) + M ( | y | + | z | )for all ( t, x, y, z ) ∈ [0 , T ] × R d × R × R d ,2H3) ϕ ( x ) ≥ h ( T, x ) for all x ∈ R d , h ∈ C ( Q T )(definitions of various function spaces used in the paper are given at the end of thesection).We prove that under (1.1) and (H1)–(H3) the obstacle problem (1.4) has at most onesolution such that u ∈ C ( ˘ Q T ) ∩ W , ,̺ ( Q T ) for some ̺ of the form ̺ ( x ) = (1+ | x | ) − α , x ∈ R d , where α ≥
0. From our existence results it follows in particular that if, in addition, ϕ ∈ L ,̺ ( R d ), g ∈ L ,̺ ( Q T ) ∩ L p,q,̺ ( Q T ), h ∈ C ( Q T ) ∩ L ,̺ ( Q T ) for some ̺ as aboveand p, q ∈ (2 , ∞ ] such that (2 /q ) + ( d/p ) <
1, and h satisfies the polynomial growthcondition, then (1.4) has a solution ( u, µ ) such that u ∈ C ([0 , T ) × R d ) ∩ W , ,̺ ( Q T ).Secondly, for each ( s, x ) ∈ [0 , T ) × R d we have( u ( t, X t ) , σ ∇ u ( t, X t )) = ( Y s,xt , Z s,xt ) , t ∈ [ s, T ] , P s,x - a.s., (1.5)where ( X, P s,x ) is a Markov process associated with A t (see [22, 24]) and Y s,x , Z s,x are the first two components of a solution ( Y s,x , Z s,x , K s,x ) of RBSDE( ϕ, f, h ) withforward driving process X . In particular, it follows that u ( s, x ) = Y s,xs , ( s, x ) ∈ Q T , (1.6)which may be viewed as a generalization of the Feynman-Kac formula. We show alsothat E s,x Z Ts ξ ( t, X t ) dK s,xt = Z Ts Z R d ξ ( t, y ) p ( s, x, t, y ) dµ ( t, y ) (1.7)for all ξ ∈ C b ( Q T ), where p stands for the transition density function of ( X, P s,x )(or, equivalently, p is the fundamental solution for A t ), which provides an additionalinformation on the process K s,x and solution ( u, µ ) of (1.4). For instance, it followsfrom (1.7) that in the linear case the solution of (1.3) admits the representation u ( s, x ) = Z R d ϕ ( y ) p ( s, x, T, y ) dy + Z Q T f ( t, y ) p ( s, x, t, y ) dy + Z Q T p ( s, x, t, y ) dµ ( t, y ) , ( s, x ) ∈ [0 , T ) × R d , which, up to our knowledge, is new (for parabolic problems). Moreover, we show that µ is absolutely continuous with respect to the Lebesgue measure λ and dµ = r dλ ifand only if K s,xt = Z ts r ( θ, X s,xθ ) dθ, t ∈ [ s, T ] . Let us remark also that the first component u of a solution of (1.4) coincides withthe solution of (1.3) in the variatonal sense.Our conditions on ϕ, g and h are similar to that used in the theory of variationalinequalities and seems to be close to the best possible. As for g , in fact we proveexistence and uniqueness of solutions of (1.4) and the representation (1.5) under theassumption that g ∈ L ,̺ ( Q T ) and E s,x Z Ts | g ( t, X t ) | dt (1.8)3s bounded uniformly in ( s, x ) ∈ K for every compact subset K of [0 , T ) × R d . Weshow also that if ϕ ∈ L ,̺ ( R d ), g ∈ L ,̺ ( Q T ) and h ∈ C ( Q T ) satisfies the polynomialgrowth condition, then there is a version of the minimal weak solution of (1.3) in thevariational sense such that if (1.8) is finite for some fixed ( s, x ) ∈ [0 , T ) × R d , thenthere exists a unique solution ( Y s,x , Z s,x , K s,x ) of RBSDE( ϕ, f, h ) and (1.5) holds true.Thus, since (1.8) is finite for a.e. ( s, x ) ∈ [0 , T ) × R d if g ∈ L ,̺ ( Q T ), (1.5) holds fora.e. ( s, x ) ∈ [0 , T ) × R d if g ∈ L ,̺ ( Q T ). What is more important, it follows from ourresult that for each ( s, x ) such that (1.8) is finite we get a probabilistic formula (1.6)for the minimal weak solution of the variational inequality associated with (1.3).In case ̺ = 1 existence of a solution of (1.4) and representation (1.5) is proved bythe method of stochastic penalization used earlier in [11]. For ̺ < u, µ ) isa solution of (1.4), u n is a solution of the Cauchy problem( ∂∂t + A t ) u n = − f u n − n ( u n − h ) − , u n ( T ) = ϕ and µ n is a measure on Q T such that dµ n = n ( u n − h ) − dλ then u n ↑ u uniformlyin compact subsets of [0 , T ) × R d and in W , ,̺ ( Q T ) ∩ C ([0 , T ] , L ,̺ ( Q T )) if ̺ < ̺ = 1. In particular, differently from the theoryof variational inequalities, we obtain strong convergence in L ,̺ ( Q T ) of gradients of u n ’s to the gradient of u . Moreover, from the proofs it follows that { µ n } convergesweakly to µ and strongly in the space dual to W , ,̺ ( Q T ), and for each ( s, x ) ∈ [0 , T ) × R d the measures ν n defined by the relation dν n /dµ n = p ( s, x, · , · ) converge weakly tothe measure ν such that dν/dµ = p ( s, x, · , · ). These results allow us to deduce someproperties of µ from properties of the sequence { µ n } .In the paper we will use the following notation. Q T = [0 , T ] × R d , ˘ Q T = (0 , T ) × R d . For E ⊂ Q T we write E t = { x ∈ R d ; ( t, x ) ∈ E } . B (0 , r ) = { x ∈ R d : | x | < r } , x + = max( x, x − = max( − x, ∇ = ( ∂∂x , . . . , ∂∂x d ).By λ we denote the Lebesgue measure. L p ( R d ) is the Banach space of measurable function u on R d having the finite norm k u k p = ( R R d | u ( x ) | p dx ) /p . L p,q ( Q T ) is the Banach space of measurable functions on Q T having the finite norm k u k p,q,T = ( R T ( R R d | u ( t, x ) | p dx ) p/q dt ) /q , L p ( Q T ) = L p,p ( Q T ), k u k p,p,T = k u k p,T .Let ̺ be a positive function on R d . By L p,̺ ( R d ) ( L p,q,̺ ( Q T )) we denote the space offunctions u such that u̺ ∈ L p ( R d ) ( u̺ ∈ L p,q ( Q T )) equipped with the norm k u k p,̺ = k u̺ k p ( k u k p,q,̺,T = k u̺ k p,q,T ). We write K ⊂⊂ X if K is compact subset of X . L locp ( R d ) = T K ⊂⊂ R d L p ( K ). By h· , ·i we denote the usual inner product in L ( R d ) andby h· , ·i ,̺ the inner product in L ,̺ ( R d ). W ,̺ ( R d ) ( W , ,̺ ( Q T )) is the Banach space consisting of all elements u of L ,̺ ( R d )( L ,̺ ( Q T )) having generalized derivatives ∂u∂x i , i = 1 , . . . , d , in L ,̺ ( R d ) ( L ,̺ ( Q T )).If ̺ ≡ W ( R d ) and W , ( Q T ). W , ,̺ ( Q T ) is thesubspace of W , ,̺ ( Q T ) consisting of all elements u having generalized derivatives ∂u∂t in L ,̺ ( Q T ), W , , ( Q T ) is the set of all function from W , ( Q T ) with compact support in Q T . W ̺ = { u ∈ L ([0 , T ] , W ,̺ ( R d )); ∂u∂t ∈ L ([0 , T ] , W − ,̺ ( R d )) } , where W − ,̺ ( R d ) is thedual space to W ,̺ ( R d ) (see [17, 18] for details); if ̺ ≡ W instead of W ̺ .4y C ( Q T ) ( C ( R d )) we denote the space of all continuous function with compactsupport on Q T ( R d ) and by C +0 ( Q T ) ( C +0 ( R d )) the set of all positive functions from C ( Q T ) ( C ( R d )).In what follows, by C (or c ) we will denote a general constant which may vary fromline to line but depends only on fixed parameters. Let Ω = C ([0 , T ] , R d ) denote the space of continuous R d -valued functions on [0 , T ]equipped with the topology of uniform convergence and let X be a canonical processon Ω. It is known that given an operator A t defined by (1.2) with a satisfying (1.1) onecan construct a weak fundamental solution p ( s, x, t, y ) for A t and then a Markov family X = { ( X, P s,x ); ( s, x ) ∈ [0 , T ) × R d } for which p is the transition density function, i.e. P s,x ( X t = x ; 0 ≤ t ≤ s ) = 1 , P s,x ( X t ∈ Γ) = Z Γ p ( s, x, t, y ) dy, t ∈ ( s, T ]for any Γ in a Borel σ -field B of R d (see [22, 24]). Theorem 2.1
For each ( s, x ) ∈ [0 , T ) × R d , if [0 , T ) × R d ∋ ( s n , x n ) → ( s, x ) then P s n ,x n ⇒ P s,x weakly in C ([0 , T ]; R d ) .Proof. Follows from the fact that X generates a strongly Feller continuous Markovtime-inhomogeneous semigroup on L ( R d ) (see [22]). (cid:3) In what follows by W we denote the space of all measurable functions ̺ : R d → R such that ̺ ( x ) = (1 + | x | ) − α , x ∈ R d , for some α ≥ E s,x denote expectation with respect to P s,x . Theorem 2.2
Let ̺ ∈ W . Then there exist < c ≤ C depending only on λ, Λ and ̺ such that (i) for any ϕ ∈ L ,̺ ( R d ) and ≤ s ≤ t < T , c Z R d | ϕ ( x ) | ̺ ( x ) dx ≤ Z R d E s,x | ϕ ( X t ) | ̺ ( x ) dx ≤ C Z R d | ϕ ( x ) | ̺ ( x ) dx, (ii) for any ψ ∈ L ,̺ ( Q T ) , c Z Tt Z R d | ψ ( θ, x ) | ̺ ( x ) dθ dx ≤ Z Tt Z R d E s,x | ψ ( θ, X θ ) | ̺ ( x ) dθ dx ≤ C Z Tt Z R d | ψ ( θ, x ) | ̺ ( x ) dθ dx, t ∈ [ s, T ] . Proof.
Both statements follow from [2, Proposition 5.1, Appendix], because by Aron-son’s estimates there exist 0 < c ≤ c depending only on λ, Λ such that c Z R d E | ϕ ( x + X c ( t − s ) ) | ̺ ( x ) dx ≤ Z R d E s,x | ϕ ( X t ) | ̺ ( x ) dx ≤ c Z R d E | ϕ ( x + X c ( t − s ) ) | ̺ ( x ) dx E denotes expectation with respect to the standard Wiener measure on Ω. (cid:3) Set F st = σ ( X u , u ∈ [ s, t ]) and define G as the completion of F sT with respect to thefamily P = { P s,µ : µ is a probability measure on B} , where P s,µ ( · ) = R R d P s,x ( · ) µ ( dx ),and define G st as the completion of F st in G with respect to P .From [23, Theorem 2.1] it follows that there exist a martingale additive functionallocally of finite energy M = { M s,t : 0 ≤ s ≤ t ≤ T } of X and a continuous additivefunctional locally of zero energy A = { A s,t : 0 ≤ s ≤ t ≤ T } of X such that X t − X s = M s,t + A s,t , t ∈ [ s, T ] , P s,x - a.s. (2.1)for each ( s, x ) ∈ [0 , T ) × R d . Moreover, the above decomposition is unique and for each( s, x ) ∈ [0 , T ) × R d the process M s, · is a ( {G st } t ∈ [ s,T ] , P s,x )-square-integrable martingaleon [ s, T ] with the co-variation process given by h M is, · , M js, · i t = Z ts a ij ( θ, X θ ) dθ, t ∈ [ s, T ] , i, j = 1 , ..., d (see [23] for details).We now formulate definitions of backward stochastic differential equation (BSDE)and reflected BSDE (RBSDE) associated with X and recall some known results on suchequations to be used further on.Write B s,t = Z ts σ − ( θ, X θ ) dM s,θ , t ∈ [ s, T ] , where M is the additive functional of the decomposition (2.1). Notice that { B s,t } t ∈ [ s,T ] is a Wiener process. Definition
A pair ( Y s,x , Z s,x ) of processes on [ s, T ] is a solution of BSDE( ϕ, f ) (associ-ated with ( X, P s,x )) if(i) Y s,x , Z s,x are {G st } -adapted,(ii) Y s,xt = ϕ ( X T ) + R Tt f ( θ, X θ , Y s,xθ , Z s,xθ ) dθ − R Tt Z s,xθ dB s,θ , t ∈ [ s, T ], P s,x -a.s.,(iii) E s,x R Ts | Z s,xt | dt < ∞ , E s,x sup s ≤ t ≤ T | Y s,xt | < ∞ . Definition
A triple ( Y s,x , Z s,x , K s,x ) of processes on [ s, T ] is a solution of RBSDE( ϕ, f, h )(associated with ( X, P s,x )) if(i) Y s,x , Z s,x , K s,x are {G st } -adapted,(ii) Y s,xt ≥ h ( t, X t ), t ∈ [ s, T ], P s,x -a.s.,(iii) Y s,xt = ϕ ( X T ) + R Tt f ( θ, X θ , Y s,xθ , Z s,xθ ) dθ + K s,xT − K s,xt − R Tt Z s,xθ dB s,θ , t ∈ [ s, T ], P s,x -a.s.,(iv) E s,x R Ts | Z s,xt | dt < ∞ , E s,x sup s ≤ t ≤ T | Y s,xt | < ∞ ,(v) K s,x is a continuous increasing process such that K s,xs = 0, E s,x | K s,xT | < ∞ and R Ts ( Y s,xt − h ( t, X t )) dK s,xt = 0, P s,x -a.s.6bserve that {G st } need not coincide with the natural filtration generated by theWiener process B s, · . Consequently, due to lack of the representation theorem for B s, · ,existence of solutions of BSDE( ϕ, f ) does not follow from known results for ,,usual”BSDEs.Existence and uniqueness of solutions of BSDE( ϕ, f ) for each starting point ( s, x ) ∈ [0 , T ) × R d was proved in [23] under the assumption that ϕ ∈ L ( R d ) and f satisfies(H2) with g ∈ L p,q ( Q T ) for some p, q such that p, q ∈ (2 , ∞ ] , q + dp < . (2.2)(see also [3] for existence results for quasi-every starting point x proved in the casewhere the forward diffusion corresponds to symmetric divergence form operator withtime-independent coefficients but not necessarily uniformly elliptic).Let us recall that u is said to be a weak solutions of the Cauchy problem ∂u∂t + A t u = − f u , u ( T ) = ϕ (2.3)(PDE( ϕ, f ) for short) if u ∈ W , ,loc ( Q T ) ∩ C ([0 , T ] , L loc ( R d )) and for any η ∈ W , , ( Q T ), Z Tt h u ( s ) , ∂η∂s ( s ) i ds + 12 Z Tt h a ( s ) ∇ u ( s ) , ∇ η ( s ) i ds = Z Tt h f u ( s ) , η ( s ) i ds + h ϕ, η ( T ) i − h u ( t ) , η ( t ) i , t ∈ [0 , T ] . It is well known that if ϕ ∈ L ( R d ), g ∈ L ( Q T ) then there exists a unique weaksolution of PDE( ϕ, f ) (see, e.g. [15]).The next theorem strengthens slightly results proved in [23]. Proposition 2.3
Assume that (H1)-(H3) are satisfied with ϕ ∈ L ( R d ) , g ∈ L ( Q T ) . (i) If ∀ K ⊂⊂ [0 ,T ) × R d sup ( s,x ) ∈ K E s,x Z Ts | g ( t, X t ) | dt < ∞ (2.4) then there exists a unique weak solution u ∈ W , ( Q T ) ∩ C ([0 , T ) × R d ) ofPDE ( ϕ, f ) and for each ( s, x ) ∈ [0 , T ) × R d the pair ( Y s,xt , Z s,xt ) = ( u ( t, X t ) , σ ∇ u ( t, X t )) , t ∈ [ s, T ] (2.5) is a unique solution of BSDE ( ϕ, f ) . (ii) There exists a version u of a weak solution of PDE ( ϕ, f ) such that if E s,x Z Ts | g ( t, X t ) | dt < ∞ (2.6) for some ( s, x ) ∈ [0 , T ) × R d then the pair (2.5) is a unique solution of BSDE ( ϕ, f ) . roof. Let ¯ u ∈ W , ( Q T ) be a weak solution of the problem (2.3) and let k ¯ u k W ( x,s,T ) = E s,x Z Ts ( | ¯ u ( t, X t ) | + |∇ ¯ u ( t, X t ) | ) dt. From the proof of [23, Theorem 6.1] it follows that under (2.4) for every K ⊂⊂ [0 , T ) × R d , sup ( s,x ) ∈ K k ¯ u k W ( x,s,T ) < ∞ . (2.7)For n, m ∈ N let u nm ∈ W , ( Q T ) ∩ C ([0 , T ) × R d ) be a weak solution of the Cauchyproblem (cid:18) ∂∂t + A t (cid:19) u nm = f +¯ u ∧ m − f − ¯ u ∧ n, u nm ( T ) = ϕ. By [23, Proposition 5.1] the pair ( u nm ( t, X t ) , σ ∇ u nm ( t, X t )), t ∈ [ s, T ], is a solution ofBSDE( ϕ, f +¯ u ∧ m − f − ¯ u ∧ n ). Using Itˆo’s formula and performing standard calculationswe conclude that there is C > n, m such that E s,x sup s ≤ t ≤ T | u nm ( t, X t ) | + E s,x Z Ts | σ ∇ u nm ( t, X t ) | dt ≤ C (cid:18) E s,x | ϕ ( X T ) | + E s,x Z Ts | g ( t, X t ) | dt + k ¯ u k W ( x,s,T ) (cid:19) . (2.8)From comparison results (see [4, Theorem 4.1.4]) and the fact that u nm are continuousit follows that for any fixed n the sequence { u nm } m ∈ N is increasing. Hence, for each n ∈ N there is u n such that u nm ↑ u n as m → ∞ . Moreover, by well known convergencetheorems (see [15, Theorem 3.4.5]), u nm → u n in W , ( Q T ) and u n is a weak solutionof the problem (cid:18) ∂∂t + A t (cid:19) u n = f +¯ u − f − ¯ u ∧ n, u n ( T ) = ϕ. If (2.4) is satisfied, then from (2.7), (2.8) and Nash’s continuity theorem (see [1])it follows that { u nm } m ∈ N is equicontinous in every compact subset of [0 , T ) × R d .Therefore the functions u n are continuous on [ s, T ) × R d . Using once again Itˆo’s formulawe deduce that for any k, l, n ∈ N , E s,x | ( u nk − u nl )( t, X t ) | + E s,x Z Ts | σ ∇ ( u nk − u nl )( t, X t ) | dt ≤ C (cid:18) E s,x Z Ts | ( f +¯ u ∧ k − f +¯ u ∧ l )( t, X t ) | dt (cid:19) / × (cid:18) E s,x Z Ts | ( u nk − u nl )( t, X t ) | dt (cid:19) / (2.9)for all t ∈ [ s, T ]. By (H2) and (2.7), (2.8) the first term on the right-hand side of (2.9)is bounded uniformly in k, l . Due to (2.7), (2.8) and the estimate | u nk | ≤ | u n | + | u n | wemay apply the Lebesgue dominated convergence theorem to conclude that the secondterm converges to zero as k, l →
0. By the above, E s,x | ( u nm − u n )( t, X t ) | + E s,x Z Ts | σ ∇ ( u nm − u n )( t, X t ) | dt → m → ∞ . Using this it is easy to see that the pair ( u n ( t, X t ) , σ ∇ u n ( t, X t )), t ∈ [ s, T ]is a solution of BSDE( ϕ, f +¯ u − f − ¯ u ∧ n ). Therefore, (2.8) holds for u nm replaced by u n and (2.9) holds for u nk , u nl replaced by u k , u l and f +¯ u replaced by f − ¯ u . Using once again(2.7) and Nash’s continuity theorem we conclude that u n is equicontinuous in everycompact subset of [0 , T ) × R d . Therefore, by comparison results, u n is decreasing andthere is u ∈ C ([0 , T ) × R d ) such that u n ↓ u . Since f +¯ u − f − ¯ u ∧ n → f ¯ u in L ( Q T ), itfollows that u is a weak solution of the Cauchy problem ( ∂∂t + A t ) u = f ¯ u , u ( T ) = ϕ .By uniqueness, u is a version of ¯ u . Finally, using the mentioned above analogues of(2.8), (2.9) we prove in much the same way as above that the pair (2.5) is a solution ofBSDE( ϕ, f ), which completes the proof of (i).To prove (ii), we first observe that using continuity of u nm and the fact that { u nm } is decreasing for every fixed n and increasing for every fixed m we can still show that { u n } is decreasing. Therefore { u n } converges pointwise to some version u of ¯ u . If (2.6)is satisfied for some ( s, x ) ∈ [0 , T ) × R d then k ¯ u k W ( x,s,T ) < ∞ . Therefore we can use(2.8), (2.9) to conclude as before that ( u ( t, X t ) , σ ∇ u ( t, X t )), t ∈ [ s, T ], is a solution ofBSDE( ϕ, f ) associated with ( X, P s,x ). (cid:3) Theorem 2.4
Assume that (H1)–(H3) are satisfied with ϕ ∈ L ( R d ) , h, g ∈ L ( Q T ) and E s,x sup s ≤ t ≤ T | h + ( t, X t ) | + E s,x Z Ts | g ( t, X t ) | dt < ∞ (2.10) for some ( s, x ) ∈ [0 , T ) × R d . Then the RBSDE ( ϕ, f, h ) associated with ( X, P s,x ) hasa unique solution ( Y s,x , Z s,x , K s,x ) . Moreover, if the pair ( Y s,x,nt , Z s,x,n ) , n ∈ N , is asolution of BSDE ( ϕ, f + n ( y − h ) + ) , then E s,x sup s ≤ t ≤ T | Y s,x,nt − Y s,xt | + E s,x Z Ts | Z s,x,nt − Z s,xt | dt + E s,x sup s ≤ t ≤ T | K s,x,nt − K s,xt | → , (2.11) where K s,x,nt = Z ts n ( Y s,x,nθ − h ( θ, X θ )) dθ, t ∈ [ s, T ] , P s,x - a.s. Finally, there is
C > depending neither on n, m ∈ N nor on s, x such that E s,x sup s ≤ t ≤ T | Y s,x,nt | + E s,x Z Ts | Z s,x,nt | dt + E s,x | K s,x,nT | ≤ C E s,x | ϕ ( X T ) | + E s,x sup s ≤ t ≤ T | h + ( t, X t ) | + E s,x Z Ts | g ( t, X t ) | dt ! (2.12) and E s,x sup s ≤ t ≤ T − δ | Y s,x,nt − Y s,x,mt | ≤ C (cid:18) E s,x | Y s,x,nT − δ − Y s,x,mT − δ | + E s,x Z T − δs ( Y s,x,nt − h ( t, X t )) − dK s,x,mt + E s,x Z T − δs ( Y s,x,mt − h ( t, X t )) − dK s,x,nt (cid:19) (2.13)9 or every δ ∈ [0 , T − s ] .Proof. From Proposition 2.3 we know that for each n ∈ N there exists a unique solutionof BSDE( ϕ, f + n ( y − h ) + ). To prove (2.11)–(2.13) it suffices to repeat step by steparguments from the proofs of corresponding results in [11]. (cid:3) Let us remark that both terms in (2.10) are bounded uniformly in ( s, x ) ∈ K forevery K ⊂⊂ [0 , T ) × R d if h, g satisfy the polynomial growth condition or h satisfies thepolynomial growth condition and g ∈ L p,q,̺ ( Q T ) with p, q satisfying (2.2) and ̺ ∈ W .The first statement is an immediate consequence of Proposition 3.2 proved in Section3. Sufficiency of the second condition on g follows from H¨older’s inequality and upperAronson’s estimate on the transition density p (see [1]).Observe also that if g ∈ L ,̺ ( Q T ) then (2.6) holds for a.e. ( s, x ) ∈ [0 , T ) × R d because by Theorem 2.2, Z T (cid:18)Z R d ( E s,x Z Ts | g ( t, X t ) | dt̺ ( x )) dx (cid:19) ds ≤ C k g k ,̺ Lemma 2.5 If ( Y s,x,it , Z s,x,it , K s,x,it ) , i = 1 , , is a solution of RBSDE ( ξ, f, h i ) then forevery δ ∈ [0 , T − s ] , E s,x sup s ≤ t ≤ T − δ | Y s,x, t − Y s,x, t | + E s,x Z T − δs | Z s,x, t − Z s,x, t | dt + E s,x sup s ≤ t ≤ T − δ | K s,x, t − K s,x, t | ≤ C E s,x sup s ≤ t ≤ T − δ | h ( t, X t ) − h ( t, X t ) | + E s,x | Y s,x, T − δ − Y s,x, T − δ | ! . Proof.
See [11]. (cid:3)
In this subsection we formulate precisely our definition of solutions of the obstacleproblem and compare it to the well known definitions of solutions in the sense of vari-ational inequalities. We prove also a priori estimates for solutions and some additionaltechnical results which will be needed in the next section.In the paper we will use the following notion of the capacity of E ⊂⊂ ˘ Q T : cap ˘ Q T ( E ) = inf { Z ˘ Q T ( | ∂η∂t ( t, x ) | + |∇ η ( t, x ) | ) dt dx : η ∈ C ∞ ( ˘ Q T ) , η ≥ E } . In the standard way we can extend the above capacity to external capacity for arbitrarysubset E ⊂ ˘ Q T . It is known that cap ˘ Q T is the Choquet capacity (see Chapter 2 in [13]).In the remainder of the paper the abbreviation “q.e.” means “except for a set ofcapacity zero”.Throughout the subsection we assume that ̺ ∈ W and (H1)–(H3) are satisfied. Definition
We say that a pair ( u, µ ), where µ is a Radon measure on Q T and u : Q T → R is a measurable function defined up to the sets of µ -measure zero, is a weak solution ofthe obstacle problem (1.3) with data ϕ, f, h (OP( ϕ, f, h ) for short) if10a) u ∈ W , ,loc ( Q T ) ∩ C ([0 , T ] , L loc ( R d )) and for any η ∈ W , , ( Q T ), Z Tt h u ( s ) , ∂η∂s ( s ) i ds + 12 Z Tt h a ( s ) ∇ u ( s ) , ∇ η ( s ) i ds = Z Tt h f u ( s ) , η ( s ) i ds + Z Tt Z R d η dµ + h ϕ, η ( T ) i − h u ( t ) , η ( t ) i , t ∈ [0 , T ] , (2.14)(b) u ≥ h on Q T ,(c) R Q T ( u − h ) ξ dµ = 0 for all ξ ∈ C +0 ( Q T ),(d) µ ( { t } × R d ) = 0 for every t ∈ [0 , T ].Some comments on the above definition are in order. In the next lemma we willshow that (a) forces µ | ˘ Q T ≪ cap ˘ Q T , which together with (d) and the well known factthat elements of W , ( Q T ) are defined up to subsets of ˘ Q T of zero capacity (see, e.g.,[6, 12, 21]) ensures that the integral R Tt R R d η dµ is correctly defined. We shall see that(a) implies that µ ( { s } × R d ) = 0 for s ∈ (0 , T ), so instead of (d) we could impose thecondition µ ( { , T } × R d ) = 0. The condition µ ( { T } × R d ) = 0 is also necessary for theterminal condition u ( T ) = ϕ ( T ) to hold, and a fortiori, for uniqueness of the solutionof the obstacle problem. Notice also that the integral in condition (c) is well definedbecause u − h ≥ K s,x ). Notice also that if the obstacle h is constant, then the abovedefinition coincides with the one adopted in [20] (in [20] exclusively constant obstaclesare considered; this implies that µ is absolutely continuous with respect to the Lebesguemeasure, so no problems arises with the definition of an obstacle problem). Lemma 2.6 If u ∈ W , ,loc ( Q T ) ∩ C ([0 , T ] , L loc ( R d )) and the pair ( u, µ ) satisfies Z T h u ( t ) , ∂η∂t ( t ) i dt + 12 Z T h a ( t ) ∇ u ( t ) , ∇ η ( t ) i dt = Z T h f u ( t ) , η ( t ) i dt + Z Q T η dµ + h ϕ, η ( T ) i (2.15) for every η ∈ W , ( Q T ) ∩ C ( Q T ) such that η (0) ≡ , then (i) µ | ˘ Q T ≪ cap ˘ Q T , (ii) µ ( { t } × R d ) = 0 for every t ∈ (0 , T ) , (iii) u ( T ) = ϕ if and only if µ ( { T } × R d ) = 0 , (iv) if µ ( { , T } × R d ) = 0 then (2.14) holds for η ∈ W , , ( Q T ) . roof. Fix E ⊂⊂ ˘ Q T and choose positive η ∈ C ∞ ( ˘ Q T ) such that η ≥ E . Then, by(2.15), µ ( E ) ≤ Z T h u ( t ) , ∂η∂t ( t ) i dt + 12 Z T h a ( t ) ∇ u ( t ) , ∇ η ( t ) i dt − Z T h f u ( t ) , η ( t ) i dt, and hence, by Gagliardo-Nirenberg-Sobolev inequality, µ ( E ) ≤ C ( cap ˘ Q T ( E )) / ( k u k ,T + k∇ u k ,T + k f u k ,T ) , which shows (i). Now, fix s ∈ (0 , T ) and consider the sequence of functions { η n,s } defined by η n,s ( t, x ) = , t ∈ [0 , s n ] , η ( s,x ) s − s n ( t − s n ) , t ∈ ( s n , s ) ,η ( t, x ) , t ∈ [ s, T ] , where { s n } ⊂ (0 , s ) is a sequence such that s n ↑ s . Observe that η n,s → [ s,T ] × R d η , ∇ η n,s → [ s,T ] × R d ∇ η and ∂η∂t n,s ( t, x ) = , t ∈ [0 , s n ] , η ( s,x ) s − s n , t ∈ ( s n , s ) , ∂η∂t ( t, x ) , t ∈ [ s, T ] . From (2.15) with η replaced by η n,s we have1 s − s n Z ss n h u ( t ) , η ( t ) i dt + Z Ts h u ( t ) , ∂η∂t ( t ) i dt + 12 Z T h a ( t ) ∇ u ( t ) , ∇ η n,s ( t ) i dt = Z T h f u ( t ) , η n,s ( t ) i dt + Z Q T η n,s dµ + h ϕ, η n,s ( T ) i . Letting n → ∞ and using the fact that u ∈ C ([0 , T ] , L loc ( R d )) we get (2.14) for every η ∈ W , ( Q T ) ∩ C ( Q T ) , t ∈ (0 , T ]. In particular, for any positive η ∈ W , ( Q T ) ∩ C ( Q T )and any 0 < h < s ≤ T we have Z ss − h Z R d η dµ = Z ss − h h u ( t ) , ∂η∂t ( t ) i dt + 12 Z ss − h h a ( t ) ∇ u ( t ) , ∇ η ( t ) i dt − Z ss − h h f u ( t ) , η ( t ) i dt − h u ( s ) , η ( s ) i + h u ( s − h ) , η ( s − h ) i , so letting h ↓ t u ( t ) in L ( R d ) we get (ii) and (iii). Toshow (iv) we assume that η ∈ W , , ( Q T ) and consider a sequence { η n } ⊂ W , ( Q T ) ∩ C ( Q T ) such that η n → η in W , ( Q T ) and quasi-everywhere in ˘ Q T . From (i) and theassumption in (iv) it follows that { η n } converges µ -a.e. in Q T as well. From (2.14)applied to | η n − η m | we conclude that { η n } is a Cauchy sequence in L ([ t, T ] × R d , µ )for every t ∈ (0 , T ]. Therefore (2.14) is satisfied for any η ∈ W , , ( Q T ) and t ∈ (0 , T ].Clearly, if µ ( { } × R d ) = 0, then it is satisfied also for t = 0. (cid:3)
12n what follows, given some function u : Q T → R d we will extend it in a naturalway to the function on [ − T, T ] × R d , still denoted by u , by putting u ( t, x ) = u ( − t, x ) , t ∈ [ − T, ,u ( t, x ) , t ∈ [0 , T ] ,u (2 T − t, x ) , t ∈ [ T, T ] . For ε > u ε ( t, x ) = 1 ε Z ε u ( t − s, x ) ds, ( t, x ) ∈ [0 , T ] × R d and note that if u ∈ C ([0 , T ] , L loc ( R d )) ∩ W , ,loc ( Q T ) then u ε ∈ W , ,loc ( Q T ), t u ε ( t ) ∈ L loc ( R d ) is differentiable, ∇ u ε → ∇ u in L loc ( Q T ) and u ε ( t ) → u ( t ) in L loc ( R d ) for every t ∈ [0 , T ]. Lemma 2.7 If ( u, µ ) satisfies (a),(d), then for any η ∈ W , , ( Q T ) and t ∈ (0 , T ) , Z Tt h u ε ( s ) , ∂η∂s ( s ) i ds + 12 Z Tt h a ( s ) ∇ u ε ( s ) , ∇ η ( s ) i ds = Z Tt h f u,ε ( s ) , η ( s ) i ds + 1 ε Z ε (cid:18)Z T − θt − θ Z R d η ( s + θ, x ) dµ ( s, x ) (cid:19) dθ + h u ε ( T ) , η ( T ) i − h u ε ( t ) , η ( t ) i (2.16) for all sufficiently small ε > .Proof. Using Fubini’s theorem and (2.14) we obtain Z Tt h u ε ( s ) , ∂η∂s ( s ) i ds = 1 ε Z ε (cid:18)Z T − θt − θ h u ( s ) , ∂η∂s ( s + θ ) i ds (cid:19) dθ = − ε Z ε (cid:18)Z T − θt − θ h a ( s ) ∇ u ( s ) , ∇ η ( s + θ ) i ds (cid:19) dθ + 1 ε Z ε (cid:18)Z T − θt − θ h f u ( s ) , η ( s + θ ) i ds + Z T − θt − θ Z R d η ( s + θ, x ) dµ ( s, x ) (cid:19) dθ + 1 ε Z ε ( h u ( T − θ ) , η ( T ) i − h u ( t − θ ) , η ( t ) i ) dθ, from which (2.16) follows. (cid:3) Proposition 2.8 If ( u, µ ) satisfies (a), (d) and u ∈ C ( ˘ Q T ) then R Q T ξ | u | dµ < ∞ forany ξ ∈ C ( Q T ) . Moreover, k u ( t ) ξ k + Z Tt h a ( s ) ∇ u ( s ) , ∇ ( uξ )( s ) i ds = k ϕξ k + 2 Z Tt h f u ( s ) , u ( s ) ξ i ds + 2 Z Tt Z R d ξ u dµ (2.17) for all t ∈ [0 , T ] . roof. Let τ ∈ (0 , T ). Write u + ε = ( u ε ) + . By (2.16) with η = ξ u + ε we have Z τt h u ε ( s ) , ξ ∂u + ε ∂s ( s ) i ds + 12 Z τt h a ( s ) ∇ u ε ( s ) , ∇ ( ξ u + ε )( s ) i ds = Z τt h f u,ε ( s ) , ξ u + ε ( s ) i ds + h u ε ( τ ) , ξ u + ε ( τ ) i − h u ε ( t ) , ξ u + ε ( t ) i + 1 ε Z ε (cid:18)Z τ − s t − s Z R d ξ u + ε ( s + s , x ) dµ ( s, x ) (cid:19) ds = Z τt h f u,ε ( s ) , ξ u + ε ( s ) i ds + k ξu + ε ( τ ) k − k ξu + ε ( t ) k + Z Q T g εξ dµ, (2.18)where g εξ ( s, x ) = 1 ε Z ε [ t − s ,τ − s ] ( s ) ξ (cid:18)Z ε u ( s + s − s , x ) ds (cid:19) + ds for s ∈ [ t, τ ) and g εξ ( τ, x ) = 0. Observe that for every ( s, x ) ∈ [ t, τ ) × R d , g εξ ( s, x ) = 1 ε Z ε ξ (cid:18)Z ε u ( s + s − s , x ) ds (cid:19) + ds for sufficiently small ε >
0. Since | a + − b + | ≤ | a − b | for every a, b ∈ R , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε Z ε ξ (cid:18)Z ε u ( s + s − s , x ) ds (cid:19) + ds − ε Z ε ξ (cid:18)Z ε u ( s, x ) ds (cid:19) + ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε Z ε ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18)Z ε u ( s + s − s , x ) ds (cid:19) + − (cid:18)Z ε u ( s, x ) ds (cid:19) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ds ≤ ε Z ε ξ (cid:12)(cid:12)(cid:12)(cid:12)Z ε u ( s + s − s , x ) ds − Z ε u ( s, x ) ds (cid:12)(cid:12)(cid:12)(cid:12) ds ≤ ε Z ε Z ε ξ | u ( s + s − s , x ) − u ( s, x ) | ds ds , and consequently, g εξ ( s, x ) → ξ u + ( s, x ) for every ( s, x ) ∈ [ t, τ ) × R d as ε →
0. Thereforefrom (2.18) we obtain12 k ξϕ + k − k ξu + ( t ) k + 12 Z τt h a ( s ) ∇ u + ( s ) , ∇ ( ξ u + )( s ) i = Z τt h f u ( s ) , ξ u + ( s ) i ds + k ξϕ + k − k ξu + ( t ) k + lim inf ε → Z τt Z R d g ξ,ε ( s, x ) dµ. (2.19)Hence, by Fatou’s lemma, Z τt Z R d ξ u + dµ ≤ k ξu ( t ) k + 12 Z τt |h a ( s ) ∇ u ( s ) , ∇ ( ξ u )( s ) i | ds + Z τt |h f u ( s ) , ξ u ( s ) i | ds + 12 k ξϕ k . t ↓ τ ↑ T we see from the above that R Q T ξ u + dµ < ∞ . Analogously,putting η = ξ u − ε we show that R Q T ξ u − dµ < ∞ , which completes the proof ofthe first part of the lemma. Since | g ξ,ε ( s, x ) | ≤ Cξ on [ t, τ ] × R d for some C >
0, using the Lebesgue dominated convergence theorem we conclude from (2.19) that(2.17) is satisfied with T replaced by τ and t ∈ (0 , τ ]. Because we know already that R Q T ξ | u | dµ < ∞ , letting τ ↑ T and t ↓ (cid:3) We now are ready to prove useful a priori estimates for solutions of an obstacleproblem.
Proposition 2.9
Assume (H1)–(H3) with ϕ ∈ L ,̺ ( R d ) , g ∈ L ,̺ ( Q T ) . If ( u, µ ) sat-isfies (a) and (d), u ∈ C ( ˘ Q T ) , ∇ u ∈ L ,̺ ( Q T ) and there is p : Q T → R such that p + ∈ W , ,̺ ( Q T ) and R Q T ( u − p ) ξdµ ≤ for all ξ ∈ C +0 ( Q T ) then there is C > suchthat sup t ∈ [0 ,T ] k u ( t ) k ,̺ + Z T k∇ u ( s ) k ,̺ ds + Z Q T | u | ̺ dµ + k µ k ( W , ,̺ ( Q T )) ∗ ≤ C k ϕ k ,̺ + sup t ∈ [0 ,T ] k p + ( t ) k ,̺ + Z T ( k ∂p + ∂s ( s ) k ,̺ + k∇ p + ( s ) k ,̺ + k g ( s ) k ,̺ ) ds (cid:19) . (2.20) Proof.
Let ξ n ∈ C ∞ ( R d ) be a function such that ξ n = 1 on B (0 , n ). By proposition2.8, (H2) and (1.1) k u ( t ) ξ n ̺ k + Z Tt h a ( s ) ∇ u ( s ) , ∇ ( uξ n ̺ )( s ) i ds = k ϕξ n ̺ k + 2 Z Tt h f u ( s ) , u ( s ) ξ n ̺ i ds + 2 Z Tt Z R d uξ n ̺ dµ ≤ k ϕξ n ̺ k + Z Tt ( k g ( s ) ξ n ̺ k + C k u ( s ) ξ n ̺ k + λ k∇ u ( s ) ξ n ̺ k ) ds + Z Tt Z R d p + ξ n ̺ dµ, Moreover, by (2.14) with η = p + ξ n ̺ we have Z Tt Z R d p + ξ n ̺ dµ ≤
12 ( k ϕξ n ̺ k + k p + ( T ) ξ n ̺ k + k u ( t ) ξ n ̺ k + k p + ( t ) ξ n ̺ k )+ Z Tt ( k g ( s ) ξ n ̺ k + k u ( s ) ξ n ̺ k + C k p + ( s ) ξ n ̺ k ) ds + Z Tt ( k ∂p + ∂s ( s ) ξ n ̺ k ) + |h a ( s ) ∇ u ( s ) , ∇ ( p + ξ n ̺ )( s ) i | ) ds By the above estimates and the fact that |∇ ̺ | ≤ α̺ there is C such that k u ( t ) ξ n ̺ k + Z Tt k∇ u ( s ) ξ n ̺ k ≤ C k ϕξ n ̺ k + sup t ∈ [0 ,T ] k p + ( t ) ξ n ̺ k Z Tt ( k ∂p + ∂s ( s ) ξ n ̺ k + k∇ p + ( s ) ξ n ̺ k + k g ( s ) ξ n ̺ k ) ds + Z Tt k u ( s ) ξ n ̺ k ds + ε n + ε n (cid:19) , where ε n = Z Tt |h a ( s ) ∇ u ( s ) , u ( s ) ̺ ∇ ξ n i | ds, ε n = Z Tt |h a ( s ) ∇ u ( s ) , p + ( s ) ̺ ∇ ξ n i | ds. Since ε n → ε n → n → ∞ , applying Gronwall’s lemma we see from the aboveestimates that sup t ∈ [0 ,T ] k u ( t ) k ,̺ + R T k∇ u ( t ) k ,̺ dt is bounded by the right-hand sideof (2.20), which when combined with (2.17) and (2.14) gives (2.20). (cid:3) For convenience of the reader we now recall definitions of solutions of an obstacleproblem in the sense of variational inequalities (see, e.g., [4, 8, 17]).
Definition
We say that u is a weak solution of OP( ϕ, f, h ) in the variational sense if u ∈ W , ,̺ ( Q T ) and for any v ∈ W , ,̺ ( Q T ) such that v ≥ h , Z T h ∂v∂t ( t ) , ( v − u )( t ) i ,̺ dt + Z T h A t u ( t ) , ( v − u )( t ) i ,̺ dt + Z T h f u ( t ) , ( v − u )( t ) i ,̺ dt ≤ k ϕ − v ( T ) k ,̺ , (2.21)where h A t u ( t ) , ( v − u )( t ) i ,̺ = − h a ( t ) ∇ u ( t ) , ∇ (( v − u )( t ) ̺ ) i . Definition
We say that u is a strong solution of OP( ϕ, f, h ) in the variational sense if u ∈ W ̺ , u ( T ) = ϕ and for any v ∈ W , ,̺ ( Q T ) such that v ≥ h , Z T h ∂u∂t ( t ) , ( v − u )( t ) i dt + Z T h A t u ( t ) , ( v − u )( t ) i ,̺ dt + Z T h f u ( t ) , ( v − u )( t ) i ,̺ dt ≤ , (2.22)where h· , ·i denote the duality pairing between W ,̺ ( R d ) and W − ,̺ ( R d ).The following proposition shows that continuous solutions of the problem (1.4)coincide with solutions of (1.3) in the variational sense. Proposition 2.10 If ( u, µ ) is a solution of OP ( ϕ, f, h ) such that u ∈ W , ,̺ ( Q T ) ∩ C ( ˘ Q T ) then u is a weak solution of the problem in the variational sense. If, in addition, u ∈ W ̺ , then u is a strong solution of OP ( ϕ, f, h ) in the variational sense.Proof. Let u ∈ W , ,̺ ( Q T ) ∩ C ( ˘ Q T ) and let ( u, µ ) be a solution of OP( ϕ, f, h ). Byproposition 2.8, 12 k u (0) k ,̺ − Z T h A t u ( t ) , u ( t ) i ,̺ dt = 12 k ϕ k ,̺ + Z T h f u ( t ) , u ( t ) i ,̺ dt + Z Q T u̺ dµ. (2.23)16n the other hand, from definition of solution of of OP( ϕ, f, h ) it follows that for any v ∈ W , ,̺ ( Q T ) we have Z T h u ( t ) , ∂v∂t ( t ) i ,̺ dt − Z T h A t u ( t ) , v ( t ) i ,̺ dt = Z T h f u ( t ) , v ( t ) i ,̺ dt + Z Q T v̺ dµ + h ϕ, v ( T ) i ,̺ − h u (0) , v (0) i ,̺ . (2.24)Combining (2.23) with (2.24) we get Z T h ∂v∂t ( t ) , ( v − u )( t ) i ,̺ dt + Z T h A t u ( t ) , ( v − u )( t ) i ,̺ dt + Z T h f u ( t ) , ( v − u )( t ) i ,̺ dt = − k u (0) k ,̺ + 12 k ϕ k ,̺ + 12 k v ( T ) k ,̺ − k v (0) k ,̺ − h ϕ, v ( T ) i ,̺ + h u (0) , v (0) i ,̺ + Z Q T ( u − v ) ̺ dµ = − k u (0) − v (0) k ,̺ + 12 k ϕ − v ( T ) k ,̺ + Z Q T ( u − v ) ̺ dµ. (2.25)Since v ≥ h , R Q T ( u − v ) ̺ d µ ≤
0, so (2.21) follows. Now, assume additionally that u ∈ W ̺ . Then by (2.24) we have − Z T h ∂u∂t ( t ) , v ( t ) i dt − Z T h A t u ( t ) , v ( t ) i ,̺ dt = Z T h f u ( t ) , v ( t ) i ,̺ dt + Z Q T v̺ dµ (2.26)for every v ∈ W , ,̺ ( Q T ). Let E ⊂⊂ ˘ Q T . From (2.26) with a positive v ∈ C ∞ ( ˘ Q T ) suchthat v ≥ E we conclude that µ ( E ) ≤ C ( cap ˘ Q T ( E )) / ( k ∂u∂t k L ([0 ,T ] ,W − ̺ ( R d )) + k f u k ,̺,T + k∇ u k ,̺,T ) , where cap ˘ Q T ( E ) = inf { Z Q T |∇ η ( t, x ) | dt dx : η ∈ C ∞ ( ˘ Q T ) , η ≥ E } . On the other hand, it is known (see [5]) that cap ˘ Q T ( E ) = R T cap R d ( E t ) dt . Therefore, if v ∈ W , ,̺ ( Q T ), then there is a version of it which is defined q.e. . Since we know alreadythat µ ≪ cap ˘ Q T , the integral R Q T v dµ is well defined for v ∈ W , ,̺ ( Q T ). Therefore, byapproximation argument, we may take as a test function in (2.26) any v ∈ W , ,̺ ( Q T ).Now from (2.26) we conclude that for any v ∈ W , ,̺ ( Q T ) such that v ≥ h , Z T h ∂u∂t ( t ) , ( v − u )( t ) i ,̺ dt + Z T h A t u ( t ) , ( v − u )( t ) i ,̺ dt + Z T h f u ( t ) , ( v − u )( t ) i ,̺ dt = Z Q T ( u − v ) ̺ dµ ≤ , (cid:3) Let us note here that in Theorem 3.14 we will prove that if ( u, µ ) is a solution of anobstacle problem, then u is the minimal solution of the same problem in the variationalsense. We begin with a general uniqueness result for continuous solutions of (1.4) satisfyingsome weak integrability assumptions.
Theorem 3.1
Assume (H1)–(H3). Then OP ( ϕ, f, h ) has at most one solution ( u, µ ) such that u ∈ C ( ˘ Q T ) ∩ W , ,̺ ( Q T ) .Proof. Suppose that ( u , µ ) , ( u , µ ) are solutions of OP( ϕ, f, h ) such that u , u ∈ C ( ˘ Q T ) ∩ W , ,̺ ( Q T ) and let u = u − u , µ = µ − µ . Let ξ n : R d → [0 , n ∈ N bea smooth function such that ξ n ( x ) = 1 if | x | ≤ n and ξ n ( x ) = 0 if | x | ≥ n + 1. By thedefinition of solution of OP( ϕ, f, h ), for any η ∈ W , , ( Q T ) we have h u ( t ) , η ( t ) i + Z Tt h u ( s ) , ∂η∂s ( s ) i ds + 12 Z Tt h a ( s ) ∇ u ( s ) , ∇ η ( s ) i ds = Z Tt Z R d η dµ + Z Tt h f u ( s ) − f u ( s ) , η ( s ) i ds, t ∈ [0 , T ] . (3.1)From proposition 2.8 we conclude that k u ( t ) ̺ξ n k + Z Tt h a ( s ) ∇ u ( s ) , ∇ ( u̺ ξ n )( s ) i ds = 2 Z Tt Z R d u̺ ξ n dµ + 2 Z Tt h f u ( s ) − f u ( s ) , u ( s ) ̺ ξ n i ds ≤ Z Tt h f u ( s ) − f u ( s ) , u ( s ) ̺ ξ n i ds, (3.2)the last inequality being a consequence of the fact that Z Tt Z R d u̺ ξ n dµ = Z Tt Z R d u ̺ ξ n dµ − Z Tt Z R d u ̺ ξ dµ (3.3) − Z Tt Z R d u ̺ ξ n dµ + Z Tt Z R d u ̺ ξ n dµ = Z Tt Z R d ̺ ξ n ( u − h ) dµ + Z Tt Z R d ̺ ξ n ( u − h ) dµ + Z Tt Z R d ̺ ξ n ( h − u ) dµ + Z Tt Z R d ̺ ξ n ( h − u ) dµ ≤ . By (3.2) and (H2), k u ( t ) ̺ξ n k + λ Z Tt k∇ u ( s ) ̺ξ n k ds − Z Tt h a ( s ) ∇ u ( s ) , u ( s ) ξ n ∇ ̺ i ds − Z Tt h a ( s ) ∇ u ( s ) , u ( s ) ̺ ∇ ξ n i ds + λ Z Tt k∇ u ( s ) ̺ξ n k ds + 2( L + λ − L ) Z Tt k u ( s ) ̺ξ n k ds. Since |∇ ̺ | ≤ α̺ , we have Z Tt |h a ( s ) ∇ u ( s ) , u ( s ) ξ n ∇ ̺ i | ds ≤ λ Z Tt k∇ u ( s ) ̺ξ n k ds + Λ α λ Z Tt k u ( s ) ̺ξ n k ds. Consequently, there is
C > n such that k u ( t ) ̺ξ n k ≤ C Z Tt k u ( s ) ̺ξ n k ds + Z Tt |h a ( s ) ∇ u ( s ) , u ( s ) ̺ n ∇ ξ n i | ds for t ∈ [0 , T ]. Letting n → ∞ we get k u ( t ) k ,̺ ≤ C Z Tt k u ( s ) k ,̺ ds, t ∈ [0 , T ]and hence, by Gronwall’s lemma, u = 0, i.e. u = u . Using this and (3.1) we see that R Q T η ( s, x ) dµ = R Q T η ( s, x ) dµ for any η ∈ W , , ( Q T ), which shows that µ = µ . (cid:3) To prove existence of a solution of the problem (1.4) and its stochastic representationwe have to impose additional integrability assumptions on g and h to ensure existenceof a solution of RBSDE( ϕ, f, h ). The assumptions must guarantee also continuity of u because we are able to prove uniqueness and a priori estimates only for continuousweak solutions of OP( ϕ, f, h ). Proposition 2.3 and Theorem 2.4 therefore suggest thatif we want the representation (1.5) to hold we should assume at least that ∀ K ⊂⊂ [0 ,T ) × R d sup ( s,x ) ∈ K ( E s,x sup s ≤ t ≤ T | h + ( t, X t ) | + E s,x Z Ts | g ( t, X t ) | dt ) < ∞ . Our assumptions on h are slightly stronger but nevertheless seems to be close to thebest possible.Now we provide a useful inequality for moments of the diffusion ( X, P s,x ). It isperhaps known but we could not find a proper reference. The inequality is given onlyfor moments greater or equal to 4, because such a form is sufficient for our purposes.
Proposition 3.2 If ( X, P s,x ) is a Markov process associated with A t then for every p ≥ , E s,x sup s ≤ t ≤ T | X t | p ≤ CE s,x | X T | p , where C depends only on λ, Λ , d and T .Proof. Let u n be a solution of PDE( ϕ n ,
0) with ϕ n ( x ) = | x | p/ B (0 ,n ) ( x ). From [23] weknow that the pair ( u n ( t, X t ) , σ ∇ u n ( t, X t )), t ∈ [ s, T ], is a solution of BSDE( ϕ n , u n ( t, X t ) = ϕ n ( X T ) − Z Tt σ ∇ u n ( θ, X θ ) dB s,θ , P s,x - a.s. u n ( s, x ) = E s,x ϕ n ( X T ). It is known that u n → u uniformly in compact subsets of Q T . By Aronson’s lower estimate, for allsufficiently large n ∈ N we have | u n ( s, x ) | = E s,x | X T | p/ B (0 ,n ) ( X T ) ≥ C Z B (0 ,n ) | y | p/ ( T − s ) − d/ exp( − | y − x | C ( T − s ) ) dy = C d/ E B (0 ,n ) ( X C ( T − s ) + x ) | X C ( T − s ) + x | p/ ≥ C d/ E B (0 ,n ) ( X C ( T − s ) + x )( | x | + 2 d X i =1 x i X iC ( T − s ) ) ! p/ ( E denotes expectation with respect to the standard Wiener measure on Ω). Letting n → ∞ we see that | u ( s, x ) | ≥ C d/ | x | p/ . By the above and known a prioriestimates for BSDE we get E s,x sup s ≤ t ≤ T | X t | p ≤ CE s,x sup s ≤ t ≤ T | u ( t, X t ) | ≤ C lim inf n →∞ E s,x sup s ≤ t ≤ T | u n ( t, X t ) | ≤ C lim inf n →∞ E s,x | ϕ n ( X T ) | ≤ CE s,x | X T | p , which completes the proof. (cid:3) Here and subsequently, we write µ n ⇒ s,x µ if for fixed ( s, x ) ∈ [0 , T ) × R d , Z Q T ξ ( t, y ) p ( s, x, t, y ) dµ n ( t, y ) → Z Q T ξ ( t, y ) p ( s, x, t, y ) dµ ( t, y )for every ξ ∈ C b ( Q T ). We use the symbol “ ⇒ ” to denote weak convergence of measures. Lemma 3.3
Let S be a Polish space and let µ , µ n , n ∈ N , be probability measureson S such that µ n ⇒ µ . If f, f n : S → R are continuous functions such that f n → f uniformly in compact subsets of S and sup n ≥ Z S | f n | dµ n < ∞ , lim α →∞ sup n ≥ Z S | f n | {| f n |≥ α } dµ n = 0 then Z S f n dµ n → Z S f dµ. Proof.
It is sufficient to modify slightly the proof of [7, Lemma 8.4.3]. We omit thedetails. (cid:3)
We now prove our main existence and representation results. For reasons to beexplained later on, we decided to consider separately the case of square-integrable data ϕ, g, h and the case where the data are square-integrable with some weight ̺ ∈ W suchthat ̺ <
1. 20 heorem 3.4
Let assumptions (H1)–(H3) hold with ϕ ∈ L ( R d ) , g ∈ L ( Q T ) andmoreover, assume that g satisfies (2.4), h ∈ L ( Q T ) ∩ C ( Q T ) , h ≤ ψ for some ψ suchthat ψ ∈ W , ( Q T ) and h ≤ c ¯ ̺ − for some c > , ¯ ̺ ∈ W . Then there exists a uniqueweak solution ( u, µ ) of OP ( ϕ, f, h ) such that (i) u ∈ C ([0 , T ) × R d ) , (ii) u n ↑ u , u n → u in W , ,loc ( Q T ) ∩ C ([0 , T ] , L loc ( Q T )) , u n → u in L ( Q T ) and µ n ⇒ µ , µ n → µ in ( W , ,loc ( Q T )) ∗ , µ n ⇒ s,x µ for every ( s, x ) ∈ [0 , T ) × R d , where dµ n = n ( u n − h ) − dλ and u n is a unique weak solution of the Cauchy problem ( ∂∂t + A t ) u n = − f u n − n ( u n − h ) − , u n ( T ) = ϕ. (3.4) Moreover, for each ( s, x ) ∈ [0 , T ) × R d , ( u ( t, X t ) , σ ∇ u ( t, X t )) = ( Y s,xt , Z s,xt ) , t ∈ [ s, T ] , P s,x - a.s. (3.5) and E s,x Z Ts ξ ( t, X t ) dK s,xt = Z Ts Z R d ξ ( t, y ) p ( s, x, t, y ) dµ ( t, y ) (3.6) for every ξ ∈ C b ( Q T ) , where ( Y s,x , Z s,x , K s,x ) is a solution of RBSDE ( ϕ, f, h ) .Proof. Step 1. We first show existence of u ∈ W , ( Q T ) and a Radon measure µ on Q T such that Z T h u ( t ) , ∂η∂t ( t ) i dt + 12 Z T h a ( t ) ∇ u ( t ) , ∇ η ( t ) i dt = Z T h f u ( t ) , η ( t ) i dt + Z Q T η dµ + h ϕ, η ( T ) i (3.7)for every η ∈ W , ( Q T ) ∩ C ( Q T ) such that η (0) ≡
0. From Proposition 2.3 we knowthat there exists a unique weak solution u n of (3.4) such that u n ∈ C ([0 , T ] , L ( R d ))and u n ∈ C ([0 , T ) × R d ). Set r n = n ( u n − h ) − and let dµ n = r n dλ . Then for any η ∈ W , , ( Q T ), Z Tt h u n ( s ) , ∂η∂s ( s ) i ds + 12 Z Tt h a ( s ) ∇ u n ( s ) , ∇ η ( s ) i ds = Z Tt h f u n ( s ) , η ( s ) i ds + Z Tt Z R d η dµ n + h ϕ, η ( T ) i − h u n ( t ) η ( t ) i , t ∈ [0 , T ] . (3.8)By Proposition 2.9 with p = ψ there is C > t ∈ [0 ,T ] k u n ( t ) k + Z T |h a ( t ) ∇ u n ( t ) , ∇ u n ( t ) i | dt ≤ C (3.9)for every n ∈ N . Since, by continuity of u n and comparison results (see [4, Theorem4.1.7]), u n ( t, x ) ≤ u n +1 ( t, x ) for every ( t, x ) ∈ [0 , T ) × R d , there is u such that u n ↑ u .21y Fatou’s lemma and (3.9), u ∈ L ( Q T ). In fact, since u ≤ u n ≤ u , it follows that u n ≤ u + u and hence, by the Lebesgue dominated convergence theorem, that u n → u in L ( Q T ). Let f n ( t, x, y ) = f u n ( t, x ) + n ( y − h ( t, x )) and let ( Y s,x,n , Z s,x,n ) be a uniquesolution of BSDE( ϕ, f n ). By results of [23], ( u n ( t, X t ) , σ ∇ u n ( t, X t )) = ( Y s,x,nt , Z s,x,nt ) P s,x -a.s. and hence, by (2.12), | u n ( s, x ) | ≤ CE s,x | ϕ ( X T ) | + Z Ts | g ( t, X t ) | dt + sup s ≤ t ≤ T | h + ( t, X t ) | ! for all ( s, x ) ∈ [0 , T ) × R d . Thus, u ( s, x ) = sup n ≥ u n ( s, x ) < ∞ and consequently, u islower semi-continuous on [0 , T ) × R d . Let K ⊂⊂ Q T and let η ∈ C ∞ ( ˘ Q T ) be a positivefunction such that η = 1 on K . Since µ n ( K ) ≤ Z Q T η dµ n = Z T h u n ( s ) , ∂η∂s ( s ) i ds + Z T h a ( s ) ∇ u n ( s ) , ∇ η ( s ) i ds − Z T h f u n ( s ) , η ( s ) i ds, we conclude from (3.9) that sup n ≥ µ n ( K ) < ∞ . Thus, by the weak compactnesstheorem for measures (see Section 1.9 in [12]), { µ n } is tight. Therefore there is asubsequence, still denoted by { n } , such that R Q T f dµ n → R Q T f dµ for every f ∈ C ( Q T ). Let ̺ ∈ W be such that R R d ( ̺ ¯ ̺ − ( x )) dx < ∞ and let K ⊂⊂ Q T . Then byTheorem 2.2, Z R d (cid:18) E ,x Z T |∇ ( u n − u m )( t, X t ) | dt (cid:19) ̺ ( x ) dx ≥ C k∇ ( u n − u m ) ̺ k . (3.10)By (2.11), for every x ∈ R d , ξ n,m ( x ) ≡ E ,x R T |∇ ( u n − u m )( t, X t ) | dt → n, m → ∞ .Moreover, by (2.12) and Proposition 3.2, | ξ n,m ( x ) | ≤ C ( E ,x | ϕ ( X T ) | + E ,x Z T | g ( t, X t ) | dt + ¯ ̺ − ( x ))for some C not depending on n, m . Therefore it follows from Theorem 2.2 and theLebesgue dominated convergence theorem that the left-hand side of (3.10) convergesto zero as n, m → ∞ and hence that k K ∇ ( u n − u m ) k → K ⊂⊂ Q T . Usingproperties of { u n } and { µ n } we have already proved we conclude from (3.8) that (3.7)holds for every η ∈ W , ( Q T ) ∩ C ( Q T ) such that η (0) = 0. Step 2. u ∈ C ([0 , T ) × R d ) ∩ C ([0 , T ] , L loc ( R d )). To see this, we first observe that u ( s, x ) = Y s,xs for ( s, x ) ∈ [0 , T ) × R d , since | u ( s, x ) − Y s,xs | ≤ n →∞ ( | ( u − u n )( s, x ) | + E s,x | Y s,x,ns − Y s,xs | ) = 0 . Hence u ( s, x ) = Y s,xs ≤ h ( s, X s ) = h ( s, x ), i.e. ( u − h ) − = 0 and, by (2.13), for any n, m ∈ N , δ > K ⊂⊂ [0 , T − δ ] × R d we have | u n ( s, x ) − u m ( s, x ) | ≤ C ( E s,x | Y s,x,nT − δ − Y s,x,mT − δ | + I s,xn,m + I s,xm,n ) , (3.11)22here I s,xn,m = E s,x Z T − δs ( Y s,x,nt − h ( t, X t )) − dK s,x,mt and K s,x,m is defined as in Theorem 2.4. By Aronson’s upper estimate, E s,x | Y s,x,nT − δ − Y s,x,mT − δ | = E s,x | ( u n − u m )( T − δ, X T − δ ) | = Z R d | ( u n − u m )( T − δ, y ) | p ( s, x, T − δ, y ) dy ≤ Cδ − d/ k u n − u m k with some C depending neither on ( s, x ) ∈ K nor on n, m ∈ N . Moreover, | I s,xn,m | ≤ E s,x sup s ≤ t ≤ T − δ | ( u n ( t, X t ) − h ( t, X t )) − | · E | K s,x,mT − δ | . In view of (2.12), sup n ≥ sup ( s,x ) ∈ K E | K s,x,nT − δ | < ∞ . By Dini’s theorem, ( u n − h ) − → ( u − h ) − = 0 uniformly in any compact subset of [0 , T ) × R d . Therefore, since | h ( t, X t ) | ≤| ¯ u ( t, X t ) | + ¯ ̺ − ( X t ), t ∈ [0 , T ], where ¯ u is a solution of PDE( ϕ, f ), it follows fromTheorem 2.1, Proposition 3.2, Lemma 3.3 and the Lebesgue dominated convergencetheorem that sup ( s,x ) ∈ K ( I s,xn,m + I s,xm,n ) → n, m → ∞ . From the above estimates and(3.11) it follows that u n → u uniformly in any compact subset of [0 , T ) × R d , whichimplies continuity of u on [0 , T ) × R d . By Theorem 2.2, Z R d E ,x sup t ∈ [0 ,T ] | Y ,x,nt − Y ,x,mt | ̺ ( x ) dx ≥ sup t ∈ [0 ,T ] Z R d E ,x | Y ,x,nt − Y ,x,mt | ̺ ( x ) dx = sup t ∈ [0 ,T ] Z R d E ,x | ( u n − u m )( t, X t ) | ̺ ( x ) dx ≥ c sup t ∈ [0 ,T ] k ( u n − u m )( t ) k ,̺ . Therefore, choosing ̺ ∈ W such that R R d ( ̺ ¯ ̺ − ( x )) dx < ∞ and arguing as in theproof of convergence of the left-hand side of (3.10) we deduce from the above thatsup t ∈ [0 ,T ] k ( u n − u m )( t ) k ,̺ → n, m → ∞ , and hence that u ∈ C ([0 , T ] , L loc ( R d )). Step 3. u is the unique weak solution of the problem OP( ϕ, f, h ). We know that E ,x Z T ξ ( t, X t ) dK ,x,nt = E ,x Z T ξn ( u n − h ) − ( t, X t ) dt = Z Q T ξ ( t, y ) p ( s, x, t, y ) dµ n ( t, y ) (3.12)for all ξ ∈ C ( Q T ). Let K ⊂⊂ R d and { ξ n } ⊂ C +0 ( Q T ) be such that ξ n ↓ { }× K . Since µ n ⇒ µ , it follows from (3.12) and (2.11) that Z R d ( E ,x Z T ξ n ( t, X t ) dK ,xt ) dx ≥ Z Q T ξ n ( t, y ) dµ ( t, y )23or n ∈ N . Letting n → ∞ and taking into account that K ,x is continuous we deducefrom the above inequality that R Q T { }× K ( θ, y ) dµ ( θ, y ) = 0. Therefore µ ( { }× K ) = 0for any K ⊂⊂ R d and hence µ ( { } × R d ) = 0. Now from Lemma 2.6 and Step 2 itfollows that µ ( { t } × R d ) = 0 for all t ∈ [0 , T ]. Using this and Lemma 2.6 we see that(2.14) is satisfied for any η ∈ W , , ( Q T ) and t ∈ [0 , T ]. Since u n − h → u − h uniformlyin compact subsets of [0 , T ) × R d , Z T − δs Z R d ξ ( u n − h )( t, x )) dµ n ( t, x ) → Z T − δs Z R d ξ ( u − h )( t, x )) dµ ( t, x )for any ξ ∈ C +0 ( R d ). Hence, since Z T − δs Z R d ξ ( u n − h )( t, x ) dµ n ( t, x )= n Z T − δs Z R d ξ ( u n − h ) · ( u n − h ) − ( t, x ) dt dx ≤ u ≥ h , it follows that R Q T ξ ( u − h ) dµ = 0, which shows that u solves OP( ϕ, f, h ).Uniqueness follows from Theorem 3.1. Step 4.
We show (3.5). From [23] we know that u n ( t, X t ) = ϕ ( X T ) + Z Tt f ( θ, X θ , u n ( θ, X θ ) , σ ∇ u n ( θ, X θ )) dθ + K s,x,nT − K s,x,nt − Z Tt h σ ∇ u n ( θ, X θ ) , dB s,θ i , P s,x - a.s. (3.13)for all n ∈ N . Since u n → u uniformly on compact sets in [0 , T ) × R d , it follows from(2.11) that u ( t, X t ) = Y s,xt , t ∈ [ s, T ], P s,x -a.s. To prove that σ ∇ u ( t, X t ) = Z s,xt , λ ⊗ P s,x -a.s. observe that for any K ⊂⊂ R d and any δ ∈ (0 , T − s ], E s,x Z Ts + δ { X t ∈ K } | σ ∇ u ( t, X t ) − Z s,xt | dt ≤ E s,x Z Ts + δ { X t ∈ K } ( | σ ∇ ( u − u n )( t, X t ) | + | Z s,x,nt − Z s,xt | ) dt ≤ Cδ − d/ Z Ts + δ Z K |∇ ( u − u n )( t, y ) | dt dy + 2 E s,x Z Ts + δ | Z s,x,nt − Z s,xt | dt, which converges to zero as n → ∞ since ∇ u n → ∇ u in L loc ( Q T ). Hence, by Fatou’slemma, E s,x R Ts | σ ∇ u ( t, X t ) − Z s,xt | dt = 0, as required. Step 5.
We show (3.6) and that µ n ⇒ s,x µ , µ n → µ in ( W , ,loc ( Q T )) ∗ for every ( s, x ) ∈ [0 , T ) × R d . Let ξ = ξ + − ξ − ∈ C b ( Q T ). By Theorem 2.4, E s,x (cid:12)(cid:12)(cid:12)(cid:12)Z Tt nξ ( u n − h ) − ( θ, X θ ) dθ − Z Tt ξ ( θ, X θ ) dK s,xθ (cid:12)(cid:12)(cid:12)(cid:12) → t ∈ [ s, T ]. Since µ n ⇒ µ on Q T and µ ( { t } × R d ) = 0 for every t ∈ [0 , T ], wesee that µ n | [ t ,t ] × R d ⇒ µ | [ t ,t ] × R d for every t ≤ t ≤ t ≤ T . Hence we have E s,x Z Ts + δ ξ + ( t, X t ) dK s,xθ = lim n →∞ E s,x Z Ts + δ ξ + ( t, X t ) dK s,x,nt = lim n →∞ Z Ts + δ Z R d ξ + ( t, y ) p ( s, x, t, y ) dµ n ( t, y )= Z Ts + δ Z R d ξ + ( t, y ) p ( s, x, t, y ) dµ ( t, y )for 0 ≤ s < T , δ >
0. Applying the monotone convergence theorem we see that E s,x Z Ts ξ + ( θ, X θ ) dK s,xθ = Z Ts Z R d ξ + ( θ, y ) p ( s, x, θ, y ) dµ ( θ, y ) (3.15)for all s ∈ [0 , T ]. In the same manner we can see that (3.15) holds for ξ − in place of ξ + and hence for ξ in place of ξ + . Consequently, (3.6) holds for s ∈ [0 , T ]. That µ n ⇒ s,x µ now follows from (2.11), (3.14). Strong convergence of { µ n } to µ in ( W , ,loc ( Q T )) ∗ follows from (3.8), (2.14) and the fact that u n → u in W , ,loc ( Q T ). (cid:3) Corollary 3.5
Under the assumption of Theorem 3.4, for any ≤ t < t ≤ T andany closed subset F of R d we have µ ([ t , t ] × F ) = Z R d E t ,x Z t t F ( X t ) dK t ,xt dx. (3.16) Proof.
Let us choose a sequence { ξ n } ⊂ C b ( Q T ) of positive functions such that ξ n ↓ [ t ,t ] × F . Since (3.14) holds for ξ n in place of ξ + , we get (3.16) letting n → ∞ andthen integrating with respect to the space variable. (cid:3) Corollary 3.6
Let assumptions of Theorem 3.4 hold. Then µ is absolutely continuouswith respect to the Lebesgue measure with density r iff K s,xt = Z ts r ( θ, X θ ) dθ, t ∈ [ s, T ] , P s,x - a.s. (3.17) Proof.
Sufficiency follows immediately from (3.6). To prove necessity, suppose that( u, rdλ ) is a weak solution of the OP( ϕ, f, h ) i.e. ( u − h ) dµ = 0, u ≥ h and ∂u∂t + A t u = − f u − r, u ( T ) = ϕ. (3.18)Set r ε = ( r ∧ ε − ) B (0 ,ε − ) and let u ε be a weak solution of PDE( ϕ, f u + r ε ), i.e. ∂u ε ∂t + A t u ε = − f u − r ε , u ε ( T ) = ϕ. k u ε ( t ) k + k σ ∇ u ε k ,T = 2 Z Tt h f u ( s ) , u ε ( s ) i ds + 2 Z Tt h r ε ( s ) , u ε ( s ) i ds ≤ Z Tt k u ε ( s ) k ds + k f u k ,T + 2 Z Tt h r ( s ) , | u ( s ) |i ds and hence, by Gronwall’s lemma, k u ε ( t ) k + k σ ∇ u ε k ,T ≤ C ( k f u k ,T + Z T h r ( t ) , | u ( t ) |i dt ) , (3.19)which is bounded, because R T h r ( t ) , | u ( t ) |i dt < ∞ by Proposition 2.8. Since { u ε } is increasing, there is ¯ u such that u ε ↑ ¯ u . Since we know that { u ε } is bounded in W , ( Q T ), u ε → ¯ u in L ( Q T ) and ∇ u ε → ∇ ¯ u weakly in L ( Q T ) from which it maybe concluded that ¯ u is a weak solution of (3.18). Therefore, u = ¯ u , by uniqueness ofsolution of PDE( ϕ, f u + r ). Now, define r n , µ n as in Theorem 3.4. Let ξ ∈ C b ( Q T ).Since µ n ⇒ s,x µ for every ( s, x ) ∈ [0 , T ) × R d and dµ = r dλ , E s,x Z Ts ξ ( t, X t ) dK s,xt = E s,x Z Ts ξ ( t, X t ) r ( t, X t ) dt. (3.20)Indeed, for every n ∈ N we have E s,x Z Ts ξ ( t, X t ) dK s,x,nt = Z R d Z Ts ξ ( t, y ) p ( s, x, t, y ) dµ n ( t, y )= Z R d Z Ts ξ ( t, y ) p ( s, x, t, y ) r n ( t, y ) dt dy, so letting n → ∞ leads to (3.20). By approximation argument, (3.20) holds for any ξ ∈ C ( Q T ) such that E s,x R Ts | ξ ( t, X t ) | dK s,xt < ∞ . In particular, it holds for ξ = u − h .Hence, letting ε ↓ I ε ≡ E s,x Z Ts ( u ε − h )( t, X t ) r ε ( t, X t ) dt (3.21) → E s,x Z Ts ( u − h )( t, X t ) r ( t, X t ) dt = E s,x Z Ts ( u − h )( t, X t ) dK s,xt = 0 . By representation results proved in [23], u ε ( t, X t ) = ϕ ( X T ) + Z Tt ( f u + r ε )( θ, X θ ) dθ − Z Tt σ ∇ u ε ( θ, X θ ) dB s,θ . (3.22)Applying Itˆo’s formula we obtain I ε ≡ E s,x | u ε ( t, X t ) | + E s,x Z Tt | σ ∇ u ε ( θ, X s,xθ ) | dt E s,x | ϕ ( X s,xT ) | + 2 E s,x Z Tt ( f u u ε + u ε r ε )( θ, X θ ) dθ ≤ E s,x | ϕ ( X s,xT ) | + E s,x Z Tt | f u ( θ, X θ ) | dθ + E s,x Z Tt | u ε ( θ, X θ ) | dθ + | I ε | + 2 E s,x Z Tt r ε h + ( θ, X θ ) dθ for every t ∈ [ s, T ]. Hence, by Gronwall’s lemma, I ε ≤ CE s,x (cid:18) | ϕ ( X s,xT ) | + Z Ts | f u ( t, X t ) | dt + | I ε | + Z Ts ( r ε h + )( t, X t ) dt (cid:19) . (3.23)For any α > E s,x Z Ts ( r ε h + )( t, X t ) dt ≤ E s,x α sup s ≤ t ≤ T | h + ( t, X t ) | + α − | Z Ts r ε ( t, X t ) dt | ! and, by (3.22), E s,x | Z Ts r ε ( t, X t ) dt | ≤ E s,x | ϕ ( X s,xT ) | + E s,x Z Ts | f u ( t, X t ) | dt + I ε . (3.24)Hence, choosing a sufficiently large α we see from (3.23) that I ε ≤ CE s,x | ϕ ( X s,xT ) | + Z Ts | f u ( t, X t ) | dt + | I ε | + sup s ≤ t ≤ T | h + ( t, X t ) | ! . (3.25)Therefore, combining (3.24) with (3.21), (3.25) and using Fatou’s lemma we concludethat E s,x ( R Ts r ( t, X t ) dt ) < ∞ . Finally, by (3.22) and Itˆo’s formula, for any ε , ε > α > E s,x | ( u ε − u ε )( t, X t ) | + E s,x Z Ts | σ ∇ ( u ε − u ε )( t, X t ) | dt ≤ CE s,x Z Ts ( r ε − r ε )( u ε − u ε )( t, X t ) dt ≤ CE s,x sup s ≤ t ≤ T | ( u ε − u ε )( t, X t ) | Z Ts | ( r ε − r ε )( t, X t ) | dt ≤ α − CE s,x sup s ≤ t ≤ T | ( u ε − u ε )( t, X t ) | + αCE s,x (cid:18)Z Ts | ( r ε − r ε )( t, X t ) | dt (cid:19) . Hence, using the Burkholder-Davis-Gundy inequality we obtain the estimate E s,x sup s ≤ t ≤ T | ( u ε − u ε )( t, X t ) | + E s,x Z Ts | σ ∇ ( u ε − u ε )( t, X t ) | dt ≤ CE s,x (cid:18)Z Ts | ( r ε − r ε )( t, X t ) | dt (cid:19) with C not depending on ε , ε . Therefore letting ε ↓ u ( t, X t ) , σ ∇ u ( t, X t ) , R t r ( t, X s,xt ) dt ), t ∈ [ s, T ], is a solution of RBSDE( ϕ, f, h ) whichin view of uniqueness completes the proof. (cid:3) emma 3.7 If ˜ u is a solution of PDE ( ϕ, f ) and ( u, µ ) is a solution of OP ( ϕ, f, h ∨ ˜ u ) then ( u, µ ) is a solution of OP ( ϕ, f, h ) .Proof. Let ( u, µ ) be a solution of OP( ϕ, f, h ∨ ˜ u ). Then u ≥ h ∨ ˜ u ≥ h . Moreover,by comparison results, for any solution ( u , µ ) of OP( ϕ, f, h ) with some h we have u ≥ ˜ u . Hence µ { h ≤ ˜ u } = 0, and consequently, Z Q T ( u − h ) dµ = Z { h ≤ ˜ u } ( u − h ) dµ + Z { h> ˜ u } ( u − h ) dµ = Z { h> ˜ u } ( u − ( h ∨ ˜ u )) dµ = 0 , which proves the lemma. (cid:3) Lemma 3.8
Let ϕ ∈ L ,̺ ( Q T ) , g ∈ L p,q,̺ ( Q T ) . Then E s,x | ϕ ( X T ) | ≤ C̺ − ( x )( T − s ) − d/ k ϕ k ,̺ and E s,x Z Ts | g ( t, X t ) | dt ≤ C̺ − ( x ) k g k p,q,̺ . Proof.
Both inequalities follows form Aronson’s estimates, because Z R d | ϕ ( y ) | p ( s, x, T, y ) dy ≤ C̺ − ( x ) Z R d | ϕ ( y ) | | ̺ ( y ) | p ( s, x, T, y ) | ̺ ( y − x ) | dy ≤ C̺ − ( x )( T − s ) − d/ k ϕ k ,̺ and, by H¨older’s inequality, Z Ts Z R d | g ( t, y ) | p ( s, x, t, y ) dt dy ≤ C̺ − ( x ) Z Ts Z R d | g ( t, y ) | ̺ ( y ) p ( s, x, t, y ) ̺ − ( y − x ) dt dy ≤ C̺ − ( x ) k g k p,q,̺ k p (0 , , · , · ) ̺ − k ( p/ ∗ , ( q/ ∗ , which is finite by Aronson’s estimate. (cid:3) Lemma 3.9 If ϕ ∈ L ,̺ ( R d ) , g ∈ L ,̺ ( Q T ) for some ̺ ∈ W and (2.6) is satisfied forevery ( s, x ) ∈ [0 , T ) × R d , then for every K ⊂⊂ [0 , T ) × R d sup ( s,x ) ∈ K E s,x | ( ϕ − ϕ n )( X T ) | → and sup ( s,x ) ∈ K E s,x Z Ts | ( g − g n )( t, X t ) | dt → as n → ∞ , where ϕ n = ϕ B (0 ,n ) , g n = g B (0 ,n ) . roof. The first assertion follows from Lemma 3.8. To prove the second, let us choose
R > K ⊂ [0 , T ) × B (0 , R ) and x ∈ B (0 , R ). Then for n ≥ R we have Z Ts Z R d | ( g n − g )( t, y ) | p ( s, x, t, y ) dt dy = Z Ts Z B c (0 , R ) | ( g n − g )( t, y ) | ̺ ( y ) p ( s, x, t, y ) ̺ − ( y ) dt dy ≤ C̺ − ( x ) Z Ts Z B c (0 , R ) | ( g n − g )( t, y ) | ̺ ( y ) ψ ( s, x, t, y ) dt dy, where ψ ( s, x, t, y ) = ( t − s ) − d/ exp( − | y − x | C ( t − s ) )(1 + | y − x | ) α . Since ψ is bounded for0 ≤ s < t ≤ T , | x − y | > R we see that E s,x Z Ts | ( g − g n )( t, X t ) | dt = Z Ts Z R d | ( g n − g )( t, y ) | p ( s, x, t, y ) dt dy ≤ C̺ − ( x ) k g n − g k ,̺,T (3.26)for ( s, x ) ∈ K , n ≥ R , which completes the proof. (cid:3) Theorem 3.10
Let assumptions (H1)-(H3) hold with ϕ ∈ L ,̺ ( R d ) , g ∈ L ,̺ ( Q T ) ,where ̺ ∈ W and ̺ < . Moreover, assume that g satisfies (2.4), h ∈ C ( Q T ) and h ≤ c ¯ ̺ − for some c > and ¯ ̺ ∈ W such that ¯ ̺ − ∈ L ,̺ ( R d ) . Then there exists aunique solution ( u, µ ) of OP ( ϕ, f, h ) such that u ∈ C ([0 , T ) × R d ) ∩ W , ,̺ ( Q T ) and (3.5),(3.6) hold for each ( s, x ) ∈ [0 , T ) × R d .Proof. We divide the proof into two steps: the case of linear and semilinear equation.
Step 1 . We first assume that f = f ( t, x ) , ( t, x ) ∈ Q T satisfies (2.4) with g replacedby f . Suppose that h ( x ) ≤ c ¯ ̺ − , where ¯ ̺ ( x ) = (1 + | x | ) − β for some c, β >
0. Set ϕ n = B (0 ,n ) ϕ , f n = B (0 ,n ) f and consider a sequence { h n } ⊂ W , ( Q T ) such that h n ≤ c ¯ ̺ − , n ∈ N , and h n → h uniformly in compact subsets of Q T . By Theorem 3.4,for each n ∈ N there is a unique solution ( u n , µ n ) of OP( ϕ n , f n , h n ), and moreover,( u n ( t, X t ) , σ ∇ u n ( t, X t )) = ( Y s,x,nt , Z s,x,nt ) , P s,x - a.s. and Z Ts Z R d ξ ( t, y ) p ( s, x, t, y ) dµ n ( t, y ) = E s,x Z Ts ξ ( t, X t ) dK s,x,nt for all ξ ∈ C b ( Q T ), where ( Y s,x,nt , Z s,x,nt , K s,x,nt ) is a solution of RBSDE( ϕ n , f n , h n ).By Lemma 2.5, E s,x sup s ≤ t ≤ T | ( u n − u m )( t, X t ) | + E s,x sup s ≤ t ≤ T | K s,x,nt − K s,x,mt | + E s,x Z Ts | σ ∇ ( u n − u m )( t, X t ) | dt ≤ E s,x sup s ≤ t ≤ T | ( h n − h m )( t, X t ) | + E s,x | ( ϕ n − ϕ )( X T ) | + E s,x Z Ts | ( f n − f m )( θ, X θ ) | dθ. (3.27)29rom this and Theorem 2.2 we deduce that k ( u n − u m )( s ) k ,̺ + k∇ ( u n − u m ) k ,̺ , T ≤ C Z R d E s,x sup s ≤ t ≤ T | ( h n − h m )( t, X t ) | ̺ ( x ) dx + k ϕ n − ϕ m k ,̺ + k f n − f m k ,̺,T (cid:1) . (3.28)Using Theorem 2.2 we also getsup ≤ t ≤ T k ( u n − u m )( t ) k ,̺ ≤ C Z R d sup ≤ t ≤ T E ,x | ( u n − u m )( t, X t ) | ̺ ( x ) dx. (3.29)Due to Lemma 3.7, without loss of generality we may assume that h n ≥ ˜ u n , where ˜ u n isa solution of PDE( ϕ n , f n ). From comparison theorem (see [11]) we know that u ≤ ˜ u n ,where u is a continuous solution of PDE( −| ϕ | , −| f | ), and that u n ց u , where u n is acontinuous solution of PDE( −| ϕ n | , −| f n | ). Since E s,x sup s ≤ t ≤ T | ( u ( t, X t )) + | ≤ E s,x sup s ≤ t ≤ T | ( u n ( t, X t )) + | and E s,x sup s ≤ t ≤ T | ( u ( t, X t )) − | = E s,x lim n →∞ sup s ≤ t ≤ T | ( u n ( t, X t )) − | ≤ lim inf n →∞ E s,x sup s ≤ t ≤ T | ( u n ( t, X t )) − | , from a priori estimates for solutions of BSDE( −| ϕ n | , −| f n | ) (see [23]) we get E s,x sup s ≤ t ≤ T | u ( t, X t ) | ≤ CE s,x (cid:18) | ϕ ( X T ) | + Z Ts | g ( t, X t ) | dt (cid:19) . (3.30)Since | h n ( t, X t ) | ≤ | u ( t, X t ) | + 2(1 + | X t | ) β and { h n } converges uniformly in compactsubsets of Q T , using (3.30), Proposition 3.2 and Lemma 3.9 we conclude that the right-hand side of (3.27) converges to zero as n, m → ∞ . From this it follows that thereis u such that u n → u pointwise in [0 , T ) × R d . Moreover, using (3.28), (3.29) andarguing as in the proof of convergence of the right-hand side of (3.10) we conclude that u n → u in W , ,̺ ( Q T ) and u n → u in C ([0 , T ] , L ,̺ ( R d )). By the definition of solutionof OP( ϕ n , f n , h n ), Z Tt h u n ( s ) , ∂η∂s ( s ) i ds + 12 Z Tt h a ( s ) ∇ u n ( s ) , ∇ η ( s ) i ds = Z Tt h f n ( s ) , η ( s ) i ds + Z Tt Z R d η dµ n + h ϕ n , η ( T ) i − h u n ( t ) , η ( t ) i (3.31)for any η ∈ W , , ( Q T ). Therefore, if K ⊂⊂ [0 , T ) × R d , then choosing η ∈ W , ( Q T ) ∩ C ( Q T ) such that η ≡ K and 0 ≤ η ≤ u n , µ n ) and p ≡ ̺ − that sup n ≥ µ n ( K ) < ∞ . Thus, { µ n } is tight. Takinga subsequence if necessary we may assume that µ n ⇒ µ , where µ is a Radon measure30n Q T . Using arguments similar to those in the proof of Step 3 of Theorem 3.4 showsthat µ ( { t } × R d ) = 0 for every t ∈ [0 , T ]. Now we will show that u is continuous on[0 , T ) × R d . By Lemma 2.5, for any 0 < δ < T / K ⊂⊂ [0 , T − δ ] × R d ,sup ( s,x ) ∈ K | ( u n − u m )( s, x ) | ≤ sup ( s,x ) ∈ K E s,x sup s ≤ t ≤ T − δ | ( h n − h m )( t, X t ) | + sup ( s,x ) ∈ K Z Ts | ( f n − f m )( t, X t ) | dt + sup ( s,x ) ∈ K E s,x | Y s,x,nT − δ − Y s,x,mT − δ | . Since h n → h uniformly in compact subsets of Q T , using Theorem 2.1, Proposition 3.2and Lemma 3.3 we conclude that the first term on the right-hand of above inequalityconverges to zero as n, m → ∞ . Convergence of the second term follows from Lemma3.9 and the third from the inequalitysup ( s,x ) ∈ K E s,x | Y s,x,nT − δ − Y s,x,mT − δ | ≤ sup ( s,x ) ∈ K C̺ − ( x ) δ − d/ k ( u n − u m )( T − δ ) k ,̺ which is a consequence of Lemma 3.8. Thus, u ∈ C ([0 , T ) × R d ). The pair ( u, µ ) isa weak solution of OP( ϕ, f, h ), because letting n → ∞ in (3.31) we get (2.14), andsimilarly, by passing to the limit we show that conditions (b), (d) of the definition of asolution are satisfied. Finally, (3.5), (3.6) we show as in the proof of Theorem 3.4. Step 2.
We consider the general semilinear case. For γ > V ( γ ) denote the Banach space consisting of elements u of W , ,̺ ( Q T ) ∩ C ([0 , T ] , L ,̺ ( R d ))equipped with the norm k u k V ( γ ) = sup ≤ s ≤ T k u γ ( s ) k ,̺ + k u γ k ,̺,T + λ k∇ u γ k ,̺,T , where u γ ( s, x ) = e sγ/ u ( s, x ). Write K n = [0 , T − /n ] × B (0 , n ). By W ( γ ) we denotethe Fr´echet space of elements of W , ( Q T ) such that sup ( s,x ) ∈ K n k u k γ,s,x < ∞ for all n ∈ N equipped with the F -norm k u k W ( γ ) = ∞ X n =0 ∧ sup ( s,x ) ∈ K n k u k γ,s,x n , where k u k γ,s,x = R Ts R R d e γt ( | u ( t, y ) | + λ |∇ u ( t, y ) | ) p ( s, x, t, y ) dt dy , and by B the Fr´echetspace of elements of C ([0 , T ) × R d ) with the F -norm k u k B = ∞ X n =1 ∧ k u K n k ∞ n . Finally, let M γ denote the Fr´echet space B ∩ V ( γ ) ∩ W ( γ ) equipped with the F -norm ||| u ||| γ = k u k V ( γ ) + k u k W ( γ ) + k u k B . Now, define the mapping Φ : M γ → M γ by putting Φ( v ) to be the first component of the solution ( u, µ ) of OP( ϕ, f v , h ). By Step 1 the definition of Φ is correct. We are going to show that Φ is contractive on M γ . Let v , v ∈ M γ and let ( u i , µ i ), i = 1 ,
2, be solutions of OP( ϕ, f v i , h ). Set31 = u − u = Φ( v ) − Φ( v ), µ = µ − µ . By the definition of a solution of theobstacle problem, h u ( t ) , η ( t ) i + Z Tt h u ( s ) , ∂η∂s ( s ) i ds + 12 Z Tt h a ( s ) ∇ u ( s ) , ∇ η ( s ) i ds = Z Tt Z R d ηdµ + Z Tt h f v ( s ) − f v ( s ) , η ( s ) i ds. Putting η ( s ) = e γs u ( s ) ̺ ξ n , where ξ n is defined as in the proof of Proposition 2.9, weobtain e γt h u ( t ) , u ( t ) ̺ ξ n i + γ Z Tt e γs h u ( s ) , u ( s ) ̺ ξ n i ds + Z Tt e γs h u ( s ) , ∂u∂s ( s ) ̺ ξ n i ds + 12 Z Tt e γs h a ( s ) ∇ u ( s ) , ∇ ( u̺ ξ n )( s ) i ds = Z Tt Z R d u ( s ) ̺ ξ n dµ + Z Tt e γs h f v ( s ) − f v ( s ) , u ( s ) ̺ ξ n i ds. By the above and (3.3), e γt k u ( t ) ̺ξ n k + γ Z Tt e γs k u ( s ) ̺ξ n k ds + 12 Z Tt e γs dds k u ( s ) ̺ξ n k ds + 12 Z Tt e γs h a ( s ) ∇ u ( s ) , ∇ ( u̺ ξ n )( s ) i ds ≤ Z Tt e γs h f v ( s ) − f v ( s ) , u ( s ) ̺ ξ n i ds. Consequently, e γt k u ( t ) ̺ξ n k + γ Z Tt e γs k u ( s ) ̺ξ n k ds + Z Tt e γs h a ( s ) ∇ u ( s ) , ∇ ( u̺ ξ n )( s ) i ds ≤ Z Tt e γs h f v ( s ) − f v ( s ) , u ( s ) ̺ ξ n i ds. Letting n → ∞ and performing computations similar to that in the proof of Theorem3.1 we get e γt k u ( t ) ̺ k + γ Z Tt e γs k u ( s ) ̺ k ds + λ Z Tt e γs k∇ u ( s ) ̺ k ds ≤ Z Tt e γs h f v ( s ) − f v ( s ) , u ( s ) ̺ i ds + Λ2 λ Z Tt e γs k u ( s ) ̺ k ds. The right-hand side of the above inequality may be estimated by2 L Z Tt e γs k ( v − v )( s ) ̺ k k u ( s ) ̺ k ds +2 L Λ Z Tt e γs k∇ ( v − v )( s ) ̺ k k u ( s ) ̺ k ds + Λ2 λ Z Tt e γs k u ( s ) ̺ k ds ≤ (4 L + 8Λ L λ + Λ2 λ ) Z Tt e γs k u ( s ) ̺ k ds + 14 Z Tt e γs k ( v − v )( s ) ̺ k ds + 14 Z Tt λ e γs k∇ ( v − v )( s ) ̺ k ds. γ = 1 + 4 L + 8 λ − Λ L + (2 λ ) − Λ we see that k Φ( v ) − Φ( v ) k V ( γ ) ≤ − k v − v k V ( γ ) . (3.32)Let ( Y s,x,i , Z s,x,i , K s,x,i ), i = 1 ,
2, denote a solution of RBSDE( ϕ, f v i , h ) and let v = v − v . We already know that ( Y s,x,it , Z s,x,it ) = ( u i ( t, X t ) , σ ∇ u i ( t, X t )), t ∈ [ s, T ].Therefore, since E s,x R Tt e γθ v ( θ, X θ ) d ( K s,x, θ − K s,x, θ ) ≤ t ∈ [ s, T ], usingItˆo’s formula we have E s,x e γt | u ( t, X t ) | + E s,x Z Tt e γθ ( γ | u ( θ, X θ ) | + | σ ∇ u ( θ, X θ ) | ) dθ ≤ E s,x Z Tt e γθ u ( f v − f v )( θ, X θ ) dθ ≤ LE s,x Z Tt e γθ u ( θ, X θ )( | v | + | σ ∇ v | )( θ, X θ ) dθ ≤ λ − Λ L εE s,x Z Tt e γθ | u ( θ, X θ ) | dθ + ε − E s,x Z Tt e γθ ( | v | + λ |∇ v | )( θ, X θ ) | dθ. Putting γ = 1 + 8Λ λ − L ε with suitably chosen ε > E s,x sup s ≤ t ≤ T e γt | u ( t, X t ) | + E s,x Z Ts λe γt ( | u ( t, X t ) | + |∇ u ( t, X t ) | ) dt ≤ − E s,x Z Ts e γt ( | v | + λ |∇ v | )( t, X t ) dt. (3.33)From this we obtain k Φ( v ) − Φ( v ) k B + k Φ( v ) − Φ( v ) k W ( γ ) ≤ − ( k v − v k B + k v − v k W ( γ ) ) , which when combined with (3.32) shows that Φ is contractive on M γ . By Banach’sprinciple, Φ has a unique fixed point u . Clearly, the solution ( u, µ ) of OP( ϕ, f u , h ) hasthe asserted properties. (cid:3) One can prove Theorem 3.10 by the method of stochastic penalization used in theproof of Theorem 3.4. To apply that method one should first generalize results of [23] onrepresentation of solutions of the Cauchy problem proved for ϕ ∈ L ( R d ), g ∈ L ( Q T )to the case ϕ ∈ L ,̺ ( R d ), g ∈ L ,̺ ( Q T ) for some ̺ ∈ W . Since detailed proof of sucha generalization does not bring new ideas and at the same time requires some efforts,we decided to present a different approach. Note, however, that the adopted approachuses some ideas from [23]. Corollary 3.11
Let assumptions of Theorem 3.10 hold. Define ( u n , µ n ) as in Theorem3.4. Then (i) u n ↑ u uniformly in compact subsets of [0 , T ) × R d , u n → u in W , ,̺ ( Q T ) ∩ C ([0 , T ] , L ,̺ ( Q T )) , µ n ⇒ µ , µ n → µ in ( W , ,̺ ( Q T )) ∗ , µ n ⇒ s,x µ for every ( s, x ) ∈ [0 , T ) × R d .Proof. Follows from Theorems 2.2, 2.4 and 3.10. (cid:3)
Let us remark that Corollaries 3.5, 3.6 hold also under the assumptions of Theorem3.10. The proof of Corollary 3.5 runs as before. In the proof of Corollary 3.6 the maindifference consists in the fact that instead of boundedness of { u ε } in W , ( Q T ) (see(3.19)) we have to prove its boundedness in W , ,̺ ( Q T ). The last assertion one can showusing arguments from the proof of Proposition 2.9. Corollary 3.12
Let assumptions of Theorem 3.4 or Theorem 3.10 hold and let ( u, µ ) be a solution of OP ( ϕ, f, h ) . (i) If g ∈ L p,q,̺ ( Q T ) then | u ( s, x ) | + k u k W ( s,x,T ) ≤ C̺ − ( x )(1 + ( T − s ) − d/ k ϕ k ,̺ + k f k p,q,̺ ) / . (ii) If | ϕ | ≤ c̺ − for some c > , ̺ ∈ W (i.e. ϕ satisfies the polynomial growthcondition) then | u ( s, x ) | + k u k W ( s,x,T ) ≤ C̺ − ( x )(1 + k f k p,q,̺ )) / . (iii) If | ϕ | + | g | ≤ c̺ − for some c > , ̺ ∈ W then | u ( s, x ) | + k u k W ( s,x,T ) ≤ C̺ − ( x ) . Proof.
It follows from (2.12), Theorem 3.10, Proposition 3.2 and Lemma 3.8. (cid:3)
It is known that the obstacle problem (1.3) with non-divergent operator A t appearas the Hamilton-Jacobi-Bellman equation for an optimal stopping time problem (see[4]) and that value functions of that stopping problem is given by the first componentof a solution of an RBSDEs with forward driving processes associated with A t (see[11]). It is worth noting that similar relations hold for divergence form operators. Corollary 3.13
Let assumptions of Theorem 3.4 or Theorem 3.10 hold and let ( u, µ ) be a solution of OP ( ϕ, f, h ) . Then for each t ∈ [ s, T ] , u ( s, x ) = sup τ ∈T st E s,x ( Z τt f ( θ, X θ , u ( θ, X θ ) , σ ∇ u ( θ, X θ )) dθ + h ( τ, X τ ) τ Theorem 3.14 Assume that (H1)–(H3) hold with ϕ ∈ L ,̺ ( R d ) , g ∈ L ,̺ ( Q T ) forsome ̺ ∈ W and with h ∈ C ( Q T ) satisfying the polynomial growth condition. Thenthere exists a version u of minimal weak solution of OP ( ϕ, f, h ) in the variational sensesuch that if (2.6) is satisfied for some ( s, x ) ∈ [0 , T ) × R d then ( Y s,xt , Z s,xt ) = ( u ( t, X t ) , σ ∇ u ( t, X t )) , t ∈ [ s, T ] , P s,x - a.s. (3.34) Proof. By [4, Theorem 4.1.6] there exists the minimal weak solution ¯ u of OP( ϕ, f, h ) inthe variational sense. Repeating arguments from the proof of Proposition 2.3 we showthat there is a version u of a weak solution of the linear OP( ϕ, f ¯ u , h ) in the variationalsense such that (3.34) holds if (2.6) is satisfied. Since k g k ,̺ < ∞ , it follows that (2.6)is satisfied for a.e. ( s, x ) ∈ [0 , T ) × R d (see remark following the proof of Theorem 2.4).Therefore, by Theorems 2.2 and 2.4, u is a limit in W , ,̺ ( Q T ) of the penalizing sequencedefined by (3.4), and hence (see the proof of [4, Theorem 4.1.6]), u is a minimal weaksolution of OP( ϕ, f ¯ u , h ) in the variational sense. Since the minimal solution is unique, u = ¯ u , and the proof is complete. (cid:3) References [1] Aronson, D.G.: Non-Negative Solutions of Linear Parabolic Equations. Ann. Sc.Norm. Super. Pisa , 607–693 (1968)[2] Bally, V., Matoussi, A.: Weak Solutions for SPDEs and Backward Doubly Stocha-stick Differential Equations. J. Theoret. Probab. , 125–164 (2001)[3] Bally, V., Pardoux, E., Stoica, L.: Backward Stochastic Differential EquationsAssociated to a Symmetric Markov Process. Potential Analysis , 17–60 (2005)[4] Bensoussan, A., Lions J.-L.: Applications of Variational Inequalities in StochasticControl. North-Holland, Amsterdam (1982)[5] Biroli, M., Mosco, U.: Wiener Estimates for Parabolic Obstacle Problem. Nonlin-ear Anal. , 1005–1027 (1987)[6] Biroli, M., Mosco, U.: Wiener Criterion and Potential Estimates for ObstacleProblems Relative to Degenerate Elliptic Operators. Ann. Mat. Pura Appl. ,255–281 (1991)[7] Bogachev, V.I: Measure Theory. Vol. II. Springer, Berlin-Heidelberg (2007)[8] Brezis, H.: Un probl`eme d’evolution avec contraintes unilat´erales d´ependant dutemps. C.R. Acad. Sci. Paris , 310–312 (1972)359] Charrier, P., Troianiello, G.M.: On strong solutions to parabolic unilateral prob-lems with obstacle dependent on time. J. Math. Anal. Appl. , 110–125 (1978)[10] Donati, F.: A penalty method approach to strong solutions of some nonlinearparabolic unilateral problems. Nonlinear Anal. , 585–597 (1982)[11] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S., Quenez, M.C.: Reflected so-lutions of backward SDEs, and related obstacle problems for PDE’s. Ann. Probab. , 702–737 (1997)[12] Evans, L.C., Gariepy, R.F.: Measure and Fine Properties of Function. CRC Press,New York (1992)[13] Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric MarkovProcesses. Walter de Gruyter, Berlin, New York (1994)[14] Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalitiesand Their Applications. Academic Press, New York, London (1980)[15] Ladyzenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-LinearEquations of Parabolic Type. Transl. Math. Monographs , Amer. Math. Soc.,Providence, R.I. (1968)[16] Leone, C.: Existence and uniqueness of solution for nonlinear obstacle problemswith measure data. Nonlinear Anal. , 199–215 (2001)[17] Lions, J.-L.: Quelques M´ethodes de R´esolutions des Probl`emes aux Limites NonLin´eaires. Dunod, Gauthier Villars, Paris (1969)[18] Malek, J., Necas, J., Rokyta, M., Ruzicka, M.: Weak and Measure-Valued Solu-tions to Evolutionary PDEs. Chapman and Hall, London (1996)[19] Mignot, F., Puel, J.P.: In´equations d’´evolution paraboliques avec conve-xes d´ependant du temps. Applications aux in´equations quasi-variationnellesd’´evolution. Arch. Ration. Mech. Anal. , 59–91 (1977)[20] Mokrane, A.: An existence Result via Penalty Method for Some NonlinearParabolic Unilateral Problems. Boll. Unione Mat. Ital. , 405–417 (1994)[21] Mosco, U.: Wiener Criterion and Potential Estimates for the Obstacle problems.Indiana Univ. Math. J. , 455–494 (1987)[22] Rozkosz, A.: Weak convergence of diffusions corresponding to divergence formoperators. Stochastics Stochastics Rep. , 129–157 (1996)[23] Rozkosz, A.: Backward SDEs and Cauchy problem for semilinear equations indivergence form. Probab. Theory Relat. Fields , 393–401 (2003)[24] Stroock, D.W.: Diffusion Semigroups Corresponding to Uniformly Eliptic Diver-gence Form Operators. Seminaire de Probabilities XXII. Lect. Notes Math.1321