arxiv:89.4447v4 [mah.oc] 15 Dec 29 The Fizparick funcion - a bridge beween convex analysis and mulivalued sochasic differenial equaions Aurel Răşcanu 1 Eduard Roensein 2 Absrac Using he Fizparick funcion, we characerize he soluions for differen classes of deerminisic and sochasic differenial equaions driven by maximal monoone operaors (or in paricular subdifferenial operaors) as he minimum poin of a suiably chosen convex lower semiconinuous funcion. Such echnique provides a new approach for he exisence of he soluions for he considered equaions. 2 Mahemaics Subjec Classificaion: 6H15, 65C3, 47H5, 47H15. Key words and phrases: maximal monoone operaors, Fizparick funcion, Skorohod problem, sochasic differenial equaions. 1 Preliminaries. Noaions The Fizparik funcion proved o be a very useful ool of he convex analysis in he sudy of maximal monoone operaors. In our paper his funcion is used for deerminisic and sochasic differenial equaions driven by mulivalued maximal monoone operaors. We will show how we can reduce he exisence problem for sochasic differenial equaions of he following ypes: The work for his paper was suppored by founds from he Gran CNCSIS nr. 1373/27 and Gran CNCSIS nr. 1156/25. 1 Deparmen of Mahemaics, Al. I. Cuza Universiy, Bd. Carol no.9-11 & Ocav Mayer Mahemaics Insiue of he Romanian Academy, Bd. Carol I, no.8, Romania, e-mail: aurel.rascanu@uaic.ro 2 Deparmen of Mahemaics, Al. I. Cuza Universiy, Bd. Carol no.9-11, Iaşi, România, e-mail: eduard.roensein@uaic.ro 1
forward case { dx +A(X )(d) F (,X )d+g(,x )dw, (1) backward case X = ξ, [,T] and { dy +A(Y )d H(,Y,Z )d Z dw, (2) Y T = ξ, [,T] o a minimizing problem for convex lower semiconinuous funcions. Usually, exisence resuls are obained via a penalized problem wih Yosida s approximaion operaor A ε := [I (I +εa) 1 ]/ε. For he forward equaion (1), by sudying firs a generalized Skorohod problem { dx()+a(x())(d) f ()d+dm(), x() = x, [,T]. he exisence of he soluion is obained (see Bensoussan & Răşcanu [4], Răşcanu [16], or Asiminoaei & Răşcanu [1]) in he general case of a maximal monoone operaor. For backward sochasic differenial equaions he exisence problem(see Pardoux & Rascanu [13]) is solved only in he case of A = ϕ (he subdifferenial of a lower semiconinuous convex funcion) and i is an open problem in he general case. Tha is he reason and he main moivaion o find an approach via convex analysis. In 1988, in he paper [1], Fizparick proved ha any maximal monoone operaor can be represened by a convex funcion; he explicily defined he minimal convex represenaion. The connecion beween maximal monoone operaors and convex funcions was also approached 13 years laer by Marinez-Legaz & Thera in [12], Burachik & Svaier in [7] and Burachik & Fizparick in [6]. Since hese las hree papers, Fizparick s resuls have been he subjec of inense research (J.P. Revalski, M. Thera, R.S. Burachik, B.F. Svaier, J.-P. Peno, S. Simons, C. Zălinescu, J.-E. Marinez-Legaz ec.). Their resuls say in he domain of nonlinear operaors: properies, characerizaions, new classes of monoone operaors. Using he idea of Fizparick funcion we can reduce he exisence problems for sochasic equaions of he form (1) or (2) o a minimizing problem of a convex lower semiconinuous funcion. Inspired by he sudies of Gyöngy & Marínez [11], we presen a new approach for solving he exisence problem for sochasic differenial equaions wih maximal monoone operaor. In his paper we will idenify he soluions of differen ypes of forward and backward mulivalued sochasic differenial equaions wih he minimum poins of a suiably chosen convex lower semiconinuous funcionals. The paper is organized as follows. In he firs secion we presen some basic properies of he Fizparick s funcion and we will inroduce he sochasic framework ha will be used. The nex secion conains a Fizparick funcion approach for he sudy of a generalized Skorohod problem as well of forward and backward sochasic differenial equaions, while Secion 3 is dedicaed o he case of forward and backward sochasic variaional inequaliies. 2
1.1 On Fizparick s funcion Le (X,. ) be a real Banach space and (X,. ) be is dual. For x X and x X we denoe x (x) (he value of x in x) by x,x or x,x. If A : X X is a poin-o-se operaor (from X o he family of subses of X ), hen Dom(A) := {x X : A(x) } and R(A) = {x : x Dom(A) s.. x A(x)}. We shall always assume ha he operaor A is proper, i.e. Dom(A). Usually he operaor A is idenified wih is graph gr(a) = {(x,x ) X X : x Dom(A), x A(x)}. The operaor A : X X is a monoone operaor (A X X is a monoone se) if x y,x y, (x,x ),(y,y ) A. A monoone operaor (se) is maximal monoone if i is no properly conained in any oher monoone operaor (se). Clearly if A is maximal monoone and (y,y ) X X hen inf y u,y u (y,y ) A. (u,u ) A Given a funcion ψ : X ],+ ] we denoe Dom(ψ) := {x X : ψ(x) < }. We say ha ψ is proper if Dom(ψ). The subdifferenial ψ : X X is defined by (x,x ) ψ if y x,x +ψ(x) ψ(y), y X. I is well known ha: if ψ is a proper convex l.s.c. funcion, hen ψ : X X is a maximal monoone operaor. Le ψ : X ],+ ] be a proper funcion. The conjugae of ψ is he funcion ψ : X ],+ ], ψ (x ) := sup{ u,x ψ(u) : u X}. Remark ha, if h : X X ],+ ], hen h : X X ],+ ] and, for any (x,x) X X, h (x,x) is well defined by idenifying X wih is image under canonical injecion of X ino X, ha is, every x X can be seen as a funcion x : X R defined by x(x ) = x (x) = x,x. For a complee sudy on maximal monoone operaors, one can consul Barbu [2] or Brézis [5]. Definiion 1 Given a monoone operaor A : X X, he associaed Fizparick funcion is defined as H = H A : X X ],+ ], (3) H(x,x ) := x,x inf{ x u,x u : (u,u ) A} = sup{ u,x + x,u u,u : (u,u ) A} Clearly H(x,x ) x,x, for all (x,x ) A and, as supremum of convex srongly (and (w,w )) coninuous funcions, H = H A : X X ],+ ] is a proper convex srongly (and (w,w )) l.s.c. funcion. Usually, we shall consider on X he srong opology and, on X he w -opology; in his case, H is also a l.s.c. funcion. Whenever is necessary, we will consider he Fizparick funcion H resriced a U V, wih U X and V X. 3
Le (x,x) H(u,u ). Then, from he definiion of a subdifferenial operaor, we have or, equivalenly, (x,x),(z,z ) (u,u ) +H(u,u ) H(z,z ), (z,z ) X X, (4) u x,u x inf{ u y,u y : (y,y ) A} z x,z x inf{ z y,z y : (y,y ) A}, (z,z ) X X. Since he operaor A is a maximal monoone one, hen consequenly, we have inf{ u y,u y : (y,y ) A} and inf{ z y,z y : (y,y ) A} =, (z,z ) A; (5) (x,x) H(u,u ) = u x,u x inf{ z x,z x : (z,z ) A}. Also, by he monooniciy of A, from (4) follows (x,x ) A = (x,x) H(x,x ). Hence, if A : X X is a maximal monoone operaor, hen H A characerizes A as follows. Theorem 2 (Fizparick) (see Fizparick [1], Simons & Zălinescu [17]) Le A : X X be a maximal monoone operaor and H is associaed Fizparick funcion. Then, for all (x,x ) X X, H(x,x ) x,x. Moreover, he following asserions are equivalen: (a) (x,x ) A; (b) H(x,x ) = x,x ; (c) H (x,x) = x,x ; (d) (u,u ) Dom( H) such ha (x,x) H(u,u ) and u x,u x = ; (e) (x,x) H(x,x ). Proof. I is no difficul o show ha (b) (a) (e) (d) (a). Moreover, using he Fenchel equaliy: (x,x) H(x,x ) H(x,x )+H (x,x) = (x,x ),(x,x), we obain ha (e)&(b) (c). The poin (c) yields (a) by using he equivalen form of he definiion of H : H (x,x) = x,x inf u +H(u,u ) u,u }. (u,u ) X X { x u,x 4
Remark 3 The funcion H A is minimal in he family of convex funcions f : X X ],+ ] wih he properies: f(x,x ) x,x for all (x,x ) X X and f(x,x ) = x,x for all (x,x ) A. Using he above ools, in he paper [17], Simons and Zălinescu give a nice proof of he famous Rockafellar s characerizaion of a maximal monoone operaor. Le H be a real separable Hilber space and A : H H be a maximal monoone operaor. Denoe for ε >, J ε,a ε : H H, he (1-, resp. 1/ε -) Lipschiz coninuous funcions J ε (x) = (I +εa) 1 (x) and Le A ε (x) = x J ε(x) ε A(J ε (x)). BV ([,T];H) = {k : [,T] H : k T <, k() = }, where k T := k BV ([,T];H). If we consider on C([,T];H) he usual norm y C([,T];H) = y T = sup{ y(s) : s T}, hen (C([,T];H)) = BV ([,T];H). We denoe he dualiy beween hese spaces by z,g := z(),dg(). Denoe by A he realizaion on C([,T];H) of he maximal monoone operaor A : H H, ha is he operaor A : C([,T];H) BV ([,T];H) defined as follows: (x,k) A if x C ( [,T];R d), k BV ([,T];H) and one of he following equivalen condiions are saisfied: (d 1 ) for all s T, s x(r) z,dk(r) z dr, (z,z ) A; (d 2 ) for all s T and for all u,u C([,T];H) such ha (u(r),u (r)) A, r [s,], s x(r) u(r),dk(r) u (r)dr ; (d 3 ) for all u,u C([,T];H) such ha (u(r),u (r)) A, r [,T], x(r) u(r),dk(r) u (r)dr. 5
A is a maximal monoone operaor since, seing ( ) x(r)+y(r) u(r) = J ε = x(r)+y(r) ( ) x(r)+y(r) εa ε 2 2 2 ( ) x(r)+y(r) ; u (r) = A ε 2 in (d 2 ) wrien for (x,k) A and respecively for (y,l) A and aking hen ε, we infer (since εa ε as ε ) ha (6) s x(r) y(r),dk(r) dl(r), s T. The maximaliy clearly follows from he definiion of A. For he realizaion of he operaor A on L r (,T;H), r 1, we use he same noaion A wihou risk of confusion since every ime we menion he space of realizaion. In his case, he operaor A : L r (,T;H) L q (,T;H), 1 r + 1 q = 1 is defined by (x,g) A if x(r) z,g(r) z dr, for all s T and for all (z,z ) A, s or (clearly), equivalenly g() A(x()), a.e. [,T]. Arguing similar o he previous siuaion, we obain ha A is a maximal monoone operaor. 1.2 Sochasic framework Le (Ω,F,P,{F } ) be a sochasic basis i.e. (Ω,F,P) is a complee probabiliy space and {F } is a filraion saisfying he usual assumpions of righ coninuiy and compleeness: N P F s F = ε> F +ε, for all s, where N P is he se of all P-null ses. Le (H, H ) be a real separable Hilber space; if F is a closed subse of H, denoe by B F he σ-algebra generaed by he closed subses of F. Denoe by S p H [,T], p, he space of progressively measurable coninuous sochasic processes X : Ω [,T] H (i.e. X (ω,) is coninuous a.s. ω Ω, and (ω,s) X(ω,s) : Ω [,T] H is ( ) F B [,],B H measurable for all [,T]), such ha X S p H [,T] = (E X p T )1 p 1 <, if p >, E[1 X T ], if p =, where X T := sup X. [,T] 6
The space (S p H [,T], S p [,T]), p 1, is a Banach space and Sp H [,T], p < 1, is a H complee meric space wih he meric ρ(z 1,Z 2 )= Z 1 Z 2 S p d [,T] (when p = he meric convergence coincides wih he probabiliy convergence). If H = R d we will denoe S p H [,T] by Sp d [,T]. Le ( ) H, H be a real separable Hilber space and B = {B (ϕ) : (,ϕ) [,T] H } L (Ω,F,P) a Gaussian family of real-valued random variables wih zero mean and covariance funcion E[B (ϕ)b s (ψ)] = ( s) ϕ,ψ H, ϕ,ψ H, s, [,T], where s = min{,s}. We call (B,{F }) a H -Wiener process if, for all [,T], we have (i) F B = σ{b s (ϕ); s [,],ϕ H } N P F and (ii) B +h (ϕ) B (ϕ) is independen of F, for all h >, ϕ H. Noe ha, given any orhonormal basis {e i ; i I N } of H, he sequence β i = {β i = B (e i ); [,T]}, i I, defines a family of independen real-valued sandard Wiener processes (Brownian moions). Moreover, if H is of finie dimension, we have B = i 1 β i e i, [,T]. In he general case his series does no converge in H, bu raher in a larger space H, H H which is such ha he injecion of H ino H is Hilber-Schmid. Moreover, B M 2 (,T; H ). By M p (,T;H), p 1, we denoe he space of H-valued coninuous, p-inegrable maringales M, ha is, he space of all coninuous sochasic processes M : Ω [,T] H saisfying, P-a.s, (m 1 ) M =, (m 2 ) E M p <, [,T], (m 3 ) E[M F s ] = M s, for all s T. M p (,T;H) is a Banach space wih respec o he norm X M p = (E X T p ) 1/p ; in he case p > 1, M p (,T;H) is a closed linear subspace of S p H [,T]. In order o define he sochasic inegral wih respec o he H -Wiener process B, we inroduceaclassofprocesses wihvalues inheseparablehilber spacel 2 (H ;H)ofHilber Schmid operaors from H ino H, i.e. he space of linear operaors F : H H saisfying F 2 HS = Fe i 2 H = TrF F = TrFF <. i=1 7
Denoe Λ p H H (,T), p [, [, he space of progressively measurable processes Z : Ω ],T[ L 2 (H ;H) such ha: [ ( ) p ]1 p 1 E Z s 2 2 HS ds, if p >, Z Λ p = [ ( ) ] 1 E 1 Z s 2 2 HS ds, if p =. The space (Λ p H H (,T), Λ p), p 1, is a Banach space and Λ p H H (,T), p < 1, is a complee meric space wih he meric ρ(z 1,Z 2 ) = Z 1 Z 2 Λ p. Consider {e i ; i I N } an orhonormal basis of H. Le Z Λ p H H (,T), wih p. The sochasic inegral I is defined by Z I I (Z), where I (Z) := Z s db s = i I Z s (e i )db s (e i ), [,T]. Noe ha i doesn depend on he choice of he orhonormal basis of H. The applicaion I : Λ p H H (,T) S p H [,T] is a linear coninuous operaor and i has he following properies: (a) EI (Z) =, if p 1, (b) E I T (Z) 2 = Z 2 Λ, if p 2, 2 (c) 1 Z p Λ E sup I c p (Z) p c p Z p Λ, if p >, p p [,T] (Burkholder-Davis-Gundy inequaliy) (d) I(Z) M p (Ω [,T];H), p 1. The definiion and he properies of he sochasic inegral can be found in Pardoux & Răşcanu [14] or Da Prao & Zabczyk [15]. If H = R k and H = R d hen {B, } is a k-dimensional Wiener process (Brownian moion); L 2 (H ;H) is he space of real marices F = (f ij ) d k and F 2 := F 2 HS = fi,j 2. i,j In his siuaion, he space Λ p H H (,T) will be denoed by Λ p d k (,T). 2 Fizparick funcion approach 2.1 A Generalized Skorohod problem Throughou his secion H is a real separable Hilber space wih he norm and he scalar produc,. 8
We sudy he mulivalued monoone differenial equaion { dx()+ax()(d) dm(), (7) x() = x,, (GSP) where we assume (H GSP ) : (i) A : H H is a maximal monoone operaor, (ii) x Dom(A), (iii) m : [, ) H is coninuous and m() =. Definiion 4 A coninuous funcion x : [,T] H is a soluion of Eq.(7) if x() Dom(A) for all T, (T arbirarily fixed) and here exiss k C([,T];H) BV ([,T];H) such ha x()+k() = x +m(), T and (8) s x(r) z,dk(r) z dr, (z,z ) A, s T. (Wihou confusion, he uniqueness of k will permi us o call he pair (x,k) soluion of he generalized Skorohod problem (GSP) and we wrie (x,k) = GSP (A;x,m).) In virue of his definiion, he (classical) Skorohod problem (for more deails, one can consul Cépa [8] or [9]) is obained for A = I E : R d R d, where E is a closed convex subse of R d, {, if x E, I E (x) = +, if x R d \E and {, if x in(e), I E (x) = ν R d : ν,y x, for all y E }, if x Bd(E),, if x / E. The definiion of he soluion can be given in a equivalen form as follows. Definiion 5 A coninuous funcion x : [,T] R d R d is a soluion of Skorohod problem in E if x() E for all T and here exiss k C ( [,T];R d) ( BV ) [,T];R d such ha (a) k = 1 x(s) Bd(E) d k s, and (b) k() = n x(s) d k s, where n x(s) N E (x(s)) and nx(s) = 1, d k s a.e. x()+k() = x +m(), [,T]. (N E (x) denoes he ouward normal cone o E a x E.) 9
Le A : C([,T];H) BV ([,T];H) be he realizaion of he maximal monoone operaor A : H H and X = {µ C([,T];H) : µ() = } he linear closed subspace of C([,T];H). For each R >, we define Y R = {k C([,T];H) : k() =, k T R}; Y R is a closed subse of C([,T];H) and, consequenly, i is a meric space wih respec o he meric fromc([,t];h). Remark ha, by Helly-Foiaşheorem (see Barbu&Precupanu [3], Theorem 3.5 & Remark 3.2), i is also a bounded w -closed subse of BV ([,T];H). Le α : R + R + a coninuous funcion such ha α() =. Denoe C α = {x X : m x (ε) α(ε) for all ε }. Here he funcion m x : R + R + represens he modulus of coninuiy of he coninuous funcion x : [,T] H and i is defined by m x (δ) = m x,t (δ) = sup{ x() x(s) : s δ,,s [,T]}. Clearly, C α is a bounded closed convex subse of X. Consider, for each (u,u ) A and ν X, he funcion J (u,u,ν) : H X Y R X R given by J (u,u,ν)(a,x,k,µ) = a x 2 + [ u(),dk() + x(),du () u(),du () ] x(),dk() +2R µ m T + and Ĵ : H X Y R X ],+ ], defined by (9) Ĵ (a,x,k,µ) = sup J (u,u,ν)(a,x,k,µ) (u,u ) A, ν C α µ() ν(),dk() R ν m T = a x 2 +H(x,k) x,k +2R µ m T + sup ν C α { µ ν,k R ν m T }, where H : C([,T];H) BV ([,T];H) ],+ ] is he Fizparick funcion associaed o he maximal monoone operaor A. Remark 6 Ĵ : H X Y R X ],+ ] is a lower semiconinuous funcion as he supremum of he coninuous funcions J (u,u,ν). Remark also ha, for µ C α, 2R µ m T + sup ν C α { µ ν,k R ν m T } R µ m T. 1
Proposiion 7 Le R > and α : R + R + a coninuous funcion such ha α() =. The funcion Ĵ has he following properies (a) Ĵ (a,x,k,µ), for all (a,x,k,µ) H X Y R C α. (b) Le (â,ˆx,ˆk,ˆµ) H X Y R C α. Then Ĵ(â,ˆx,ˆk,ˆµ) = iff â = x, ˆµ = m and ˆk A(ˆx). (c) The resricion of Ĵ o he closed convex se K = {(a,x,k,µ) H X Y R C α : x+k = a+µ} is a convex lower semiconinuous funcion; for (â,ˆx,ˆk, ˆµ) K, we have Ĵ(â,ˆx,ˆk,ˆµ) = iff â = x, ˆµ = m and (ˆx,ˆk) = GSP (A;x,m). Proof. The poins (a) and (b) clearly are consequences of he properies of he Fizparick funcion H. Le us prove (c). We have (a,x,k,µ) K and Ĵ (a,x,k,µ) = a x 2 +H(x,k) x,k +2R µ m T + sup ν C α { µ ν,k R ν m T } and he convexiy of Ĵ follows. = a x 2 +H(x,k)+ 1 2 x(t) µ(t) 2 1 2 a 2 +2R µ m T + sup ν C α { µ ν,k R ν m T } = x 2 2 a,x + 1 2 a 2 +H(x,k)+ 1 2 x(t) µ(t) 2 +2R µ m T + sup ν C α { ν,k R ν m T } µ(s),dk(s) In he sequel we prove he exisence and uniqueness of he soluion of he mulivalued monoone differenial equaion (7). Our proof is srongly conneced wih he one from Răşcanu [16]. Firs highligh some properies of a soluion (x,k) = GSP (A;x,m). Consider M a bounded and equiconinuous subse of C([,T];H) and we denoe M T = sup{ y T : y M} and m M,T (δ) = sup{m y,t (δ) : y M} Proposiion 8 Fix T >. Le he assumpion (H GSP ) be saisfied and in(dom(a)). Then, here exiss a posiive consan C M such ha (a) If m M and (x,k) = GSP (A;x,m) hen (1) x 2 T + k T C M(1+ x 2 ). 11
(b) If m, ˆm M, (x,k) = GSP (A;x,m) and (ˆx,ˆk) = GSP(A;ˆx, ˆm) hen (11) x ˆx T C M (1+ x + ˆx )( x ˆx + m ˆm 1/2 T ). In paricular, he uniqueness follows, ha is, if x = ˆx and m = ˆm hen (x,k) = (ˆx,ˆk). Proof. (a) In he sequel we fix arbirary u H and < r 1 such ha and B(u,r ) Dom(A) A # u,r := sup{ û : û A(u +r v), v 1} <. If in (8) we consider z = u +r v, v 1 and z A(z), hen z A # u,r and we infer (12) r d k x() u,dk() +A # u,r [r + x() u ]d. Le δ = δ,m > be defined by By Energy Equaliy δ +m M,T (δ ) = r 4. x() m() u 2 +2 and, using (12), we obain x() m() u 2 +2r k x u 2 +2 x(r) u,dk(r) = x u 2 +2 m(r),dk(r) m(r),dk(r) +2A # u,r [r + x(r) u ]dr. T Le n = δ and consider he pariion = < 1 <... < n =, i+1 i = n δ, i =,n 1 ( a is he smalles ineger greaer or equal o a R). Then Hence m(r),dk(r) = n 1i+1 i= i n 1 n 1 m(r) m( i ),dk(r) + m( i ),k( i+1 ) k( i ) m M,T (δ ) k + m( i ),m( i+1 ) x( i+1 )+u m( i )+x( i ) u i= r 4 k +2(n +1) m x u m. x() m() u 2 + 3r 2 k x u 2 + [ 4(n +1) m +2A # u,r ] x u m +2(+ m )A # u,r, i= 12
which implies (1), where C M = C(T,u,r,A # u,r,δ, M T ). (b) By ordinary differenial calculus and (1) we infer x() m() ˆx()+ ˆm() 2 +2 = x ˆx 2 +2 x(r) ˆx(r),dk(r) dˆk(r) m(r) ˆm(r),dk(r) dˆk(r) x ˆx 2 +2 m ˆm T [ k T + ˆk T ] x ˆx 2 +4C M m ˆm T (1+ x 2 + ˆx 2 ). On he oher hand, x() m() ˆx()+ ˆm() 2 1 2 x() ˆx() 2 m ˆm 2 T Combining hese las wo inequaliies wih (6), we deduce 1 2 x() ˆx() 2 2 M T m ˆm T x() ˆx() 2 2 x ˆx 2 +4 M T m ˆm T +8C M m ˆm T (1+ x 2 + ˆx 2 ) and (11) easily follows, wih a consan ĈM; he wo relaions (1) and (11) can be wrien wih a common consan C M := max{c M,ĈM}. Theorem 9 Under he assumpions (H GSP ), if we have also in(dom(a)), hen he generalized convex Skorohod problem (7) has a unique soluion (x, k) and esimaes (1) and (11) hold. Proof. The uniqueness and esimaes (1) and (11) have been obained in he above resul. I suffices o prove he exisence on an arbirary fixed inerval [,T]. Le x,n Dom(A) and m n C ([,T];H) be such ha x,n x in H and m n m in C([,T];H). Noice ha M = {m,m 1,m 2,...} is a bounded equiconinuous subse of C([,T];H). We se α(ε) = m M,T (ε) and le Ĵ (resp. Ĵ n ): H X Y R X ],+ ] be he funcions defined by (9) associaed o (x,m,a) (and resp. (x,n,m n,a)). Then Ĵ (a,x,k,µ) = Ĵn(a,x,k,µ) a x,n 2 2R µ m n T + a x 2 + sup ν C α { µ ν,k R ν m T } sup ν C α { µ ν,k R ν m n T } Ĵn(a,x,k,µ) a x,n 2 2R µ m n T + a x 2 +R sup ν C α { ν m n T ν m T } Ĵn(a,x,k,µ) a x,n 2 2R µ m n T + a x 2 +R m m n T. 13
In paricular, (13) Ĵ (x,n,x,k,m n ) Ĵn(x,n,x,k,m n )+ x,n x 2 +R m m n T. By a classical resul (see Barbu [2], Theorem 2.2) here exis x n C([,T];H) and h n L 1 (,T;H), h n () Ax n (), a.e. [,T], such ha (14) x n ()+ If we denoe k n () = h n (s)ds = x,n +m n (). h n (s)ds, hen (x n,k n ) A andherefore, by Fizparick s Theorem, H(x n,k n ) = x n,k n. Then, using Proposiion 8, here exiss a posiive consan C, no depending on n, such ha, for all n,j N, x n 2 T + k n T C and x n x j T C( x,n x,j + m n m j 1/2 T ). Hence, here exiss x C([,T];H) such ha, as n, Le We deduce ha and clearly follows x n x in C([,T];Dom(A)). k() = x +m() x(). k n = x,n +m n x n k in C([,T];H) k BV ([,T];H), k T C. Seing R = C, he quaniies Ĵ (x,n,x n,k n,m n ) and Ĵn(x,n,x n,k n,m n ) are well defined. Moreover, by Proposiion 7, Ĵ n (x,n,x n,k n,m n ) =. Passing o liminf in (13), he lowersemiconinuiy of Ĵ n + implies Ĵ (x,x,k,m) liminfĵ (x,n,x n,k n,m n ) =, n + ha is, here exiss a minimum poin for which Ĵ is zero. By Proposiion 7 (-(c)) we infer ha he generalized convex Skorohod problem (7) has a soluion. Remark 1 We highligh ha he exisence problem is reduced o he minimizaion of a specific l.s.c. convex funcion on a bounded closed convex subse of H X BV ([,T];H) X. Indeed, via Proposiion 7 (-(c)), he minimizaion of Ĵ is on he se H ρ X R Y R C α, where H ρ = {h H : h ρ := sup{ x, x,n : n N }}, X R = {x X : x T R} and R = C. Classical resuls (see Zeidler [18], Theorem 38.A) esablish sufficien condiions for a funcional defined on a subse of a reflexive Banach space o aain is minimum. 14
We noe ha, in he framework of Hilber spaces, he assumpion in(dom(a)) from he above resuls is fairly resricive. One can renounce a his condiion, bu we have o consider a sronger assumpion on m and, moreover, o weaken he noion of soluion for he generalized Skorohod problem (7). Therefore, along H, we consider (V, V ) a real separable Banach space wih separable dual (V, V ) such ha V H = H V, whereheembeddingsareconinuous, wihdenserange(hedualiyparing(v,v)isdenoed also by,, and, for k : [, ) V, k() =, we use he adequae noaion k T = k BV ([,T];V ) ). Reconsider he mulivalued monoone differenial equaion (7) under he assumpions H GSP : { HGSP : (i) and (ii), (iii ) m : [, ) V is coninuous and m() =. Definiion 11 A coninuous funcion x : [, ) H is a soluion of Eq.(7) if (i) here exis he sequences {x,n } Dom(A) and m n : [, ) V, m n () = of C 1 coninuous funcions saisfying, for all T >, x,n x + m n m C([,T];V), as n, (ii) here exis x n C([, );Dom(A)), k n C([, );H) BV,loc (R + ;V ), k n () =, and a funcion k such ha and, for all T >, x n ()+k n () = x,n +m n (), (a) x n x T + k n k T, as n, (b) (c) sup k n T <, n N s x n (r) z,dk n (r) z dr, (z,z ) A, s T. (Wihou confusion, he uniqueness of k will permi us o call he pair (x,k) soluion of he generalized Skorohod problem (7) and we wrie (x,k) = GSP (A;x,m).) Remark 12 If (x,k) = GSP (A;x,m) hen we clearly have (iii) x() Dom(A), for all, (iv) k C([, );H) BV,loc (R + ;V ), k() = and 15
(v) x()+k() = x +m(),. Replacing now he condiion in(dom(a)) we obain (see, for example, Răşcanu [16], Theorem 2.3) he following resul of exisence and uniqueness of a soluion for he generalized Skorohod problem (7). Theorem 13 Under he hypohesis ( HGSP ), if here exis h H and r,a 1,a 2 > such ha (15) r z V z,z h +a 1 z 2 +a 2, (z,z ) A hen he differenial equaion (7) has a unique soluion (x,k) in he sense of Definiion 11. Moreover, for all T >, (a) if (x,k) = GSP (A;x,m) and (ˆx,ˆk) = GSP (A;ˆx, ˆm), hen here exiss a posiive consan C such ha [ ] x ˆx 2 T C x ˆx 2 + m ˆm 2 T + m ˆm C([,T];V) k ˆk T and (b) for every equiuniform coninuous subse M C([, T]; V), m M, here exiss C = C (r,h,a 1,a 2,T,N M ) > for which x 2 T + k T C [ 1+ x 2 + m 2 T]. (Here N M is he consan of equiuniform coninuiy given by sup{ f () f (s) V : s T/N M } r /4, f M.) From Răşcanu [16] we menion hree siuaions when he relaion (15) is saisfied: (a) A = A + ϕ, where A : H H is a coninuous monoone operaor on H and ϕ : H ],+ ] is a proper convex l.s.c. funcion for which here exis h H, R >, a > such ha ϕ(h +x) a, x V, x V R. (b) There exissa separable Banach space U such ha U H U densely and coninuously and U V is dense in V, A : H H is a maximal monoone operaor wih Dom(A) U, a,λ R, a >, such ha for all (x 1,y 1 ), (x 2,y 2 ) A (y 1 y 2,x 1 x 2 )+λ x 1 x 2 2 a x 1 x 2 2 V, h U, r,a > such ha h +r e Dom(A) and for all e U V, e V = 1, where A x := Pr Ax. A (h +r e) U r, (c) A is a maximal monoone wih in(dom(a)) and V = H. 16
2.2 Maximal monoone SDE wih addiive noise Consider now he following sochasic differenial equaion (for shor SDE), where by B we denoe he H -Wiener process defined in Secion 1.2, (16) { dx +AX (d) G db, X = ξ, [,T], where (H MSDE ) : (i) A : H H is a maximal monoone operaor, (ii) ξ L (Ω,F,P;Dom(A)), (iii) G Λ 2 H H. Seing X = L 2 (Ω;C([,T];H)), he space L 2 (Ω;BV ([,T];H)) is a linear subspace of he dual of X and, he naural dualiy (X,K) E X,dK beween hese wo suggess o use he noaion X for L 2 (Ω;BV ([,T];H)), even i is no heenire dual space. OnXwe shall consider he srong opologyand onx he w -opology. Le A he realizaion of A on X X. Definiion 14 By a soluion of Eq.(16) we undersand a pair of sochasic processes (X,K) L (Ω;C([,T];H)) [ L (Ω;C([,T];H)) L (Ω;BV ([,T];H)) ], saisfying, P-a.s. ω Ω, for all s T, Clearly, where M = (c 1 ) X Dom(A), (c 2 ) X +K = ξ + (c 3 ) s G s db s and X r u,dk r vdr, (u,v) A. (X(ω, ),K(ω, ))= GSP (A;ξ(ω),M(ω, )), P-a.s. ω Ω, G s db s M 2 (,T;H). Consequenly, under he hypohesis (H MSDE ), if in(dom(a)) hen by Theorem 9 here exiss a unique soluion (X,K) (in he sense of Definiion 14) for Eq.(16). Moreover, if ( ) 2 E ξ 4 +E G 2 HS d < + 17
hen X L 4 (Ω;C([,T];H)) X and K X X (see for example Pardoux & Răşcanu [14], Proposiion 4.22). In he sequel we define a convex funcional whose minimum poin coincide wih he soluion of Eq.(16). Le S = L 2 (Ω,F,P;H) X X Λ 2 H H. by Define, for each (U,U ) A, J (U,U ) : S R J (U,U )(η,x,k,g) = 1 2 E η ξ 2 + 1 2 E g G 2 HS d +E [ U,dK + X,dU U,dU X,dK ] and Ĵ : S ],+ ] Ĵ (η,x,k,g) = sup J (U,U )(η,x,k,g) (U,U ) A = 1 2 E η ξ 2 +H(X,K) X,K + 1 2 E g G 2 HS d, where H : X X ],+ ] is he Fizparick funcion associaed o he maximal monoone operaor A. I is clear ha Remark 15 Ĵ : S ],+ ] is a lower semiconinuous funcion as supremum of coninuous funcions. Since H(X,K) X,K, hen we easily deduce Proposiion 16 Ĵ has he following properies: (a) Ĵ (η,x,k,g), for all (η,x,k,g) S. (b) Ĵ (η,x,k,g) = iff η = ξ, g = G and K A(X). (c) Le R >. The resricion of Ĵ o he bounded closed convex se { L = (η,x,k,g) S : X +K = η + g s db s, [,T], E η 2 +E X 2 X +E K X +E g s 2 HS ds R } is a convex l.s.c. funcion and soluion of he SDE (16). Ĵ (η,x,k,g) = iff η = ξ, g = G and (X,K) is he 18
Proof. The poins (a) and (b) clearly are consequences of he properies of he Fizparick funcion H. Le us prove (c). Since, by Energy Equaliy hen 1 2 E X T 2 +E X,dK = 1 2 E η 2 + 1 2 E g 2 HS d Ĵ (η,x,k,g) = 1 2 E η ξ 2 +H(X,K) X,K + 1 2 E g G 2 HS d = 1 2 E ξ 2 E η,ξ +H(X,K)+ 1 2 E X T 2 E and he convexiy of Ĵ on he se L follows. g,g d+ 1 2 E G 2 HS d To complee his secion, we will siuae in he exended framework inroduced in he final par of Subsecion 2.1. We will consider once again he spaces H and V and we assume ha V H = H V, where he embeddings are coninuous wih dense range. Concerning he SDE (16), he hypohesis (H MSDE ) will be replaced by A : H H is a maximal monoone operaor and ( ) (i) here exis h H and r,a 1,a 2 > such ha HMSDE : r z V z,z h +a 1 z 2 +a 2, (z,z ) A (ii) ξ L 2 (Ω,F,P;Dom(A)), (iii) G Λ 2 H H (,T;L 2 (H,H)). Definiion 17 Le M := G sdb s. A sochasic process X L ad (Ω;C([,T];H)) ha saisfies, P-a.s., X = ξ and X Dom(A), [,T] is a (generalized) soluion of mulivalued SDE (16) if here exis K L ad(ω;c([,t];h)) L (Ω;BV (,T;V )),K = P-a.s and a sequence of sochasic processes {M n } n N saisfying (17) { M n L 2 ad (Ω;C([,T];V)) M2 (,T;H), M n M in M 2 (,T;H) such ha, denoing for a.s. ω Ω, (X n (ω, ),K n (ω, )) = GSP (A;ξ(ω),M n (ω, )), we have X n X, K n K in L ad (Ω,C([,T];H)) as n and supe K n T < +. n (Wihou confusion, he uniqueness of K permis us o call he pair (X, K) a generalized soluion of he mulivalued SDE (16).) 19
Recall, from Răşcanu [16], he following exisence resul which is a consequence of he corresponding deerminisic case here above. Theorem 18 Under he assumpion ( H MSDE ) he problem (16) has a unique generalized soluion (X, K). Moreover he soluion saisfies ] (18) E sup X 2 +E sup K 2 +E K T C [1+E ξ 2 +E G 2 HS d, [,T] [,T] where C = C (T,r,h,a 1,a 2 ) >. If (X,K) and ( X, K) are wo soluions of (16) corresponding o (ξ,g) and, respecively, ( ξ, G) hen [ (19) E sup X X 2 C(T) E ξ ξ 2 +E [,T] G G 2 HS d ]. Proof. Since he process M does no have V-valued coninuous rajecories, we use he deerminisic resul approximaing he sochasic inegral by he sequence M n := n M,e i e i, i=1 where {e i ; i N } V is an orhonormal basis in H. By Theorem 13, here exiss (X n (ω),k n (ω)) = GSP(A;ξ(ω),M n (ω)), P-a.s. ω Ω. I is no difficul o prove ha he following inequaliies hold [ E sup X n 2 +E sup K n 2 +E K n T C 1+E ξ 2 +E MT n 2] [,T] [,T] and, if ( X n (ω), K n (ω)) = GSP(A; ξ(ω), M n (ω)), hen E sup X n X n 2 +E sup K n K n 2 C(T) [E ξ ξ 2 +E M nt M nt ] 2. [,T] [,T] So (replacing M n by M n ), here exis X,K L 2 ad (Ω;C([,T];H)) such ha Xn X and K n K in L 2 ad (Ω;C([,T];H)) as n. The inequaliies (18) and (19) are immediae consequences and, as a by-produc, (X, K) is a soluion of Eq.(16). For more deails, we invie he ineresed reader o consul Răşcanu [16]. 2.3 Backward sochasic A represenaion Le (Ω,F,P,{F } ) be a sochasic basis, where {F } is he sandard filraion associaed o a H -Wiener process {B }. 2
By he represenaion heorem, for ξ L 2 (Ω,F T,P;H) here exiss a unique Z Λ 2 H H (,T) such ha ξ = Eξ + Z s db s and, for each (ξ,h) L 2 (Ω,F T,P;H) Λ 2 H (,T), here exiss a unique pair such ha (Y,Z) S 2 H[,T] Λ 2 H H (,T) Y + H s ds = ξ Z s db s and he mapping (ξ,h) (Y,Z) : L 2 (Ω,F T,P;H) Λ 2 H (,T) S2 H [,T] Λ2 H H (,T) is linear and coninuous. (Y,Z) is defined as ( ) ( ) Y = E ξ H s ds F and ξ H s ds = E ξ H s ds + Z s db s. Denoe Y = C (ξ,h) and Z = D (ξ,h). Remark ha, by he Energy Equaliy, we have (2) E Y 2 +E Z s 2 HS ds = E ξ 2 +2E Y s,f s ds. If A : H H is a maximal monoone operaor hen he realizaion of A on Λ 2 H (,T) is he maximal monoone operaor A : Λ 2 H (,T) Λ2 H (,T) defined by H A(Y) iff H (ω) A(Y (ω)), dp d-a.e. (ω,) Ω ],T[. The inner produc in Λ 2 H (,T) is given by U,V = E U,V d. Consider he backward sochasic differenial equaion (21) where { dy +A(Y )d Z db, [,T], Y T = ξ, { (i) A : H H is a maximal monoone operaor and (ii) ξ L 2 (Ω,F T,P;Dom(A)). Definiion 19 Y SH 2 [,T] is a soluion of Eq.(21) if here exis H Λ2 H (,T) and Z Λ 2 H H (,T) such ha Y + H s ds = ξ Z s db s and H A(Y) (ha is, H (ω) A(Y (ω)), dp d-a.e. (ω,) Ω ],T[). 21
Le R > and he ball F R = { η L 2 (Ω,F T,P;H) : E η 2 R }. For (U,U ) A and ζ F R define by J (ζ,u,u ) : L 2 (Ω,F T,P;H) Λ 2 H(,T) Λ 2 H(,T) R J (ζ,u,u )(η,y,h) = 1 2 E η ξ 2 +E [ U,H + Y,U U,U Y,H ]d + 1 [ E ζ η 2 E ζ ξ 2] 2 and Ĵ : L2 (Ω,F T,P;H) Λ 2 H (,T) Λ2 H (,T) ],+ ], Ĵ (η,y,h) = sup { } (22) J (ζ,u,u )(η,y,h) : (U,U ) A, ζ F R = 1 2 E η ξ 2 +H(Y,H) Y,H + 1 2 sup ζ F R [ E ζ η 2 E ζ ξ 2], where H : Λ 2 H (,T) Λ2 H (,T) ],+ ] is he Fizparick funcion associaed o he maximal monoone operaor A. Remark 2 Ĵ : L2 (Ω,F T,P;H) Λ 2 H (,T) Λ2 H (,T) ],+ ] is a l.s.c. funcion as he supremum of he coninuous funcions J (ζ,u,u )(η,y,h). If ξ F R hen and clearly follows 2R 2 +2E η 2 sup ζ F R ( E ζ η 2 E ζ ξ 2) E η ξ 2 Proposiion 21 Le R > and ξ F R. Ĵ has he following properies: (a) Ĵ (η,y,h) H(Y,H) Y,H, for all (η,y,h) L 2 (Ω,F T,P;H) Λ 2 H (,T) Λ 2 H (,T). (b) Le (ˆη,Ŷ,Ĥ) F R Λ 2 H (,T) Λ2 H (,T). Then Ĵ(ˆη,Ŷ,Ĥ) = iff ˆη = ξ, Ĥ A(Ŷ). (c) The resricion of Ĵ o he closed convex se K = { (η,y,h) F R Λ 2 H (,T) Λ2 H (,T) : Y = C (η,h), [,T] } (ˆη,Ŷ,Ĥ) K he following asser- is a convex lower semiconinuous funcion and for ions are equivalen: (c 1 ) inf Ĵ (η,y,h) = Ĵ(ˆη,Ŷ,Ĥ) =. (η,y,h) F R Λ 2 H (,T) Λ2 H (,T) 22
(c 2 ) ˆη = ξ and (Ŷ,Ĥ,Ẑ), wih Ẑs = D s (ξ,ĥ), is he soluion of he BSDE (21). Proof. (Skech) Since he poins (a) and (b) are obvious, we focus on (c). The convexiy of Ĵ on K is obained as follows. By Energy Equaliy we have 1 T 2 C (η,h) C (ζ,) 2 +E Y s C s (ζ,),h s ds+ 1 2 E D s (η,h) D s (ζ,) 2 ds Then J (ζ,u, Ũ) (η,y,h) = 1 2 E η ξ 2 +E [ U,H + Y,U U,U Y,H ]d+ 1 2 = 1 2 E η ζ 2. = 1 2 E η ξ 2 +[ U,H + Y,U U,U ]+ 1 2 C (η,h) C (ζ,) 2 + C(ζ,),H + 1 2 D(η,H) D(ζ,) 2 E ζ ξ 2 Hence (η,y,h) Ĵ (η,y,h) = 1 2 E η ξ 2 +H(Y,H)+sup ζ [ E ζ η 2 E ζ ξ 2] { 1 2 C (η,h) C (ζ,) 2 + C(ζ,),H + 1 2 D(η,H) D(ζ,) 2 E ζ ξ 2 } is, clearly, a convex lower semiconinuous funcion. Then, he equivalence beween (c 1 ) and (c 2 ) easily follows. Proving he exisence of a soluion for he backward sochasic differenial equaion (21) is herefore equivalen o solving a problem on convex analysis. More precisely, i is sufficien o show ha he funcional defined by he formula (22) aains a minimum and is value in ha poin is zero. Unforunaely, his is sill an open problem, bu we esimae ha he perspecive and he ools inroduced along his paper will lead us o he desired resul. 3 Fizparick ype mehod for SVI and BSVI In he following secions we will consider he finie dimensional case H = R d and H = R k. Le {B, } be a k-dimensional Brownian moion wih respec o a given complee sochasic basis (Ω,F,P,{F } ). 23
3.1 Sochasic variaional inequaliy 3.1.1 Known resuls Le (23) F : Ω [,+ [ R d R d, G : Ω [,+ [ R d R d k. Consider he sochasic variaional inequaliy (for shor SVI) { dx + ϕ(x )(d) F(,X )d+g(,x )db,, where will assume X = ξ, (24) (H ) : ξ L (Ω,F,P;Dom(ϕ)) and (25) (H ϕ ) : { (i) ϕ : R d ],+ ] is a convex l.s.c. funcion, (ii) in(dom(ϕ)). Definiion 22 A pair (X,K) Sd S d, K =, is a soluion of he sochasic variaional inequaliy (23) if he following condiions are saisfied, P-a.s.: (d 1 ) X Dom(ϕ), a.e. > and ϕ(x) L 1 loc (, ), (26) (d 2 ) K T <, T >, (d 3 ) X +K = ξ + (d 4 ) s y(r) X r,dk r + F(s,X s )ds+ s ϕ(x r )dr G(s,X s )db s,, s ϕ(y(r))dr, y : R + R d coninuous funcion and s. Noaion 23 The noaion dk ϕ(x )(d) will be used o say ha (X,K) saisfy (d 1 ),(d 2 ) and (d 4 ). The SDE (23) will be wrien, also, in he form X +K = ξ + F(s,X s )ds+ G(s,X s )db s,, dk ϕ(x )(d). Remark (see Asiminoaei & Răşcanu [1]) ha he condiion (d 4 ) from Definiion 22 is equivalen o each of he following condiions, for any fixed T >, (a 1 ) (a 2 ) (a 3 ) s s z X r,dk r + s ϕ(x r )dr ( s)ϕ(z), z R d, s T, X r z,dk r z dr, (z,z ) ϕ, s T, y(r) X r,dk r + ϕ(x r )dr 24 ϕ(y(r))dr, y C([,T],R d ).
Hence, he condiion (d 4 ) means ha (X (ω),k (ω)) ϕ, P-a.s., where ϕ is he realizaion of ϕ on C ( [,T];R d), ha is ϕ : C([,T];R d ) ],+ ], ϕ(x())d, if ϕ(x) L 1 (,T), (27) ϕ(x) = +, oherwise. Noaion 24 We inroduce he noaion: F # R () := esssup{ F(,x) : x R}. We recall he basic assumpions on F and G under which we will sudy he mulivalued sochasic equaion (23): he funcions F (,,x) : Ω [,+ [ R d and G(,,x) : Ω [,+ [ R d k are progressively measurable sochasic processes for every x R d, here exis µ L 1 loc (, ) and l L2 loc (, ;R +), such ha dp d-a.e.: Coninuiy: (C F ) : x F (,x) : R d R d is coninuous, Monooniciy condiion: (28) (H F ) : (M F ) : x y,f(,x) F(,y) µ() x y 2, x,y R d, and (B F ) : Boundedness condiion: F # R (s)ds <, for all R,T. Lipschiz condiion: (L G ) : G(,x) G(,y) l() x y, x,y R d, (29) (H G ) : Boundedness condiion: (B g ) : G(,) 2 d <. Clearly (H F ) and (H G ) yield F(,,X ) L 1 loc( R+ ;R d) and G(,,X ) Λ d k for all X S d. Theorem 25 If he assumpions (24), (25), (28) and (29) are saisfied, hen he SDE (23) has a unique soluion (X,K) Sd S d (in he sense of Definiion 22). Moreover, if here exis p 2 and u in(dom(ϕ)) such ha, for all T, ( p ( p/2 (3) E ξ p +E F (,u ) d) +E G(,u ) d) 2 < +, hen ( p/2 E( X p T + K p/2 T + K p/2 T )+E ϕ(x r ) dr) <. 25
(For he proof see Pardoux & Răşcanu [14], Theorem 4.14.) 3.1.2 Fizparick approach In his subsecion, assumpions (H F ) and (H G ) are replaced by (i) he funcions F (,,x) : Ω [,+ [ R d and G(,,x) : Ω [,+ [ R d k are progressively measurable sochasic processes for every x R d and, dp d-a.e., (ii) x F (,x) : R d R d and x G(,x) : R d R d k are coninuous, (iii) for all x,y R d (31) 2 x y,f(,x) F(,y) + G(,x) G(,y) 2 and (iv) here exiss b > such ha, for all x R d, (32) F(,x) + G(,x) b(1+ x ). Remark 26 If µ() + 1 2 l2 (), for every, hen he assumpions (28-M F ) and (29-L G ) implies ha (31) holds. Denoe S BV [,T] = { K S d [,T] : K =, E K 2 T < }, wih he w -opology, ha means K n K if lim E X n,dk n = E X,dK, for all X L 2 (Ω;C([,T];R d )). Le Φ : S 2 d (33) Φ(X) = [,T] ],+ ] defined by E ϕ(x )d, if ϕ(x) L 1 (Ω ],T[), +, oherwise. Since ϕ : R d ],+ ] is a proper convex l.s.c. funcion hen Φ is also a proper convex l.s.c. funcion. Le S := L 2 (Ω,F,P;Dom(ϕ)) S 2 d [,T] S BV [,T] Λ 2 d k(,t) and, for each U Dom(Φ) = {X Sd 2 [,T] : Φ(X) < }, we consider he mapping J U : S ],+ ], defined by J U (η,x,l,g) = 1 2 E η ξ 2 +E [ U s X s,f (s,u s ) + 12 ] g s G(s,U s ) 2 ds (34) +E U s X s,dl s +Φ(X) Φ(U) and Ĵ : S ],+ ] Ĵ (η,x,l,g) := sup J U (η,x,l,g). U Dom(Φ) 26
Remark 27 Ĵ : S ],+ ] is a lower semiconinuous funcion as supremum of lower semiconinuous funcions. We now have Proposiion 28 Ĵ has he following properies: (a) Ĵ (η,x,l,g), for all (η,x,l,g) S and Ĵ is no idenically +. (b) Le (ˆη, ˆX, ˆL,ĝ) S. Then Ĵ(ˆη, ˆX, ˆL,ĝ) = iff ˆη = ξ, ĝ = G(, ˆX ), ˆL+ F(s, ˆX s )ds Φ( ˆX). (c) The resricion of Ĵ o he closed convex se { L = (η,x,l,g) S : X +L = η + } g s db s, [,T] is a convex l.s.c. funcion. If (ˆη, ˆX, ˆL,ĝ) L, hen Ĵ(ˆη, ˆX, ˆL,ĝ) = iff ˆη = ξ, ĝ = G(, ˆX ) and ( ˆX, ˆL+ F(s, ˆX s )ds) is a soluion of he SVI (23). Proof. (a) If X / Dom(Φ) hen Ĵ (η,x,l,g) = +. If X Dom(Φ) hen Ĵ (η,x,l,g) = sup J U (η,x,l,g) U Dom(Φ) J X (η,x,l,g) = 1 2 E η ξ 2 + 1 2 E g s G(s,X s ) 2 ds. Ĵ is a proper funcion since, for v ϕ(u ) and η = ξ, X = u, L = v F (s,u )ds, g s = G(s,u ), we have (using he assumpion (31)) ha J U ( η,x,l,g ), for all U Dom(Φ). (b) If Ĵ(ˆη, ˆX, ˆL,ĝ) =, hen ˆX Dom(Φ) and by he calculus from he proof of (a) we infer ˆη = ξ, ĝ = G(, ˆX ) and J U (ˆη, ˆX, ˆL,ĝ), for all U Dom(Φ). Hence E U s ˆX s,f (s,u s )ds+dˆl s +Φ( ˆX) Φ(U), for all U Dom(Φ). 27
Le V Dom(Φ) and λ ],1[ be arbirary. Since Dom(Φ) is a convex se, we can replace U by (1 λ) ˆX +λv. I follows λe V s ˆX s,f(s, ˆX s +λ(v s ˆX s ))ds+dˆl s +Φ( ˆX) Φ((1 λ) ˆX +λv) (1 λ)φ( ˆX)+λΦ(V), which is equivalen o E V s ˆX s,f(s, ˆX s +λ(v s ˆX s ))ds+dˆl s +Φ( ˆX) Φ(V), for all V Dom(Φ). By he coninuiy of x F (,x) and assumpion (32) we can pass o limi under he las inegral, and i follows ha ˆL+ F(s, ˆX s )ds Φ( ˆX). Conversely, using (31), we have J U (ξ, ˆX, ˆL,G(, ˆX. )) = 1 2 E G(s, ˆX s ) G(s,U s ) 2 ds+e +E U s ˆX s,f(s, ˆX s )ds+dˆl s +Φ( ˆX) Φ(U) U s ˆX s,f (s,u s ) F(s, ˆX s )ds and, consequenly, Ĵ(ξ, ˆX, ˆL,G(, ˆX. )) =. (c) The second par of his poin is easy o observe, and, herefore, ( ˆX, ˆL+ F(s, ˆX s )ds) is a soluion of he SVI (23). I remains o prove he convexiy of Ĵ on L. By he Energy Equaliy we have 1 2 E X T 2 +E X s,dl s = 1 2 E η 2 + 1 2 E g s 2 ds and, using i in he formula (34), he funcional J U (η,x,l,g) becomes J U (η,x,l,g) = 1 2 E η ξ 2 +E U s X s,f (s,u s )ds + 1 2 E g s G(s,U s ) 2 ds [ + E U s,dl s 1 2 E η 2 1 2 E g s 2 ds+ 1 2 E X T ]+Φ(X) Φ(U) 2 = E η,ξ + 1 2 E ξ 2 +E + 1 2 E X T 2 + 1 2 E G(s,U s ) 2 ds E U s X s,f (s,u s )ds +E U s,dl s g s,g(s,u s ) ds+φ(x) Φ(U). I clearly follows ha J U is convex and lower semiconinuous for U Dom(Φ). Consequenly, he mapping (η,x,l,g) Ĵ (η,x,l,g) = sup U Dom(Φ)J U (η,x,l,g) has he same properies. The proof is now complee. 28
3.2 Backward sochasic variaional inequaliy In his secion we suppose ha he filraion {F : } is he naural filraion of he k dimensional Brownian moion {B : }, i.e., for all, 3.2.1 Known resuls F = F B := σ({b s : s }) N P. Consider he backward sochasic variaional inequaliy (for shor BSVI) (35) { dy + ϕ(y )d F (,Y,Z )d Z db, < T, Y T = ξ, or, equivalenly, Y + We assume H s ds = ξ + H (ω) ϕ(y (ω)), dp d-a.e. (H ξ ) : ξ : Ω R d is a F T -measurable random vecor, F (s,y s,z s )ds Z s db s, [,T], P-a.s., (H ϕ ) : ϕ is he subdifferenial of he proper convex l.s.c. funcion ϕ : R d ],+ ], (H F ) : F : Ω [, [ R d R d k R d saisfies he funcion F (,,y,z) : Ω [,T] R d is a progressively measurable sochasic process for every (y,z) R d R d k, here exis some deerminisic funcions µ L 1 (,T;R) and l L 2 (,T;R), such ha, (i) for all y,y R d, z,z R d k, dp d-a.e. : (36) Coninuiy: (C y ) : y F (,y,z) : R d R d is coninuous, Monooniciy condiion: (M y ) : y y,f(,y,z) F(,y,z) µ() y y 2, Lipschiz condiion: (L z ) : F(,y,z ) F(,y,z) l() z z, (ii) Boundedness condiion: (B F ) F # R 29 ()d <, P-a.s., R,
where F # R () = sup{ F(,y,) : y R}. Definiion 29 A pair (Y,Z) Sd [,T] Λ d k (,T) of sochasic processes is a soluion of he backward sochasic variaional inequaliy (35) if here exiss a progressively measurable sochasic process H such ha, P-a.s., and, for all [,T], (37) Y + (a) H d+ F(,Y,Z ) d <, (b) (Y (ω),h (ω)) ϕ, a.e. [,T] H s ds = η+ F (s,y s,z s )ds Z s db s. (Wihou confusion, he uniqueness of he sochasic process H will permi o call he riple (Y,Z,H) a soluion of Eq.(35).) We inroduce now a supplemenary assumpion (A) : There exis p 2, a posiive sochasic process β L 1 (Ω ],T[), a posiive funcion b L 1 (,T) and a real number κ, such ha for all (u,û) ϕ and z R d k û,f (,u,z) 1 2 û 2 +β +b() u p +κ z 2, dp d-a.e. Theorem 3 Le assumpions (H ξ ), (H ϕ ), (H F ) and (A) be saisfied. If here exiss u Dom( ϕ) such ha ( p (38) E ξ p +E ϕ(ξ) +E F(s,u,) ds) <, hen he BSVI (35) has a unique soluion (Y,Z) S p d [,T] Λp d k (,T). Moreover, uniqueness holds in S 1+ d [,T] Λ d k (,T), where S 1+ d [,T] := p>1s p d [,T]. (For he proof see Pardoux & Răşcanu [14], Theorem 5.13.) 3
3.2.2 Fizparick approach In his subsecion he assumpions (H F ) are replaced by (i) he funcion F (,,y,z) : Ω [,+ [ R d is a progressively measurable sochasic processes for every (y,z) R d R d k, (ii) (y,z) F (,y,z) : R d R d k R d is coninuous dp d-a.e., (iii) for all y,y R d and z,z R d k (39) y y,f(,y,z) F(,y,z ) 1 2 z z, dp d-a.e., (iv) here exiss b > such ha, for all y R d, F(,y,z) b(1+ y + z ), dp d-a.e. Remark ha, if µ()+ 1 2 l2 (), a.e., hen he assumpions (H F ) implies (i) (iii). Denoe by Φ : Sd 2 [,T] ],+ ] he proper convex lower semiconinuous funcion defined by E ϕ(x )d, if ϕ(x) L 1 (Ω ],T[), Φ(X) := +, oherwise For each we inroduce he funcion by (U,V) D := Dom(Φ) L 2( Ω [,T];R d) J (U,V) : S := L 2 (Ω,F T,P,R d ) Λ 2 d (,T) S2 d (,T) Λ2 d k (,T) R J (U,V) (η,g,y,z) := 1 2 E η ξ 2 +E U Y,F(,U,V ) G d 1 2 E Z V 2 d+φ(y) Φ(U) and consider he funcional Ĵ : S ],+ ], Ĵ(η,G,Y,Z) := sup J (U,V) (η,g,y,z). (U,V) D 31
Remark 31 Ĵ : S ],+ ] is a lower semiconinuous funcion as supremum of lower semiconinuous funcions. We now have Proposiion 32 The mapping Ĵ has he following properies: (a) Ĵ(η,G,Y,Z), (η,g,y,z) S and Ĵ is no idenical +. (b) Le (ˆη,Ĝ,Ŷ,Ẑ) S. Then Ĵ(ˆη,Ĝ,Ŷ,Ẑ) = iff ˆη = ξ, F(Ŷ,Ẑ) Ĝ Φ(Ŷ). (c) The resricion of Ĵ o he closed convex se { K = (η,g,y,z) S : Y = η+ G s ds } Z s db s, [,T] is a convex lower semiconinuous funcion. If (ˆη,Ĝ,Ŷ,Ẑ) K, hen Ĵ(ˆη,Ĝ,Ŷ,Ẑ) = iff ˆη = ξ and (Ŷ,Ẑ,Ĥ),wih Ĥ = F(Ŷ,Ẑ) Ĝ is a soluion of he BSVI (35). Proof. (a) If Y / Dom(Φ) hen J (U,V) (η,g,y,z) = + and if Y Dom(Φ), we have Ĵ(η,G,Y,Z) J (Y,Z) (η,g,y,z). Moreover, Ĵ is a proper funcion since for v ϕ(u ) and η = ξ, Y = u, Z =, G = F (,u,) v we have (using he assumpion (39)) ha ( Ĵ (U,V) η,g,y,z ), for all (U,V) D. (b) If Ĵ(ˆη,Ĝ,Ŷ,Ẑ) = hen J (U,V) (ˆη,Ĝ,Ŷ,Ẑ), U Dom(Φ), V L2( Ω [,T];R d). So, for all (U,V) D, 1 T 2 E ˆη ξ 2 +E U Ŷ,F(U,V ) Ĝ d 1 2 E 2 Ẑ V d+φ(ŷ) Φ(U), which yields Ŷ Dom(Φ); aking in paricular U = Ŷ and V = Ẑ, we infer ˆη = ξ, P-a.s. Hence, for all (U,V) D, d+φ(ŷ) (4) E U Ŷ,F(U,V ) Ĝ 1 2 E Ẑ V 2 d+φ(u). 32
Since D is a convex se, we can replace (U,V) by ((1 λ)ŷ + λu,(1 λ)ẑ +λv), where λ (,1). The convexiy of Φ leads o he following inequaliy E U Ŷ,F((1 λ)ŷ +λu,(1 λ)ẑ +λv ) Ĝ d λ 2 E Ẑ V 2d+Φ(U) Φ(Ŷ). Passing o liminf, we deduce λ E U Ŷ,F(Ŷ,Ẑ) Ĝ ha is Conversely, using assumpion (39) we have d+φ(ŷ) Φ(U), U Dom(Φ), F(Ŷ,Ẑ) Ĝ Φ(Ŷ). J (U,V) (ξ,ĝ,ŷ,ẑ) T = E U Ŷ,F(U,V ) Ĝ d 1 2 E Ẑ V 2 d+φ(ŷ) Φ(U) E +E U Ŷ,F(U,V ) F(Ŷ,Ẑ) d 1 2 E U Ŷ,F(Ŷ,Ẑ) Ĝ d+φ(ŷ) Φ(U) and, consequenly, Ĵ(ˆη,Ĝ,Ŷ,Ẑ) =. and (c) If, moreover, (ˆη,Ĝ,Ŷ,Ẑ) K, hen Y + (F(Ŷs,Ẑs) Ĝs)ds = ˆη + F(Ŷ,Ẑ) Ĝ Φ(Ŷ), F(Ŷs,Ẑs)ds Ẑ V 2 d Z s db s ha is, (Ŷ,Ẑ,F(Ŷ,Ẑ) Ĝ) is soluion of he SVI (35). The convexiy of Ĵ on K is obained as follows: by he Energy Equaliy we have Y 2 +E Z s 2 ds = E η 2 +2E Y s,g s ds 33
and J (U,V) (η,g,y,z) becomes J (U,V) (η,g,y,z) = 1 2 E η ξ 2 +E +E Y,G d 1 2 E = 1 2 E ξ 2 E η,ξ +E U Y,F(U,V ) d E U,G d Z V 2 d+φ(y) Φ(U) U Y,F(U,V ) d E U,G d +E Z,V d 1 2 E V 2 d+ 1 2 E Y 2 +Φ(Y) Φ(U). Hence Ĵ is a convex l.s.c. funcion as supremum of convex l.s.c. funcions. The proof is now complee. Acknowledgemen 33 The auhors are graeful o he referees for he aenion in reading his paper and for heir very useful suggesions. References [1] Asiminoaei, I.; Răşcanu, A. - Approximaion and Simulaion of Sochasic Variaional Inequaliies - Spliing Up Mehod, Numer. Func. Anal. and Opimiz., 18 (3&4), pp. 251-282, 1997. [2] Barbu, V. - Nonlinear Semigroups and Differenial Equaions in Banach Spaces, Noordhoff, Leyden, 1975. [3] Barbu, V.; Precupanu, Th.- Convexiy and opimizaion in Banach spaces, Mahemaics and is Applicaions (Eas European Series), 1. D. Reidel Publishing Co., Dordrech; Ediura Academiei Republicii Socialise România, Buchares, xviii, 1986. [4] Bensoussan, A.; Răşcanu, A. - Sochasic Variaional Inequaliies in Infinie Dimensional Spaces, Numer. Func. Anal. and Opimiz., 18 (1&2), pp. 19-54, 1997. [5] Brézis, H. - Opéraeurs Maximaux Monoones e Semigroupes de Conracions Dans les Espaces de Hilber, Norh-Holland, Amserdam, 1973. [6] Burachik, R.S.; Fizparick, S. - On a family of convex funcions associaed o subdifferenials, J. Nonlinear Convex Anal., 6(1), pp. 165 171, 25. [7] Burachik, R.S.; Svaier, B.F. - Maximal monoone operaors, convex funcions and a special family of enlargemens, Se-ValuedAnal., 1(4), pp. 297 316, 22. 34
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