Backward stochastic differential equations and integral partial differential equations

Transcription

1 Backward ochaic differenial equaion and inegral parial differenial equaion Guy BARLS, Rainer BUCKDAHN 1 and ienne PARDOUX 2 June 30, 2009 Abrac We conider a backward ochaic differenial equaion, whoe he daa (he final condiion and he coefficien are given funcion of a jump diffuion proce. We prove ha under mild condiion he oluion of he BSD provide a vicoiy oluion of a yem of parabolic inegral parial differenial equaion. Under an addiional aumpion, ha yem of equaion i proved o have a unique oluion, in a given cla of coninuou funcion. Keyword : backward ochaic differenial equaion, jump diffuion procee, inegral parial differenial operaor, vicoiy oluion of yem of parial differenial equaion. AMS claificaion : 60H10, 60G44, 35K55 Running head : BSD and PD Inroducion Backward ochaic differenial equaion (in hor BSD are new ype of ochaic differenial equaion, whoe value i precribed a he final ime T, ee Pardoux, Peng [8]. I ha been noed in Pardoux, Peng [9] ha BSD provide a probabiliic formula for he oluion of cerain clae of yem of quai linear parabolic PD of econd order, and in paricular ha BSD are naurally aociaed wih vicoiy oluion of PD. The aim of hi paper i o generalize he above reul o he cae of BSD wih repec o boh Brownian moion and a Poion random meaure. The aociaed yem of parabolic PD i hen a yem of inegro parial differenial equaion. We prove, under appropriae aumpion, ha our BSD ha a unique oluion. We hen pu u in a Markovian framework, in which cae a cerain funcion defined hrough he oluion of he BSD i he unique vicoiy oluion of a yem of parabolic inegro parial differenial equaion. 1 The reearch of hi auhor ha been done during a vii a he Univerié de Provence, and wa uppored by a gran of he German Deuche Forchunggeellchaf. 2 Member of he Iniu Univeriaire de France. 1

2 1 A jump diffuion proce Le (Ω, F, (F 0, P be a ochaic bai uch ha F 0 conain all P null elemen of F, and F + = F +ε = F, 0, and uppoe ha he filraion i generaed by he following ε>0 wo muually independen procee : a d dimenional andard Brownian moion {W } 0, and a Poion random meaure µ on IR +, where = IR l \{0} i equipped wih i Borel field, wih compenaor ν(d, de = dλ(de, uch ha { µ([0, ] A = (µ ν([0, ] A} 0 i a maringale for all A aifying λ(a <. λ i aumed o be a σ finie meaure on (, aifying (1 e 2 λ(de < +. Le b : IR d IR d, σ : IR d IR d d be globally Lipchiz and β : IR d IR d be meaurable and uch ha for ome real K, and for all e, β(x, e K(1 e, x IR d, β(x, e β(x, e K x x (1 e, x, x IR d. We now conider he following SD : dx = b(x d + σ(x dw + β(x, e µ(d, de. (1 We denoe by {X (x} he unique oluion of equaion (1 aring from x a ime =, and define X = X 0 (x 0, 0, for ome x 0 IR d. xience and uniquene, a well a he properie of he oluion of (1 are aed in he following reul, which follow from Theorem 2.2 and 2.3 in Fujiwara, Kunia [5] : Propoiion 1.1 For each 0, here exi a verion of {X (x ; x IR d, } uch ha X i a C 2 (IR d valued càdlàg proce. Moreover (i X and X 0 have he ame diribuion, 0 ; (ii X 0 1, X 1 2,..., X n 1 n are independen, for all n IN, 0 0 < 1 < < n ; (iii X r = X r X, 0 < < r. Fuhermore, for all p 2, here exi a real M p uch ha for all 0 <, x, x IR d, (iv ( up Xr(x x p r ( up Xr(x Xr(x (x x p r M p ( (1 + x p M p ( x x p 2

3 2 BSD wih repec o Brownian moion and Lévy proce. xience and uniquene of a oluion From now on, we fix a erminal ime T > 0. We define ome pace of procee. Le S 2 denoe he e of F adaped càdlàg k dimenional procee {Y, 0 T } which are uch ha Y S 2 = up 0 T Y L 2 (Ω <. Le L 2 (W be he e of F progreively meaurable k d dimenional procee {Z, 0 T } which are uch ha Z L 2 (W ( T 1/2 = Z d 2 <. 0 By L 2 ( µ we denoe he e of mapping U : Ω [0, T ] IR k which are P meaurable 1 and uch ha ( T 1/2 U L 2 ( µ = U (e 2 λ(de d <. 0 Finally, we define B 2 = S 2 L 2 (W L 2 ( µ. A proof of he nex reul can be found in Tang, Li [12], Lemma 2.4. Theorem 2.1 Le Q (L 2 (Ω, F T, P k and f : Ω [0, T ] IR k IR k d L 2 (,, λ ; IR k IR k be P B k B k d B(L 2 (,, λ ; IR k meaurable and aify : (A.1.i (A.1.ii here exi K > 0 uch ha T 0 f (0, 0, 0 2 d < ; f (y, z, u f (y, z, u K( y y + z z + u u, for all 0 T, y, y IR k, z, z IR k d, u, u L 2 (,, λ ; IR k. Then here exi a unique riple (Y, Z, U B 2 which olve he B. S. D.. Y = Q + T f (Y, Z, U d T Z dw T U (e µ(d, de, 0 T. (2 We now eablih a coninuiy reul, which will be ueful in he equel. 1 P denoe he σ algebra of F predicable ube of Ω [0, T ]. 3

4 Propoiion 2.2 Le (Q, f and (Q, f be wo daa aifying he aumpion of heorem 2.1 and le (Y, Z, U denoe he oluion of he BSD wih daa (Q, f and (Y, Z, U ha of he BSD wih daa (Q, f. Define ( Q, f = (Q Q, f f and ( Y, Z, U = (Y Y, Z Z, U U. Then here exi a conan c uch ha [ T T ] up Y 2 + Z 2 d + U (e 2 λ(de d 0 T [ c Q T 0 0 ] f(y, Z, U 2 d Proof : I follow from Iô formula ha [ T T ] Y 2 + Z 2 d + U (e 2 λ(de d = [ [ Q Q 2 + c T T + 1 T 2 Z 2 d + 1 T 2 < Y, f (Y, Z, U + f (Y, Z, U f (Y, Z, U > d [ Y 2 + f (Y, Z, U 2 ] d ] U(, e 2 λ(de d The inequaliy which we wan o prove, bu wih up ouide he expecaion, follow 0 T from Gronwall lemma. We apply again Iô formula, yielding Y 2 + T Z 2 d + T T U (e 2 λ(de d = Q < Y, f (Y, Z, U f (Y, Z U > d T T 2 < Y, Z dw > ( Y + U (e 2 Y 2 µ(d, de. The reul now follow from he la ideniy, Doob inequaliy and he previou eimae. In he equel, we are concerned by a pecific cla of BSD where boh Q and for each, y, z, u, he proce {f (y, z, u, T } are given funcion of he proce X (x conruced in ecion 1. More preciely, we are given wo coninuou funcion ] g : IR d IR k, f : [0, T ] IR d IR k IR k d IR k IR k and a meaurable funcion γ : IR d IR k uch ha, for each 1 i k, f i (, x, y, z, q depend on he marix z only hrough i i h column z i, and on he vecor q only hrough i 4

5 i h coordinae q i. The fir one of hee aumpion i eenial for he noion of vicoiy oluion of he yem of inegro parial differenial equaion o be conidered below o make ene. The econd rericion i le eenial, bu will be ueful for proving he uniquene reul. We aume pecifically ha (A.2i f(, x, 0, 0, 0 C(1 + x p, g(x C(1 + x p, for ome C, p > 0 ; (A.2ii f = f(, x, y, z, q i globally Lipchiz in (y, z, q, uniformly in (, x ; (A.2iii for each (, x, y, z [0, T ] IR d IR k IR k d, 1 i k, he funcion p f i (, x, y, z, p i non-decreaing ; (A.2iv here i ome real C > 0 uch ha, for all 1 i k, 0 γ i (x, e C(1 e, x IR d, e γ i (x, e γ i (x, e < C x x (1 e, x, x IR d, e. Under he aumpion (A.2i (A.2iv, for each [0, T ] and x IR d, we conider he BSD Y,i(x = g i (XT (x (3 T + f i (r, Xr(x, Yr (x, Zr,i(x, Ur,i(x, eγ i (Xr(x, eλ(de dr T T Zr,i(x dw r Ur,i(x, e µ(drde, T, 1 i k. Noe ha in (2.2 and in he equel, f depend on U in a very pecific way. The main reaon for hi rericion i ha we hall need he comparion heorem (propoiion 2.6 below, which would no be rue in general, a i i explained in remark 2.7. I follow readily from Theorem 2.1 Corollary 2.3 For each [0, T ], x IR d, he BSD (3 ha a unique oluion (Y (x, Z (x, U (x, B 2, and (, x Y (x define a deerminiic mapping from [0, T ] IR d ino IR k. Remark 2.4 From Propoiion 1.1 (ii and (iii, and from he uniquene of he oluion of he BSD (3 we obain ha, for any 0 < T, x IR d, (Y X (x, Z X (x, U (X (x, T = (Y (x, Z(x, U(x, T in B 2. 5

6 Le u denoe he deerminiic funcion Y (x by u(, x and indicae ome of i baic properie. Propoiion 2.5 Under he aumpion (A.2, he funcion u : [0, T ] IR d IR k i coninuou in (, x and for ome real C and p, u(, x C(1 + x p, (, x [0, T ] IR d. Moreover, if g and f(,, y, z, q are uniformly coninuou, uniformly wih repec o (, y, z, q and bounded, hen u i uniformly coninuou and bounded. Proof : From Corollary 3.2 in [3], we know ha, for any q 2, here exi C q > 0 uch ha, for all 0 T, x IR d, [ Y (x q ] + ( T T q/2 Zr(x 2 dr + Ur(x, e 2 λ(de dr C q (1 + x q. Chooing =, we obain he wihed eimae for he growh of u(, x. We now define Y,x for all [0, T ] by chooing Y,x = Y,x for 0. In order o prove he coninuiy of u, we ue propoiion 2.2 o eimae Y (x Y (x 2 = Y0 (x Y0 (x 2 ( c + up 0 T [ Y (x Y (x 2 g(x T (x g(x T (x 2 T 0 1 [,T ](f(,, X(x, Y (x, Z(x, U(x 1 [,T ](f(, X (x, Y (x, Z (x, U (x ] 2 d and he coninuiy of u follow from he aumpion (A.2v and he polynomial growh of g and f in all heir variable (excep, which reul from (A.2i, (A.2ii and (A.2iv. If moreover g and f(,, y, z, q are uniformly coninuou, hen he uniform coninuiy of u follow from he ame eimae and he boundedne i immediae. Finally, in preparaion o he nex ecion, we provide a comparion heorem for one dimenional BSD. Propoiion 2.6 Le h : Ω [0, T ] IR IR d IR be P B B d B meaurable and aify T (i [ h(, 0, 0, 0 2 d] < +, 0 6

7 (ii h(, y, z, q h(, y, z, q K( y y + z z + q q, for any y, y IR, z, z IR d, q, q IR, [0, T ] and for ome real K > 0; (iii q h(, y, z, q i non-decreaing, for all (, y, z [0, T ] IR IR d. Fuhermore, le γ : Ω [0, T ] IR be P meaurable and aify We e 0 γ (e C(1 e, e. f(, ω, y, z, ϕ = h(, ω, y, z, ϕ(eγ (ω, eλ(de, for (, ω, y, z, ϕ [0, T ] Ω IR IR d L 2 (,, λ. Le Q, Q L 2 (Ω, F T, P and le denoe by (Y, Z, U B 2 (rep. (Y, Z, U B 2 he unique oluion of eq. (2 wih final condiion Q (rep. Q. Then, if Q Q, i follow Y Y, for 0 T. The proof uing Iô formula follow he argumen of he proof of he comparion heorem [9] for BSD wihou jump. Remark 2.7 If we impoe on f only he aumpion (A.1i and (A.1ii, hen, in general, he comparion heorem doe no hold. We give now a couner-example. Le = IR {0}, λ(de = δ 1 (de and f(, ω, y, z, ϕ = 2ϕ(1. Then N = µ(dde, 0 T, i andard Poion proce, and if we chooe hen 0 Q = N T and Q = 0, (Y, Z, U = (N (T, 0, I {e=1}, (Y, Z, U = (0, 0, 0. Clearly, Q Q, bu P {Y < Y } > 0, for all 0 < T. 7

8 3 Aociaed inegral parial differenial equaion. xience and uniquene. We conider he yem of inegral parial differenial equaion of parabolic ype u i(, x Lu i (, x f i (, x, u(, x, ( u i σ(, x, B i u i (, x = 0, (, x [0, T ] IR d, 1 i k u i (T, x = g i (x, x IR d, 1 i k, (4 where he econd-order inegral differenial operaor L i of he form wih L = A + K, Aϕ(x = 1 ( 2 T r a(x 2 ϕ x (x + < b(x, ϕ(x >, 2 ϕ C 2 (IR d, a ij (x = (σ(xσ(x i,j, Kϕ(x = (ϕ(x + β(x, e ϕ(x < ϕ(x, β(x, e >λ(de, ϕ C 2 (IR d, and B i i an inegral operaor defined a B i ϕ(x = (ϕ(x + β(x, e ϕ(xγ i (x, eλ(de, ϕ C 1 (IR d. The funcion σ, b and β are uppoed o aify he aumpion made in ecion 1, f, g and γ hall aify (A.2i (A.2iv. For uch a yem (4, we inroduce he noion of vicoiy oluion. Definiion 3.1 We ay ha u C([0, T ] IR d ; IR k i (i a vicoiy uboluion of (4 if u i (T, x g i (x, x IR d, 1 i k and if, for any 1 i k, ϕ C 2 ([0, T ] IR d, wherever (, x [0, T [ IR d i a global maximum poin of u i ϕ, ϕ (, x Aϕ(, x Kδ (u i, ϕ(, x f i (, x, u(, x, ( ϕσ(, x, B δ i (u i, ϕ(, x 0. for any δ > 0 where K δ (u i, ϕ(, x = (ϕ(, x + β(x, e ϕ(, x < ϕ(, x, β(x, e > λ(de δ + (u i (, x + β(x, e u i (, x < ϕ(, x, β(x, e > λ(de c δ 8

9 and B δ i (u i, ϕ(, x = wih δ = {e ; e < δ}. (ii a vicoiy uperoluion of (4 if (ϕ(, x + β(x, e ϕ(, x γ i (x, eλ(de δ + (u i (, x + β(x, e u i (, x γ i (x, eλ(de c δ u i (T, x g i (T, x,, x IR d, 1 i k, and, for any 1 i k, ϕ C 2 ([0, T ] IR d, whenever (, x [0, T [ IR d i a global minimum poin of u i ϕ, ϕ (, x Aϕ(, x Kδ (u i, ϕ(, x f i (, x, u(, x, ( ϕσ(, x, B δ i (u i, ϕ(, x 0. (iii a vicoiy oluion of (4 if i i boh a ub and a uperoluion of (4. Remark 3.2 The inroducion of he operaor K δ and B δ i in Definiion 3.1 i neceary : indeed, ince we aume only u o be coninuou in x, Ku i and B i u i are no well defined becaue of he ingulariy of λ(de a 0. On he conrary, ince ϕ i a C 2 funcion and ϕ(, x + β(x, e ϕ(, x < Dϕ(, x, β(x, e > C β(x, e 2 ϕ(, x + β(x, e ϕ(, x γ i (x, e C β(x, e γ i (x, e for ome conan C and herefore he wo fir erm in he definiion of K δ and Bi δ have a ene. Non linear ellipic and parabolic equaion wih inegro differenial erm have been udied uing vicoiy oluion heory by A. Sayah [10] and H.M. Soner [11] (ee alo O. Alvarez and A. Tourin [1]. They conider eiher a differen cla of oluion or a differen ype of inegro differenial erm. We borrow here ome of he argumen of hee work bu i i worh menionning ha he paricular form of he yem (4 linear erm + Lipchiz coninuou nonlineariie allow u o provide a impler proof of uniquene. We fir give an equivalen definiion of vicoiy oluion which will be ueful laer on. Lemma 3.3 In he definiion of u being a vicoiy ub (rep. uper oluion of (4, we can replace K δ (u i, ϕ(, x by Kϕ(, x, B δ i (u i, ϕ(, x by B i ϕ(, x. In he ame way, we can replace global maximum poin or global minimum poin by ric global maximum poin or ric global minimum poin. The proof of hi claim i very imple and we leave i a an exercice for he reader. 9

10 Proof : We rea only he uboluion cae. If (, x i a global maximum poin of u i ϕ, we have for all (, y [0, T ] IR d. Therefore for any y IR d and hi yield, for any δ > 0, u i (, y ϕ(, y u i (, x ϕ(, x, u i (, y u i (, x ϕ(, y ϕ(, x, K δ (u i, ϕ(, x Kϕ(, x, B δ i (u i, ϕ(, x B i ϕ(, x. From he inequaliy given by Definiion 3.1, uing aumpion (A.2iii, we deduce eaily ha ϕ (, x Aϕ(, x Kϕ(, x f i(, x, u(x,, ( ϕσ(, x, B i ϕ(, x 0. (5 I remain o how ha hi la condiion implie ha of he definiion. Changing ϕ ino ϕ (ϕ(, x u i (, x, we may aume ha u i (, x = ϕ(, x, u i ϕ. Moreover we may aume w.l.o.g. ha, for all α > 0, here i ome η α > 0, η α 0 a α 0, uch ha ϕ(, y u i (, y η α, for all (, y [0, T ] IR d wih (, y (, x > α. Bu we will how below ha, under hee circumance, here exi a equence (ϕ ε ε of elemen of C 2 ([0, T ] IR d uch ha (i ϕ ε (, y = ϕ(, y, if (, y (, x ( δ 2, 1 ε ; (ii u i (, y < ϕ ε (, y ϕ(, y, if δ 2 < (, y (, x < 1 ε ; (iii ϕ ε (, y u i (, y, a ε 0, for all (, y [0, T ] IR d uch ha (, y (, x δ. In paricular, we have ϕ ε (, x = ϕ(, x, ϕ ε ϕ (, x = (, x, D2 ϕ ε (, x = D 2 ϕ(, x. Since moreover ϕ ε (, x = u i (, x and ϕ ε u i, i follow from (5 ha ϕ ε(, x Aϕ ε (, x Kϕ ε (, x f i (, x, u(, x, ( ϕ ε σ(, x, B i ϕ ε (, x 0. 10

11 Then he propery (ii above ogeher wih (A.2iii yield ϕ(, x Aϕ(, x Kδ (ϕ ε, ϕ(, x f i (, x, u(, x, ( ϕσ(, x, B δ i (ϕ ε, ϕ(, x 0. Now, uing (ii and (iii, we deduce from he Lebegue dominaed convergence heorem ha lim ε 0 Kδ (ϕ ε, ϕ(, x = K δ (u i, ϕ(, x, lim ε 0 Bδ i (ϕ ε, ϕ(, x = Bi δ (u i, ϕ(, x, and leing ε end o 0 in he above relaion yield he deired reul. We now prove he exience of a equence (ϕ ε ε having he required properie. For convenience bu wihou lo of generaliy of he mehod we forge abou he variable and uppoe ha we have a ϕ C 2 (IR d and a u i C(IR d uch ha (a ϕ(x = u i (x (b for all α > 0 here i a η α > 0 uch ha η α 0 a α 0 and ϕ(y u i (y η α, for all y IR d wih y x α. Fix now ε (0, 4 and inroduce he non-negaive funcion 3δ ψ ε (y = (ϕ(y u i (y η 2 1 { 3δ 4 x y 1 }, ε y IRd, where η = η α given by (b wih α = 3δ/4. Le X be a non-negaive elemen of C (IR d wih uppor in he uni ball of IR d uch ha X(y dy = 1, and we e X µ (y = µ d X(µ 1 y, for µ > 0. Since ϕ u i i coninuou, we can find ome µ ε (0, δ/4 uch ha (ϕ u i (y (ϕ u i (y z η 4, for all y, z IR d wih x y 2 ε and z µ ε. Finally we define he funcion ϕ ε (y = ϕ(y X IR d µ ε (y zψ ε (z dz, y IR d. One check eaily ha he following properie hold 11

12 (i ϕ ε (y = ϕ(y, for all y IR d wih y x / ( 3δ 4 µ ε, 1 ε + µ ε ; (ii u i (y + η 4 ϕ ε(y ϕ(y, for y x ( δ 2 µ ε, + ; (iii ϕ ε (y u i (y, a ε 0, for all y IR d. Thi complee he proof. We now prove ha our BSD provide a vicoiy oluion of (4. Theorem 3.4 The funcion u(, x = Y (x, (, x [0, T ] IR d, inroduced in Secion 2, i a vicoiy oluion of eq. (4. Proof : Due o Propoiion 2.5, he funcion u belong o C([0, T ] IR d ; IR k. I clearly aifie he boundary condiion a ime = T. We now how ha i i a vicoiy uboluion of eq. (4. A imilar argumen would how ha i i a vicoiy uperoluion of eq. (4. Le 1 i k, ϕ C 2 ([0, T ] IR d, (, x [0, T ] IR d uch ha ϕ u i, ϕ(, x = u i (, x. Taking ino accoun he correponding properie of u i, we can aume addiionally ha ϕ and i derivaive have a mo polynomial growh a y, uniformly in [0, T ]. Nex we noe ha from uniquene of he oluion of our BSD, for any T, x IR d, Y (x = Y (X(x = u(, X(x. Y Chooe h > 0 uch ha + h T. I follow from he la remark ha for + h,,i(x = u i ( + h, X +h(x + +h Z r,i(x db r +h +h f i (r, X r(x, Y U r,i(x, e µ(drde. r (x, Zr,i(x, U r,i(x, eγ i (X r(x, eλ(de dr If y IR, z IR k, we denoe by (y, z i he k dimenional vecor whoe i h componen equal y, and all he oher componen equal he correponding one of z. Conider he one dimenional BSD Y h (x = ϕ( + h, X+h(x +h + f i (r, Xr(x, (Y h r, Ỹ r,i(x, Z h r, U h r (eγ i (Xr(x, eλ(de dr +h Z h +h r db r U h r (e µ(drde, + h. Taking ino accoun ha ϕ u i, i follow from Propoiion 2.6 ha Y h Y,i(x, + h, a.e., 12

13 and, in paricular, Fuhermore, puing Y h u i (, x = ϕ(, x. we have by Iô formula, ϕ(, X (x = ϕ( + h, X +h(x ψ(, y = ϕ(, y + Lϕ(, y, Φ(, y, e = ϕ(, y + β(y, e ϕ(, y, +h Define, for + h, e, +h ψ(r, X r(x dr Φ(r, Xr (x, e µ(drde. Ŷ h = Y h ϕ(, X (x, Ẑ h = Z h ( ϕσ(, X (x, Û h (e = U h (e Φ(, X (x, e. +h ( ϕσ(r, X r(x db r I follow from he above idenifie ha (Ŷ h, Ẑh, Û h i he unique oluion of he following one dimenional BSD : where Ŷ h = +h [ψ(r, Xr(x + f i (r, Xr(x, Ỹ r h, Z r h, Ũ r h ] dr +h +h Ẑr h db r Ûr h (e µ(drde, Ỹr h = (ϕ(r, Xr(x + Ŷ r h, Ỹ r,i(x Z r h = ( ϕσ(r, Xr(x + Ẑh r Ũr h = (Φ(r, Xr (x, e + Û r h (eγ i (Xr(x, eλ(de Hence, by uing andard echnique o eimae he quared norm of he oluion (Ŷ h, Ẑh, Û h, we obain for + h, +h [ Ŷ h 2 ] + [ c[ Ẑh r 2 dr] + [ +h +h Ŷ r h (1 + Ŷ r h + Ẑh r + ( Û h r (e 2 λ(de dr] Û h r (e 2 λ(de 1/2 dr], 13

14 i.e., for ome c > 0, Hence, in paricular [ Ŷ h 2 ] + 1 +h 2 ([ Ẑh r 2 dr] + [ c[ and conequenly for ome C > 0 On he oher hand, 1 +h 2 ([ Ẑh r 2 dr] + [ and herefore, for ome C > 0 +h +h ([Ŷ h r + Ŷ h r 2 dr] +h [ Ŷ h 2 ] 2c(h + [ Ŷ r h 2 dr], +h ( 1 +h [ h Ẑh r 2 dr] + [ We uppoe now ha [ Ŷ h 2 ] Ch. Û h r (e 2 λ(de dr] +h Û r h (e 2 λ(de dr] c[ ( Ŷ r h + Ŷ r h 2 dr], +h Û h r (e 2 λ(de dr] C h, h > 0. ϕ(, x + Lϕ(, x + f i(, x, u(, x, ( ϕσ(, x, B i ϕ(, x < 0, and find a conradicion by uing he above eimae. Indeed, under he above aumpion, here exi ome δ > 0 and ome h 0 > 0 uch ha, for all 0 < h h 0, ξ h := 1 +h h [ψ(r, Xr(x +f i (r, X r(x, (ϕ(r, X r(x, ũ i (r, X r(x, ( ϕσ(r, X r(x, B i ϕ(r, X r(x] dr δ. Now we have ha Ŷ h 0, hence 0 h 1Ŷ h = 1 +h h [ψ(r, Xr(x + f i (r, Xr(x, (ϕ(r, Xr(x + Ŷ r h, ũ i (r, Xr(x, ( ϕσ(r, Xr(x + Ẑh r, B i ϕ(r, Xr(x + Ûr h (eγ i (Xr(x, eλ(de] dr. 14

15 Therefore, for all 0 < h h 0, δ h 1Ŷ h ξ h c[ 1 h c ch 1/4. +h up +h ( Ŷ h + Ẑh + ( ( 1 [ Ŷ h 2 ] 1/2 + Thi i impoible, conequenly Û h (e 2 λ(de 1/2 d] +h h [ Ẑh r 2 dr] 1/2 + ( 1 +h h [ ϕ(, x Lϕ(, x f i(, x, u(, x, ( ϕσ(, x, B i ϕ(, x 0. In view of Lemma 3.3, hi complee he proof. Û r h (e 2 λ(de dr] 1/2 Now we give a uniquene reul for (4. Thi reul i obained under more rericive aumpion ha he exience one : namely we need he wo following addiional aumpion (A.2 v f i (, x, r, p, q f i (, y, r, p, q m i R( x y (1 + p for 1 i k, where m i R( 0 when 0 +, for all [0, T ], x, y R, r R, p IR d, q IR ( R <. For γ, we aume in addiion (A.2 vi γ i (x, e γ i (y, e C 1 x y (1 e 2 for 1 i k for ome conan C 1 > 0 and for any x, y IR d, e. Our reul i he Theorem 3.5 Aume ha f, g and γ aify (A2. Then here exi a mo one vicoiy oluion u of (4 uch ha 2 lim u(, x e Ã[log( x ] = 0, (6 x + uniformly for [0, T ], for ome à > 0. In paricular, he funcion u(, x = Y (x i he unique vicoiy oluion of (4 in he cla of oluion which aify (6 for ome à > 0. Remark 3.6 Noice ha, by Propoiion 2.5, u(, x = Y (x ha a mo a polynomial growh a infiniy and herefore i aifie (6. The growh condiion (6 i opimal o ge uch a uniquene reul for (4. conider he equaion Indeed, u x2 2 u 2 x x u 2 2 x = 0 in (0, T (0, + ; (7 15

16 hen u i a oluion of (7 if and only if he funcion v(, y = u(, e y i a oluion of he Hea quaion v 1 2 v = 0 in (0, T IR. (8 2 x2 Bu i i well-known ha, for he Hea quaion, he uniquene hold in he cla of oluion v aifying lim v(, y + y e à y 2 = 0, (9 uniformly for [0, T ], for ome à > 0. And (9 give back (6 for (7 ince y = log(x. Le u finally menion ha, in our cae, he growh condiion (6 i mainly a conequence of he aumpion on he coefficien of he differenial operaor and in paricular on a = (a i,j i,j ; under he aumpion of Theorem 3.5, he marix a ha a priori a quadraic growh a infiniy. If a i aumed o have a linear growh a infiniy, an eay adapaion of he proof of Theorem 3.5 how ha he uniquene hold in he cla of oluion aifying uniformly for [0, T ], for ome à > 0. lim u(, x + x e à x = 0, Proof of Theorem 3.5 : Le u and v be wo vicoiy oluion of (4. The proof coni in wo ep. We fir how ha u v and v u are vicoiy uboluion of an inegral parial differenial yem; hen we build a uiable equence of mooh uperoluion of hi yem o how ha u v = 0 in [0, T ] IR d. Here and below, we denoe by he up norm in IR k. Lemma 3.7 Le u be a uboluion and v a uperoluion of (4. Then he funcion ω := u v i a vicoiy uboluion of he yem ω i Lω i K [ ω + ω i σ + (B i ω i +] = 0 in [0, T ] IR d (10 for 1 i k, where K i he Lipchiz conan of f in (r, p, q. Proof : Le ϕ C 2 ([0, T ] IR d and le ( 0, x 0 (0, T IR d be a ric global maximum poin of ω i ϕ for ome 1 i k. We inroduce he funcion ψ ε,α (, x,, y = u i (, x v i (, y x y 2 ( 2 ϕ(, x ε 2 α 2 where ε, α are poiive parameer which are devoed o end o zero. Since ( 0, x 0 i a ric global maximum poin of u i v i ϕ, by a claical argumen in he heory of vicoiy oluion, here exi a equence (, x,, y uch ha 16

17 (i (, x,, y i a global maximum poin of ψ ε,α in ([0, T ] B R 2 where B R i a ball wih a large radiu R. (ii (, x, (, y ( 0, x 0 a (ε, α 0. (iii x y 2 ε 2, ( 2 α 2 are bounded and end o zero when (ε, α 0. We have dropped above he dependence of, x, and y in ε and α for he ake of impliciy of noaion. I follow from Theorem 8.3 in [4] ha here exi X, Y S d uch ha ( a + ϕ (, x, p + Dϕ(, x, X (a, p, Y D 2, v i (, y ( X 0 0 Y 4 ( I I ε 2 I I + D 2,+ u i (, x ( D 2 ϕ(, x where 2( 2(x y a = and p =. α 2 ε 2 Modifying if neceary ψ ε,α by adding erm of he form χ(x and χ(y wih uppor in BR/2 c, we may aume ha (, x,, y i a global maximum poin of ψ ε,α in ([0, T ] IR d 2. Since u and v are repecively ub and uperoluion of (4, we have for δ mall enough a ϕ (, x 1 T r(a(xx < b(x, p + Dϕ(, x > 2 β(x, e δ 2 λ(de (ϕ(, x + β(x, e ϕ(, x < Dϕ(, x, β(x, e >λ(de ε 2 δ (u i (, x + β(x, e u i (, x < p + Dϕ(, x, β(x, e >λ(de c δ f i (, x, u(, x, (p + Dϕ(, xσ(x, B δ i 0 where B δ i = δ ( < p, β(x, e > + β(x, e 2 γ ε 2 i (x, eλ(de (ϕ(, x + β(x, e ϕ(, x γ i (x, eλ(de δ + (u i (, x + β(x, e u i (, x γ i (x, eλ(de c δ 17

18 and where a 1 T r(a(yy < b(y, p > 2 β(y, e + δ 2 λ(de (v ε 2 i (, y + β(y, e v i (, y < p, β(y, e >λ(de δ c B δ i = δ f i (, y, v(, y, pσ(y, Bδ i 0 ( β(y, e 2 < p, β(y, e > γ ε 2 i (y, eλ(de + (v i (, y + β(y, e v i (, yγ i (y, eλ(de. δ c I i worh noicing ha he χ erm we have o add o ψ ε,α o have a global maximum poin do no appear in he wo inequaliie above becaue β i bounded and hey have a uppor which i included in B c R/2 for R large. Of coure, we are going o ubrac hee inequaliie and we need o eimae difference beween erm of he ame ype. Fir, if (e 1... e d i an orhonormal bai of IR d, T r(a(xx T r(a(yy = T r(σ (xxσ(x T r(σ (yy σ(y d = [< Xσ(xe i, σ(xe i > < Y σ(ye i, σ(ye i >] i=1 To eimae hi um, we ue he marix inequaliy above ogeher wih he Lipchiz coninuiy of σ. We ge x y 2 T r(a(xx T r(a(yy C + T r(a(xd 2 ϕ(, x ε 2 for ome conan C. Then x y 2 < b(x, p > < b(y, p > C 1 x y p C ε 2 becaue of he Lipchiz coninuiy of b. To eimae he difference of he inegro differenial erm, we rongly ue he fac ha (, x,, y i a global maximum poin of ψ ε,α in B R/2. From he inequaliy we deduce ψ ε,α (, x + β(x, e,, y + β(y, e ψ ε,α (, x,, y [u i (, x + β(x, e u i (, x] [v i (, y + β(y, e v i (, y] < p, β(x, e β(y, e > 1 ε 2 β(x, e β(y, e 2 ϕ(, x + β(x, e ϕ(, x. 18

19 Therefore c δ + (u i (, x + β(x, e u i (, x < p + Dϕ(, x, β(x, e >λ(de c δ c δ c δ (v i (, y + β(y, e v i (, y < p, β(y, e >λ(de [ϕ(, x + β(x, e ϕ(, x < Dϕ(, x, β(x, e >]λ(de β(x, e β(y, e 2 λ(de. ε 2 Noice ha he la erm of he righ hand ide i eimaed by wih C independen of δ, becaue of he aumpion on β. In he ame way, Bi δ Bδ i + c δ + + δ δ ( β(x, e 2 < p, β(x, e > + γ i (x, eλ(de ( < p, β(y, e > + ε 2 β(y, e 2 ε 2 [ϕ(, x + β(x, e ϕ(, x]γ i (x, eλ(de + γ i (y, eλ(de [v i (, y + β(y, e v i (, y](γ i (x, e γ i (y, eλ(de c δ C x y 2 ε 2 [< p, β(x, e β(y, e > + 1 ε 2 β(x, e β(y, e 2 ]γ i (x, eλ(de Becaue of he aumpion on β and γ, he la inegral i eimaed by C x y 2 ε 2 where C i independen of δ and he preceeding one i eimaed by C x y ince v i coninuou (and herefore locally bounded and becaue of he addiional aumpion (A2.vi made on γ in he aemen of Theorem 3.5. Finally, we conider he difference beween he nonlinear erm f i (, x, u(, x, (p + Dϕ(, xσ(x, B δ i f i (, y, v(, y, pσ(y, Bδ i ρ ε,δ ( + m i ( x y (1 + pσ(y + K u(, x v(, y + K p(σ(x σ(y + Dϕ(, xσ(x + K( B δ i Bδ i +. The fir erm in he righ hand ide come from he coninuiy of f i in : ρ ε,δ ( 0 when 0 + for fixed ε and δ. The econd erm come from (A.2 v : we have denoed by m i he modulu m i R which appear in hi aumpion for R large enough. The hree la erm come from he Lipchiz coninuiy of f i w.r.. he hree la variable and he fac ha i i non-decreaing wih repec o he la one. 19

20 We noice ha x y 2 p(σ(x σ(y C ε 2 becaue of he Lipchiz coninuiy of σ and ha x y 2 x y pσ(y C. ε 2 Now we ubrac he vicoiy inequaliie for u and v : hank o he above eimae, we can wrie he obained inequaliy in he following way ϕ (, x Aϕ(, x Kϕ(, x K u(, x v(, y K Dϕ(, xσ(x K (B i ϕ(, x + ρ ε,δ ( + ω 1 (ε, α + ω ε 2(δ x y 2 where we have gahered in he ω 1 (ε, α erm, all he erm of he form and x y ; ε 2 ω 1 (ε, α 0 when (ε, α end o 0. The ω2(δ ε erm conain all he remaining inegral on 2 δ. To conclude we fir le α go o zero : ince i bounded, 0 and we ge rid of he fir erm of he righ hand ide above. Then we le δ go o zero keeping ε fixed and finally we le ε 0. Since (, x, (, y ( 0, x 0, we obain : α 2 ϕ ( 0, x 0 Lϕ( 0, x 0 K ω( 0, x 0 K Dϕ( 0, x 0 σ(x 0 K(B i ϕ + ( 0, x 0 0 and herefore ω i a uboluion of he deired equaion by Lemma 3.3. Now we are going o build uiable mooh uperoluion for he equaion (10. Lemma 3.8 For any à > 0, here exi C 1 > 0 uch ha he funcion where aifie χ(, x = exp [ (C 1 (T + Ãψ(x] ψ(x = [ log ( ( x /2 + 1 ] 2, χ Lχ Kχ K Dχσ K(B i χ + > 0 in [ 1, T ] IR d for 1 i k where 1 = T Ã/C 1. 20

21 Proof : We fir eimae he erm Kχ and B i χ, he main poin being heir dependence in x. For he ake of impliciy of noaion, we denoe below by C all he poiive conan which ener in hee eimae. Thee conan depend only on à and on he bound on he coefficien of he equaion. We fir give eimae on he fir and econd derivaive of ψ : eay compuaion yield Dψ(x 2[ψ(x]1/2 ( x /2 4 in IRd, and D 2 ψ(x C(1 + [ψ(x]1/2 x Thee eimae imply ha, if [ 1, T ] in IR d. and, in he ame way Dχ(, x (C 1 (T + Ãχ(, x Dψ(x [ψ(x] 1/2 Cχ(, x ( x 2 + 1, 1/2 D 2 χ(, x Cχ(, x ψ(x x I i worh noicing ha, becaue of our choice of 1, he above eimae do no depend on C 1. Then, ince β i bounded and ince ψ i lipchiz coninuou in IR d, ediou bu raighforward compuaion imply and χ(, x + β(x, e χ(, x < Dχ(x,, β(x, e > Cχ(, x ψ(x x β(x, e 2, [ψ(x] 1/2 χ(, x + β(x, e χ(, x Cχ(, x β(x, e. ( x /2 Since σ and b grow a mo linearly a infiniy, we have χ (, x Lχ(, x Kχ(, x K Dχ(, xσ(x K(B i χ + [ χ C 1 ψ(x Cψ(x + C[ψ(x] 1/2 C ψ(x x K 1/2 C K[ψ(x] C K ] [ψ(x]1/2 ( x /2 Since ψ(x 1 in IR d, by uing Cauchy-Schwarz inequaliy, i i clear enough ha for C 1 large enough he quaniy in he bracke i poiive and he proof i complee. 21

22 To conclude he proof, we are going o how ha ω = u v aifie ω(, x αχ(, x in [0, T ] IR d for any α > 0. Then we will le α end o zero. To prove hi inequaliy, we fir remark ha becaue of (6 lim ω(, +1 1/2 ] 2 x + x e Ã[log(( x 2 = 0 uniformly for [0, T ], for ome à > 0. Thi implie, in paricular, ha ω i αχ i bounded from above in [ 1, T ] IR d for any 1 i k and ha M = max max 1 i k [ 1,T ] IR d( ω i αχ(, xe K(T i achieved a ome poin ( 0, x 0 and for ome i 0. We fir remark ha, ince i he up norm in IR k, we have M = max [ 1,T ] IR d( ω αχ(, xe K(T and ω i0 ( 0, x 0 = ω( 0, x 0. We may aume w.l.o.g. ha ω i0 ( 0, x 0 > 0, oherwie we are done. Then wo cae : eiher ω i0 ( 0, x 0 > 0 or ω i0 ( 0, x 0 < 0. We rea he fir cae, he econd one i reaed in a imilar way ince he role of u and v are ymmeric. From he maximum poin propery, we deduce ha ω i0 (, x αχ(, x (ω i0 αχ( 0, x 0 e K( 0 and hi inequaliy can be inerpreed a he propery for he funcion ω i0 φ o have a global maximum poin a ( 0, x 0 where φ(, x = αχ(, x + (ω i0 αχ( 0, x 0 e K( 0 Since ω i a vicoiy uboluion of (10, if 0 [ 1, T [, we have φ ( 0, x 0 Lφ( 0, x 0 K ω( 0, x 0 K Dφ( 0, x 0 σ(x 0 K(B i φ + 0. Bu he lef hand ide of hi inequaliy i nohing bu [ α χ ] ( 0, x 0 Lχ( 0, x 0 Kχ( 0, x 0 K Dχ( 0, x 0 σ(x 0 K(B i χ + ( 0, x 0 ince ω i0 ( 0, x 0 = ω( 0, x 0 ; o, by Lemma 3.8, we have a conradicion. Therefore 0 = T and ince ω(t, x = 0, we have ω(, x αχ(, x 0 in [ 1, T ] IR d. 22

23 Leing α end o zero, we obain ω(, x = 0 in [ 1, T ] IR d. Applying ucceively he ame argumen on he inerval [ 2, 1 ] where 2 = ( 1 Ã/C 1 + and hen, if 2 > 0 on [ 3, 2 ] where 3 = ( 2 Ã/C ec. We finally obain ha and he proof i complee. ω(, x = 0 in [0, T ] IR d Remark 3.9 The aumpion (A.2 vi on γ i ued in he proof o eimae he difference ( B δ i Bδ i + : if u or v i aumed o be locally Lipchiz coninuou hi addiional aumpion i no neceary anymore o obain he reul of Theorem 3.5. Moreover if he funcion f i aify r l f i (, x, r, p, q i non decreaing for l i for any [0, T ], x IR d, p IR d, q IR and (r 1,..., r l 1, r l+1,..., r k in IR k 1 hen eay modificaion in he proof how ha a comparion reul i rue for (4. More preciely, if u, v are repecively vicoiy ub and uperoluion of (4 aifying (6 and hen u(t, x v(t, x in IR d u(, x v(, x in [0, T ] IR d. Our proof, which avoid hi aumpion, i inpired from H. Ihii and S. Koike [7]. Finally, we wan o menion ha, under he above monooniciy aumpion on he f i, he cae of emiconinuou oluion can alo be reaed. We refer o O. Alvarez and A. Tourin [1] for a complee decripion of he properie of emiconinuou oluion for uch inegral parial differenial equaion. Reference [1] O. Alvarez and A. Tourin : Vicoiy oluion of nonlinear inegro differenial equaion, Annale de l Iniu Henri Poincaré. Analye non Linéaire, à paraîre. [2] G. Barle : Soluion de vicoié de équaion de Hamilon Jacobi, Mahémaique e Applicaion 17, Springer, [3] R. Buckdahn,. Pardoux : BSD wih jump and aociaed inegral ochaic differenial equaion, preprin. [4] M.G. Crandall, H. Ihii, P.L. Lion : Uer guide o vicoiy oluion of econd order parial differenial equaion, Bull. Amer. Soc. 27, 1 67,

24 [5] T. Fujiwara, H. Kunia : Sochaic differenial equaion of Jump ype and Lévy procee in diffeomorphim group, J. Mah. Kyoo Univ. 25, 1, , [6] N. Ikeda, S. Waanabe : Sochaic Differenial quaion and Diffuion Procee, Norh Holland/Kodanka [7] J. Ihii, S. Koike : Vicoiy oluion for monoone yem of econd order ellipic PDe, Commun. in Parial Diff. qu., 16 (6 & 7, , [8]. Pardoux, S. Peng : Adaped oluion of a backward ochaic differenial equaion, Sy. Conrol Le. 14, 55 61, [9]. Pardoux, S. Peng : Backward ochaic differenial equaion and quailinear parabolic parial differenial equaion, in : B.L. Rozovkii, R.B. Sower (ed Sochaic parial differenial equaion and heir applicaion (Lec. Noe Conrol Inf. Sci. 176, Berlin, Heidelberg New York : Springer [10] A. Sayah : quaion d Hamilon Jacobi du premier ordre avec erme inégro différeniel : Parie I : Unicié de oluion de vicoié ; Parie II : xience de oluion de vicoié ; Comm. in Parial Diff. q. 16 (6 & 7, , [11] H. M. Soner : Opimal conrol of jump-markov procee and vicoiy oluion, in Sochaic Differenial Syem, Sochaic Conrol Theory and Applicaion (W.H Fleming and P.L Lion, ed, IMA Mah. Appl. vol 10, Springer, Berlin, [12] S. Tang, X. Li : Neceary condiion for opimal conrol of ochaic yem wih random jump, SIAM J. Conrol and Opim. 32, , Guy BARLS Faculé de Science e Technique Univerié de Tour Déparemen de Mahémaique e Informaique Scienifique 7, Parc Grandmon F TOURS Rainer BUCKDAHN Univerié de Breagne Occidenale Faculé de Science e Technique Déparemen de Mahémaique 6, avenue Le Gorgeu BP 809 F BRST CDX 24

25 ienne PARDOUX LATP Cenre de Mahémaique e d Informaique Univerié de Provence 39, rue Jolio Curie F MARSILL CDX 13 25