ONTHEPATHWISEUNIQUENESSOF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTS

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1 OCKY MOUNTAIN JOUNAL OF MATHEMATICS Volume 43, Numbe 5, 213 ONTHEPATHWISEUNIQUENESSOF STOCHASTIC PATIAL DIFFEENTIAL EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTS DEFEI ZHANG AND PING HE ABSTACT. In hi pape, we pove a pahwie uniquene eul of a cla of ochaic paial diffeenial equaion diven by pace-ime whie noie whoe coefficien aify non-lipchiz condiion. 1. Inoducion. Many mahemaician and phyici have inveigaed he uniquene of he following ochaic paial diffeenial equaion (SPDE): (1.1) ν (x) =Δν (x)+σ(ν (x))ẇ (, x), ν = μ, whee Ẇ i he pace-ime whie noie. I i a vey impoan model which wa popoed by Dawon in 1972 a follow: (1.2) ν (x) =Δν (x)+σ ν (x)ẇ (, x), ν = μ. In hi cae, he uniquene of he oluion of he SPDE (1.2) i only poved in he weak ene uing ha of he maingale poblem. The difficuly in poving pahwie uniquene in (1.2) aie fom he fac ha ν(, x) i non-lipchiz. Fo a moe deailed decipion he 21 AMS Mahemaic ubjec claificaion. Pimay 6H1, 6H2. Keywod and phae. Backwad doubly ochaic diffeenial equaion, ochaic paial diffeenial equaion, pahwie uniquene, non-lipchiz coefficien. Thi wok wa uppoed by The Naional Baic eeach Pogam of China (973 Pogam) (No. 27CB81491) and The Scienific eeach Foundaion of Yunnan Povince Educaion Commiee (No. 21Y167), Naional Science Foundaion of China (No ), Naual Science Foundaion of Yunnan Povince (No. 213FZ116) and Scienific eeach Foundaion of Yunnan Povince Educaion Commiee (No. 211C12), Naional Science Foundaion of China gan , Naual Science Foundaion of Yunnan Povince (No. 213FZ116) and Scienific eeach Foundaion of Yunnan Povince Educaion Commiee (No. 211C12). eceived by he edio on Decembe 1, 21, and in evied fom on Mach 6, 211. DOI:1.1216/MJ Copyigh c 213 ocky Mounain Mahemaic Conoium 1739

2 174 DEFEI ZHANG AND PING HE eade i efeed o [1 7, 12 14]. We will no eolve he pahwie uniquene queion fo SPDE (1.2) bu will ucceed in olving he SPDE (1.1) wih non-lipchiz noie coefficien by uing a new cla of backwad doubly ochaic diffeenial equaion (BDSDE) diven by pace-ime whie noie wih non-lipchiz coefficien. Mynik [8] and Mynik, Pekin and Sum [9] obained he pahwie uniquene fo ochaic hea equaion wih non-lipchiz coefficien, bu we will inveigae hi poblem by uing backwad mehod. BSDE and BDSDE wee fi inoduced by Padoux and Peng, epecively, in [1, 11]. We will eablih he connecion beween BDSDE and SPDE diven by pace-ime whie noie. Conide he SPDE a follow: (1.3) ν (y) =μ(y)+ σ(u, ν (y))w (ddu)+ ν (y) d, whee σ aifie he following condiion (1.4) σ(u, y 1 ) σ(u, y 2 ) 2 du ρ( y 1 y 2 2 ), whee ρ i a concave and nondeceaing funcion fom + o + uch ha ρ() =, ρ(u) > fou>and + (du)/ρ(u) =. Unde aumpion (1.4), we obain he pahwie uniquene fo SPDE (1.3). 2. Backwad doubly ochaic diffeenial equaion diven by pace-ime whie noie. We conide a new fom of BDSDE a follow: (2.5) y = ξ + g(u, y )W ( ddu) z db, T, whee ξ i an FT B-meauable andom vaiable, whee F T B = σ(b : T ), B i a Bownian moion, W, independen of B, i a paceime whie noie in (, ) 2 and he noaion W ( ddu) and fo he backwad Iô inegal. Definiion 2.1. The pai of pocee (y,z ) i a oluion o he BDSDE (2.5) if hey ae G -adaped and, fo each [,T], he ideniy

3 STOCHASTIC PATIAL DIFFEENTIAL EQUATIONS 1741 (2.5) hold almo uely, whee G = σ(b, ; W ([, T ] A), [, T ],A B()). We hall need he following exenion of he well-known Iô fomula. Lemma 2.1. Le g :[,T] Ω be a G -adaped andom field, and le x be given by (2.6) x = x + g(, u)w ( ddu)+ z db. Then, fo any f Cb 2 (), we have f(x )=f(x)+ f (x )g(, u)w ( ddu)+ f (x )z db (2.7) 1 f (x )g 2 (, u) du d + 1 f (x )z d. The poof i imila o he poof of Lemma 1.3 in Padoux and Peng [11]. Wih he help of he above lemma, we can now pove he following heoem. Theoem 2.1. Suppoe funcion g aifie he aumpion (1.4). Then he BDSDE (2.5) ha a mo one oluion. Poof. Le(y.,z. ), (y.,z.) be wo oluion of BDSDE (2.5) and β>. By viue of Lemma 2.1, we have E y y 2 e β + E β y y 2 e β d + E z z 2 d = E e β g(u, y ) g(u, y ) 2 du d. Fom he aumpion (1.4), fo all [,T], we deive (2.8) E y y 2 + E z z 2 d e βt ρ(e[ y y 2 ]) d.

4 1742 DEFEI ZHANG AND PING HE Theefoe, E y y 2 e βt ρ(e[ y y 2 ]) d. Then we can ge E y y 2 =, [,T]; hi mean ha y = y, almo uely. I hen follow fom (2.8) ha z = z, almo uely, fo any [,T]. 3. Connecion beween SPDE and BDSDE. We conide he SPDE a follow: (3.9) ν (y) =μ(y)+ g(u, ν (y))w (ddu)+ ν (y) d. Fo T fixed, we define he andom field u (y) = ν T (y), fo all [,T], y. We alo inoduce he new noie W by W (, x) =W (T,x) W (T, x), fo all [,T],x. Then he SPDE (3.9) i conveed o i backwad veion a follow: (3.1) u (y) =μ(y)+ g(v, u (y))w ( ddv)+ u (y) d. I i clea ha he SPDE (3.9) and (3.1) have he ame uniquene popey. Denoe X,x = x + B B, fo all T. Conide he following BDSDE: (3.11) Y,x = μ(x,x T )+ g(v, Y,x )W ( ddv) Z,x db, T. Theoem 3.1. If {u (x)} i a oluion o (3.1) aifying up E[ x u(, x) 2 ] <, (,x) [,T ] hen u(, x) =Y,x,wheeY,x i a oluion o he BDSDE (3.11).

5 STOCHASTIC PATIAL DIFFEENTIAL EQUATIONS 1743 Poof. Fi, we mooh he funcion u (y) by uing he Bownian emigoup. Fo any ε >, le u ε (y) = T ε u (y), whee T ε f(x) = P ε(x y)f(y) dy and P ε (x) =1/ 2πε exp( x 2 /2ε). Applying he Bownian emigoup o boh ide of (3.1), we ge u ε (y) =T ε μ(y)+ P ε (y z)g(v, u (z)) dzw ( ddv) (3.12) + u ε (y) d. Le Y,x,ε = u ε (X,x )andz,x,ε = x u ε (X,x ). Le = < 1 < < n = T be a paiion of [, T ]. We ue Lemma 2.1 fo u ε i and he SPDE (3.12) wih y = X i+1 ;hen (3.13) u ε (X,x ) T εμ(y) n 1 = (u ε i (X,x i ) u ε i (X,x i+1 )) i= n 1 + = (u ε i (X,x i+1 ) u ε i+1 (X,x i= n 1 i+1 i= i n 1 i+1 i= i n 1 i+1 + i= i n 1 i+1 + i= i x u ε i (X,x ) db u ε i (X,x ) d u ε i (X,x ) d Takinghemehizeo,wehave (3.14) u ε (X,x ) T ε μ(y) = + x u ε (X,x ) db P ε (X,x i+1 )) P ε (X,x z)g(v, u (z))w ( ddv). i+1 z)g(v, u (z)) dzw ( ddv).

6 1744 DEFEI ZHANG AND PING HE A ε, we noe ha (3.15) E and (3.16) E x u ε (X,x ) db. P ε (X,x x u (X,x ) db 2 E x u ε (X,x ) x u (X,x ) 2 d 1 T ε x u x u 2 L 2 () d, z)g(v, u (z)) dzw ( ddv) g(v, u (X,x ))W ( ddv) E T ε g(v, u (X,x 2 )) g(v, u (X,x )) 2 d dv Finally, we ake ε on boh ide of (3.14). Then (3.11) follow fom (3.14). Theoem 3.2. Le he funcion g aify aumpion (1.4), andleμ be bounded in (3.9). Then he pahwie uniquene hold fo he SPDE (3.9). Poof. Becaue μ i bounded, hi implie up E[ x ν(, x) 2 ] <, (,x) [,T ] and he poof immediaely follow fom Theoem 2.1 and 3.1. Moe geneally, we can conide he following SPDE: (3.17) ν (y) =μ(y)+ σ(, u, ν (y))w (ddu)+ ( ν (y)+b(, ν (y))) d,

7 STOCHASTIC PATIAL DIFFEENTIAL EQUATIONS 1745 whee b aifie (3.18) b(, y 1 ) b(, y 2 ) 2 ρ(, y 1 y 2 2 ), [,T]; and σ aifie (3.19) σ(, u, y 1 ) σ(, u, y 2 ) 2 du ρ(, y 1 y 2 2 ), [,T], whee ρ :[,T] + + aifie: fo fixed [,T], ρ(, ) i a concave and nondeceaing funcion uch ha ρ() =. Fo fixed v, ρ(, v) d <. Fo any C>, he following ODE (3.2) { v = Cρ(, v), v(t )=. ha a unique oluion v() =, [,T]. We fuhe aume ha μ i bounded. Then we have: Theoem 3.3. Unde aumpion (3.18) and (3.19), hespde (3.17) ha a mo one oluion. Poof. The poof of Theoem 3.3 i imila o ha of Theoem 3.2; o, we omi i. emak 3.1. Ou mehod can be applied in he SPDE diven by coloed noie; he poof i imila o ha of Theoem 3.2, we omi i. emak 3.2. Unde he uiable aumpion abou geneao g, we can pove he exience of BSDE o ge he exience of SPDE. Acknowledgmen. The auho hank Pof. Zengjing Chen fo helpful dicuion and valuable uggeion and he efeee fo a caeful eading.

8 1746 DEFEI ZHANG AND PING HE EFEENCES 1. D.A. Dawon, Sochaic evoluion equaion and elaed meaue-valued pocee, J. Muliva. Anal. 5 (1975), , Meaue-valued Makov pocee, Écol. Pobab. S.-Flou 21, Lec. Noe Mah. 1541, Spinge, Belin, E. Dynkin, Supepocee and paial diffeenial equaion, Ann. Pobab. 21 (1993), E. Dynkin, S.E. Kuzneov and A.V. Skookhod, Banching meaue-valued pocee, Pobab. Theo. el. Field 99 (1994), J. Lampei, Coninuou ae banching pocee, Bull. Ame. Mah. Soc. 73 (1967), X. Mao, Adaped oluion of backwad ochaic diffeenial equaion wih non-lipchiz coefficien, Soch. Poc. Appl. 58 (1995), A. Maoui and M. Scheuzow, Sochaic PDE diven by nonlinea noie and backwad doubly SDE, J. Theo. Pobab. 15 (22), L. Mynik, Weak uniquene fo he hea equaion wih noie, Ann. Pobab. 26 (1998), L.Mynik,E.PekinandA.Sum,On pahwie uniquene fo ochaic hea equaion wih non-lipchiz coefficien, Ann. Pobab. 34 (26), E. Padoux and S. Peng, Adaped oluion of a backwad ochaic diffeenial equaion, Sy. Con. Le. 14 (199), , Backwad doubly ochaic diffeenial equaion and yem of quailinea SPDE, Pobab. Theoy el. Field 98 (1994), S. Waanabe, A limi heoem of banching pocee and coninuou ae banching pocee, J. Mah. Kyoo Univ. 8 (1968), J. Xiong, Supe-Bownian moion a he unique ong oluion o an SPDE, Ann. Pobab. 41 (213), T. Yamada and S. Waanabe, On he uniquene of oluion of ochaic diffeenial equaion, J. Mah. Kyoo Univ. 11 (1971), Depamen of Mahemaic, Honghe Univeiy, Mengzi , China and Depamen of Mahemaic, Shandong Univeiy, Jinan 251, China adde: zhdefei@163.com Depamen of Mahemaic, Honghe Univeiy, Mengzi 6611, China adde: hepingky@163.com

Degree of Approximation of a Class of Function by (C, 1) (E, q) Means of Fourier Series

Degree of Approximation of a Class of Function by (C, 1) (E, q) Means of Fourier Series IAENG Inenaional Jounal of Applied Mahemaic, 4:, IJAM_4 7 Degee of Appoximaion of a Cla of Funcion by C, E, q Mean of Fouie Seie Hae Kihna Nigam and Kuum Shama Abac In hi pape, fo he fi ime, we inoduce