Backward doubly stochastic di erential equations with quadratic growth and applications to quasilinear SPDEs

Transcription

1 Backward doubly sochasic di erenial equaions wih quadraic growh and applicaions o quasilinear SPDEs Badreddine MANSOURI (wih K. Bahlali & B. Mezerdi) Universiy of Biskra Algeria La Londe 14 sepember 2007 B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 1 / 28

2 Inroducion Y = ξ + f (s, Y s, Z s ) ds Z s dw s B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 2 / 28

3 Inroducion Y = ξ + f (s, Y s, Z s ) ds Z s dw s u + Lu F (, x, u, σ (, x) Du) = 0 in (0, T ) Rn u (T, x) = g (x) in R n. Lu = 1 2 n a ij (, x) i,j=1 2 u x i x j n b i (, x) u, i,j=1 x i B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 2 / 28

4 Inroducion Y = ξ + f (s, Y s, Z s ) ds + g (s, Y s, Z s ) db s Z s dw s B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 3 / 28

5 Inroducion Y = ξ + f (s, Y s, Z s ) ds + g (s, Y s, Z s ) db s Z s dw s u (, x) = h (x) + s [L s u (s, x) + f (x, u (s, x), (ruσ) (s, x))] ds + g (x, u (s, x), (ruσ) (s, x)) db s s where u 2 R k (Lu) (, x) = (Lu i ) (, x) and L = 1 2 d (σσ 2 ) ij i,j=1 x i x j + 1 i k d b i i=1 x i B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 3 / 28

6 Inroducion Boccardo, Mura, Puel (83), Exeisence de soluions faibles pour des équaions ellipiques quasi-linéaires à croissance quadraique. B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 4 / 28

7 Inroducion Boccardo, Mura, Puel (83), Exeisence de soluions faibles pour des équaions ellipiques quasi-linéaires à croissance quadraique. Boccardo, Mura, Puel (88), Exeisence resuls for some quasilinear parabolic equaiopn. B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 4 / 28

8 Inroducion Boccardo, Mura, Puel (83), Exeisence de soluions faibles pour des équaions ellipiques quasi-linéaires à croissance quadraique. Boccardo, Mura, Puel (88), Exeisence resuls for some quasilinear parabolic equaiopn. Barles, Mura (95), Uniqueness and he maximum principale for quasilinear ellipic equaion wih quadraic growh condiions. B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 4 / 28

9 Inroducion Boccardo, Mura, Puel (83), Exeisence de soluions faibles pour des équaions ellipiques quasi-linéaires à croissance quadraique. Boccardo, Mura, Puel (88), Exeisence resuls for some quasilinear parabolic equaiopn. Barles, Mura (95), Uniqueness and he maximum principale for quasilinear ellipic equaion wih quadraic growh condiions. M. Kobylanski, Backward sochasic di erenial equaions and parial di erenial equaions wih quadraic growh, Ann of probabiliy vol 28 (2000) No 2, B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 4 / 28

10 Inroducion Backward doubly sochasic di erenial equaion (BDSDE) are equaion of he following ype : Y = ξ + f (s, Y s, Z s ) ds + g (s, Y s ) db s Z s dw s, where he dw is a forward Iô inegral and he db is a backward Iô inegral. 0 T (1.1) Pardoux-Peng (1994) are inroduced his ype of equaions, and hey have proved he exisence and uniqueness of soluions for BDSDEs when g and f is uniformly Lipschiz. B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 5 / 28

11 Inroducion Backward doubly sochasic di erenial equaion (BDSDE) are equaion of he following ype : Y = ξ + f (s, Y s, Z s ) ds + g (s, Y s ) db s Z s dw s, where he dw is a forward Iô inegral and he db is a backward Iô inegral. 0 T (1.1) Pardoux-Peng (1994) are inroduced his ype of equaions, and hey have proved he exisence and uniqueness of soluions for BDSDEs when g and f is uniformly Lipschiz. Y. Shi, Y. Gu and K. Liu (2005), have proved he exisence of soluion for BDSDE when g is uniformly lipischiz and f coninuous, wih sub-linear growh in y and z. B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 5 / 28

12 Inroducion Backward doubly sochasic di erenial equaion (BDSDE) are equaion of he following ype : Y = ξ + f (s, Y s, Z s ) ds + g (s, Y s ) db s Z s dw s, where he dw is a forward Iô inegral and he db is a backward Iô inegral. 0 T (1.1) Pardoux-Peng (1994) are inroduced his ype of equaions, and hey have proved he exisence and uniqueness of soluions for BDSDEs when g and f is uniformly Lipschiz. Y. Shi, Y. Gu and K. Liu (2005), have proved he exisence of soluion for BDSDE when g is uniformly lipischiz and f coninuous, wih sub-linear growh in y and z. In he presen noe, we consider he case where g is uniformly lipischiz and f is coninuous and quadraic growh in z. We prove he exisence a minimal and a maximal soluion. B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 5 / 28

13 BDSDE wih quadraic growh Le f : Ω [0, T ] R R d! R g : Ω [0, T ] R! R be measurable and such ha for any (y, z) 2 R R d, f (., y, z) 2 M 2 0, T, R d g (., y) 2 M 2 (0, T, R) Moreover, we assume ha here exis consans C > 0 and K > 0, such ha for any (ω, ) 2 Ω [0, T ] and (y, z) 2 R R d, 8 >< jf (, y, z)j C 1 + jzj 2 jg (, y 1 ) g (, y 2 )j C jy 1 y 2 j (H1) >: g (, y) K le us assume ha ξ 2 S T ([0, T ], R). B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 6 / 28

14 BDSDE wih quadraic growh De niion A pair of process (y, z) : Ω [0, T ]! R R d is soluion of BDSDE (1.1) if (y, z) 2 S T ([0, T ] ; R) M2 T 0, T ; Rd, and sais es BDSDE (1.1). B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 7 / 28

15 Exisence resul Theorem Under Assumpion (H1), BDSDE (1.1) has a soluion (Y, Z ) 2 S T ([0, T ], R) M2 T 0, T, Rd. Moreover, his equaion admis a minimal soluion (Y, Z ) and a maximal soluion (Y, Z ). B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 8 / 28

16 Uniqueness resul We say ha he coe cien f sais es condiion (H1)-(H2) wih some consans C, C 1 > 0, if for every 2 R +, 8 < jf (, y, z)j C 1 + jzj 2, P p.s : f z (, y, z) (H2) C (1 + jzj) Theorem f (, y, z) y C jzj 2, P p.s. (H3) Under Assumpions (H2)-(H3), BDSDE (1.1) has a uinque soluion in S T ([0, T ], R) M2 T 0, T, Rd. B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 9 / 28

17 BDSDE and quasilinear SPDEs Le b 2 C 3 R d, R d, and σ 2 C 3 R d, R d d. For (, x) 2 [0, T ] R d, le fxs,x, s T g denoe he unique soluion of he following SDE : Le h 2 C 2 f(y,x s BDSDE:, Z,x s dx,x s = b X,x s X,x s = x, ds + σ X,x s dws, s T (1.2) R d, R. For (, x) 2 [0, T ] R d, le ) ; s T g denoe he unique soluion of he following s = h X,x + f Y,x T s s Zr,x dw s r, X,x r, Y,x r, Zr,x dr + g s r, X,x r, Yr,x dbr (1.3) B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 10 / 28

18 BDSDE and quasilinear SPDEs We will relae BDSDE (1.3) o he following quasilinear backward sochasic parial di erenial equaions: u (, x) = h (x) + [Lu (s, x) + f (s, x, u (s, x), (ruσ) (s, x))] ds + g (s, x, u (s, x)) db s (1.4) where u : R + R d! R, wih L = 1 2 d i,j=1 (σσ ) ij (, x) 2 x i x j Theorem + d i=1 b i (, x) x i. Le f, g saisfy assumpion (H1). Le fu (, x), (, x) 2 [0, T ] R d g be a random eld such ha u(, x) is F,T B measurable for each (, x), u 2 C 0,2 [0, T ] R d, R a.s, and u sais es equaion (1.4). Then u (, x) = Y,x, where f(ys,x, Zs,x ) ; s T g 0,x 2R d is he unique soluion of he BDSDE (1.3). B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 11 / 28

19 Proof of he exisence Proposiion Le (Y, Z ) 2 ST (R) M2 T Rd be a soluion of BDSDE (1.1), wih parameers (f, g, ξ), and suppose ha f sais es (H1). Then for all 0 T, ( resp. Y E " # + Y E sup Y T /F + C (T ) a.s Ω " inf Ω Y T /F # C (T ) a.s). Moreover, here exiss a consan ec depending on ky k and C such ha E 0 jz s j 2 ds ec. B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 12 / 28

20 Proof of he exisence Consider he soluion ϕ of he sochasic di erenial equaion : + ϕ = sup Y T + Cds + g (s, ϕ s ) db s C > 0. Ω Our aim is o prove ha Y ϕ. Applying Iô s formula o he process Y ϕ, wih he increasing C 2 funcion G given by e 2Cu 1 2Cu 2C G (u) = 2 u 2 for u 2 [0, M] 0 for u 2 [ M, 0] where M = ky k + kϕk, Applying Gronwall s lemma we ge: EG (Y ϕ ) = 0, 8 2 [0, T ], We obain Y ϕ P a.s. Hence, " # + Y E sup Y T /F + C (T ) = M +. Ω B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 13 / 28

21 Proof of he exisence The second inequaliy will be proved by using similar argumens. Indeed, le ψ be he soluion of he SDE : ψ = inf Y T Ω Cds + g (s, ψ s ) db s. This implies ha Y ψ, P a.s, and hen " # Y E inf Y T /F C (T ) = M. P a.s. Ω We shall prove he inequaliy E R T 0 jz s j 2 ds ec. Le M = ky k and he funcion H de ned on [ M, M] by H (y) = 1 h 2C (y +M ) 2C 2 e (1 i + 2C (y + M)), B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 14 / 28

22 Proof of he exisence Io s formula applied o H (Y ) i holds ha: EH (Y 0 ) + E 0 jz s j 2 ds H (M) + M 2 Ce 4CM T, which leads E R T 0 jz s j 2 ds H (M) + M 2 Ce 4CM T = ec B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 15 / 28

23 Proof of he exisence 1 K. Bahlali, S. Hamadene and B, Mezerdi, Backward sochasic di erenial equaion wih wo re ecing barriers and coninuous wih quadraic growh coe cien, Sochasic processes and heir applicaions 115 (2005) pp B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 16 / 28

24 Proof of he exisence 1 K. Bahlali, S. Hamadene and B, Mezerdi, Backward sochasic di erenial equaion wih wo re ecing barriers and coninuous wih quadraic growh coe cien, Sochasic processes and heir applicaions 115 (2005) pp M. Kobylanski, Backward sochasic di erenial equaions and parial di erenial equaions wih quadraic growh, Ann of probabiliy vol 28 (2000) No 2, B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 16 / 28

25 Proof of he exisence 1 K. Bahlali, S. Hamadene and B, Mezerdi, Backward sochasic di erenial equaion wih wo re ecing barriers and coninuous wih quadraic growh coe cien, Sochasic processes and heir applicaions 115 (2005) pp M. Kobylanski, Backward sochasic di erenial equaions and parial di erenial equaions wih quadraic growh, Ann of probabiliy vol 28 (2000) No 2, J.P. Lepelier, J. San-Marin, Exisence for BSDE wih superlinear-quadraic coe cien, Sochasic Sochasic Rep. 63 (1998) B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 16 / 28

26 Proof of he exisence Le us consider he following BDSDE X = η + F (s, X S, Λ s ) ds + G (s, X s ) db s Λ s dw s. (1.5) Theorem Assume he following hypoheses. (i) η be bounded and F T measurable random variable wih values in R. (ii) F : Ω [0, T ] ]0, [ R d! R a P measurable funcion, coninuous in (x, λ) and saisfying he following srucure condiion : 9C > 0 el que 8, x, λ P p.s 2C 2 x C jλj 2 F (s, x, λ) 2C 2 x and jg (, x)j 2CK jxj. Then BDSDE (1.5) has a maximal soluion (X, Λ ). B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 17 / 28

27 Proof of he exisence Following Proposiion 3, we have M Y M +, hen m = e 2cM X e 2cM + = M Sep 1 Exisence of X. Le γ p : R d! R be a smooh funcion saisfying : le us de ne he funcion γ p (λ) = 1 si jλj 1 0 si jλj 1 F p (, x, λ) = 2c 2 x 1 γ p (λ) + γ p (λ) F (, x, λ). we have lim p! & F p = F. B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 18 / 28

28 Proof of he exisence Applying heorem he exisence in Y. Shi e al. he BDSDE associaed wih (F p, G, η) has a soluion (X p, Λ p ). Again by applying he comparison heorem proved in Y. Shi e al., i follows ha M X p X p+1 m. Since F p+1 F p, here exiss a process X such ha X = lim X p 8 T, P p.s. p! In addiion P a.s, for all T, M X m. B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 19 / 28

29 Proof of he exisence Sep 2. There exiss a subsequence of (Λ p ) p1 which converges in M 2. The sequence (Λ p ) p1 is bounded in M 2 by proposiion 3, hen here exiss a subsequence of (Λ p ) p1, which we sill denoe by (Λ p ) p1, which converges weakly in M 2 o a process Λ := (Λ ) T. Le θ = max 1 m, 4c2 M, φ (x) = e 12θx 1 12θ x and p q, hen we have X p X q. On he oher hand, by using Iô s formula wih φ (X p X q ) he fac (Λ p ) p1 converge weakly in M 2, from which we deduce ha lim E jλ p p! s Λ s j 2 ds = 0. Hence (Λ p s ) p1 converges o Λ s. B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 20 / 28

30 Proof of he exisence Sep 3. The process X = (X ) T is coninuous. We shall prove ha (X p ) converges uniformly o X in L 2. Le p q, applying Iô s formula o (X p X q ) 2, and applying Burkholder-Davis-Gundy inequaliy, aqnd he fac F and F p are coninuous, and (F p ) is a decreasing sequence converging o F, hen by Dini s heorem (F p (,.,.)) converges o F (,.,.) uniformly. On he oher hand, since (Λ p ) converge in M 2 o Λ, hen here exiss eλ 2 M 2 and a subsequence, which we sill denoe (Λ p ) p1 such ha (Λ p ) converge o Λ and sup p1 jλ p j eλ. Then F p (s, Xs p, Λ p s ) converges o F (s, Xs p, Λ p s ), d dp p.s. Moreover for consan C 1. F p (s, Xs p, Λ p s ) C jλ p j 2 C Λ e p 2 B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 21 / 28

31 Proof of he exisence Finally, since he sequence (X p ) p1 is bounded, he Lebesgue dominaed convergence heorem implies ha 0 E (Xs p Xs q ) (F p (s, Xs p, Λ p s ) F q (s, Xs q, Λ q s )) ds! 0 p,q! 0 Hence, he sequence (X p ) p1 converges uniformly o X in L 2 and hen X is coninuous. B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 22 / 28

32 Proof of he exisence Sep 4. Maximal and minimal soluions Le (X, Λ) and (X, Λ ) wo soluions of (1.5). For p 1 and n 1, le F n p be he funcion de ned by : F n p (, x, λ) = sup ff p (, u, v) n (ju xj + jv λj)g. u,v 2R 2 Since we have jf p (, x, λ)j C jλj 2, he funcion F n p is Lipschiz in (x, λ) and converges of p as n!. Now le X n p, Λ n p be a soluion of he BDSDE associed wih F n p, G, η. Since F n p F p F, X n p X for all n, p 1, hen for all p 1 we have lim n! X n p = X p. Therefore X p X and naly X X which implies ha he soluion considered is maximal. The exisence of a minimal soluion is based on he same argumens. Indeed, le (X, Λ ) anoher soluion of (1.5). For all p 1 and n 1, le F n p be he funcion de ned by F n p (, x, λ) = inf ff p (, u, v) + n (ju xj + jv λj)g. u,v 2R 2 B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 23 / 28

33 Proof of he exisence Since we have jf p (, x, λ)j C p 2, he funcion Fp n is Lipschiz in (x, λ) and converges o F p as n!. Now le Xp n, Λ n p be he soluion of he BDSDE associed wih Fp n, G, η. Since Fp n F p, Xp n X for all n, p 1. herefore for all p 1, we have lim n! Xp n = X p. Hence X p X and naly X X, which implies ha he soluion considered is minimal. B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 24 / 28

34 Proof of he exisence Proof of heorem (1.4) We ransform equaion (1.1) by he exponenial funcion X = e 2cY, we are led o solving he following BDSDE : X = η + F (s, X S, Λ s ) ds + G (s, X s ) db s Λ s dw s. (1.5) Applying Iô s formula o X = e 2cY, we ge X = X T cX s f (s, Y s, Z s ) ds 4c 2 X s jg (s, Y s )j cX s g (s, Y s ) db s + 2cX s Z s dw 4c 2 X s jz s j 2 ds, B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 25 / 28

35 Proof of he exisence where F (s, x, λ) = 2cx G (s, x) = 2cxg " f s, ln x s, ln x 2c 2c, λ 2cx, + c g s, ln x 2 2c # jλj 2 4cx 2 and X s = e 2cY s. This implies in paricular ha Y s = ln X s 2c, Z s = Λ s 2cX s η = e 2cξ. I is no di cul o show ha he generaor F (s, x, λ) sais es he srucure condiion, and hen we are in a posiion o apply heorem 5, o prove exisence of a maximal soluion o he BDSDE (1.5) and hen he same can be claimed for equaion (1.1) wih Y s = ln X s 2c, Z s = Λ s 2cX s. and B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 26 / 28

36 References K. Bahlali, S. Hamadene and B, Mezerdi, Backward sochasic di erenial equaion wih wo re ecing barriers and coninuous wih quadraic growh coe cien, Sochasic processes and heir applicaions 115 (2005) pp G. Barles and F. Mura, Uniqueness and he maximum principle for quasilinear ellipic equaions wih quadraic growh condiions, Arch. Raional Mech. Anal. 133 (1995) M. Eddahbi, Y. El Qalli, From empirical observaions o modeling The Forward Rae via SPDE. Preprin, universié Cadi Ayyad, Marrakech, Maroc. Submied M. Kobylanski, Backward sochasic di erenial equaions and parial di erenial equaions wih quadraic growh, Ann of probabiliy vol 28 (2000) No 2, E. Pardoux, S. Peng, Backward souchasic di erenial equaions and quasilinear parabolic parial di erenial equaion, In rozuvskii, B.L., B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 27 / 28

37 Thank you for your aenion B. Mansouri (Biskra-Algeria) BDSDE wih quadraic growh 14/09 28 / 28