Existence and uniqueness of solution for multidimensional BSDE with local conditions on the coefficient

Transcription

1 1/34 Exisence and uniqueness of soluion for mulidimensional BSDE wih local condiions on he coefficien EL HASSAN ESSAKY Cadi Ayyad Universiy Mulidisciplinary Faculy Safi, Morocco ITN Roscof, March 18-23, 21 Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 1 / 34

2 2/34 1. BSDEs Inroducion (Ω, F, (F ) 1, P) be a complee probabiliy space F = σ(b s, s ) N be a filraion Consider he following ODE { dy =, [, T ], Y T = ξ IR. (1) Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 2 / 34

3 2/34 1. BSDEs Inroducion (Ω, F, (F ) 1, P) be a complee probabiliy space F = σ(b s, s ) N be a filraion Consider he following erminal value problem { dy =, [, T ], Y T = ξ L 2 (Ω, F T ; IR). (1) Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 2 / 34

4 2/34 1. BSDEs Inroducion (Ω, F, (F ) 1, P) be a complee probabiliy space F = σ(b s, s ) N be a filraion Consider he following erminal value problem { dy =, [, T ], Y T = ξ L 2 (Ω, F T ; IR). (1) We wan o FIND F -ADAPTED soluion Y for equaion (1). Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 2 / 34

5 2/34 1. BSDEs Inroducion (Ω, F, (F ) 1, P) be a complee probabiliy space F = σ(b s, s ) N be a filraion Consider he following erminal value problem { dy =, [, T ], Y T = ξ L 2 (Ω, F T ; IR). (1) We wan o FIND F -ADAPTED soluion Y for equaion (1). This is IMPOSSIBLE, since he only soluion is which is no F adaped. Y = ξ, for all [, T ], (2) Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 2 / 34

6 2/34 1. BSDEs Inroducion (Ω, F, (F ) 1, P) be a complee probabiliy space F = σ(b s, s ) N be a filraion Consider he following erminal value problem { dy =, [, T ], Y T = ξ L 2 (Ω, F T ; IR). (1) We wan o FINDF -ADAPTED soluion Y for equaion (1). This is IMPOSSIBLE, since he only soluion is Y = ξ, for all [, T ], (2) which is no F adaped. A naural way of making (2) F adaped is o redefine Y. as follows Y = IE(ξ F ), [, T ]. (3) Then Y. is F adaped and saisfies Y T = ξ, bu no equaion (1). Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 2 / 34

7 3/34 1. BSDEs Inroducion MRT = here exiss an F adaped process Z square inegrable s. Y = Y + Z sdb s. (4) I follows ha Combining (4) and (5), one has Y T = ξ = Y + Z sdb s. (5) Y = ξ Z sdb s, (6) whose differenial form is { dy = Z db, [, T ], Y T = ξ. (7) Comparing (1) and (7), he erm Z db has been added. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 3 / 34

8 4/34 1. BSDEs Inroducion BSDE is an equaion of he following ype: Y = ξ + f (s, Y s, Z s)ds Z sdb s, T. (8) T : TERMINAL TIME f : Ω [, T ] IR d IR d n IR d : GENERATOR or COEFFICIENT ξ : TERMINAL CONDITION F T adaped process wih value in IR d. UNKNOWNS ARE : Y IR d and Z IR d n. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 4 / 34

9 5/34 1. BSDEs Inroducion Denoe by L he se of IR d IR d n valued processes (Y, Z) defined on IR + Ω which are F adaped and such ha: (Y, Z) 2 = IE The couple (L,. ) is hen a Banach space. ( sup Y 2 + Z s 2 ds T Definiion ) < +. A soluion of equaion (8) is a pair of processes (Y, Z) which belongs o he space (L,. ) and saisfies equaion (8). Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 5 / 34

10 6/34 2. BSDEs wih Lipshiz coefficien Consider he following assumpions: For all (y, z) IR d IR d n : (ω, ) f (ω,, y, z) is F progressively measurable f (.,, ) L 2 ([, T ] Ω, IR d ) f is Lipschiz : K > and y, y IR d, z, z IR d n and (ω, ) Ω [, T ] s. ( ) f (ω,, y, z) f (ω,, y, z ) K y y + z z. ξ L 2 (Ω, F T ; IR d ) Theorem : Pardoux and Peng 199 Suppose ha he above assumpions hold rue. Then, here exiss a unique soluion for BSDE (15). Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 6 / 34

11 6/34 2. BSDEs wih Lipshiz coefficien Consider he following assumpions: For all (y, z) IR d IR d n : (ω, ) f (ω,, y, z) is F progressively measurable f (.,, ) L 2 ([, T ] Ω, IR d ) f is Lipschiz : K > and y, y IR d, z, z IR d n and (ω, ) Ω [, T ] s. ( ) f (ω,, y, z) f (ω,, y, z ) K y y + z z. ξ L 2 (Ω, F T ; IR d ) Theorem : Pardoux and Peng 199 Suppose ha he above assumpions hold rue. Then, here exiss a unique soluion for BSDE (15). Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 6 / 34

12 7/34 3. APPLICATIONS OF BSDE : FINANCE & PDE Consider a marke where only wo basic asses are raded. BOND : STOCK : Consider a European call opion whose payoff is (X T K) +. The opion pricing problem is : fair price of his opion a ime =? Suppose ha his opion has a price y a ime =. Then he fair price for he opion a ime = should be such a y ha he corresponding opimal invesmen would resul in a wealh process Y saisfying Y T = (X T K) +. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 7 / 34

13 7/34 3. APPLICATIONS OF BSDE : FINANCE & PDE Consider a marke where only wo basic asses are raded. BOND : dx = rx d STOCK : Consider a European call opion whose payoff is (X T K) +. The opion pricing problem is : fair price of his opion a ime =? Suppose ha his opion has a price y a ime =. Then he fair price for he opion a ime = should be such a y ha he corresponding opimal invesmen would resul in a wealh process Y saisfying Y T = (X T K) +. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 7 / 34

14 7/34 3. APPLICATIONS OF BSDE : FINANCE & PDE Consider a marke where only wo basic asses are raded. BOND : dx = rx d STOCK : dx = bx d + σx db Consider a European call opion whose payoff is (X T K) +. The opion pricing problem is : fair price of his opion a ime =? Suppose ha his opion has a price y a ime =. Then he fair price for he opion a ime = should be such a y ha he corresponding opimal invesmen would resul in a wealh process Y saisfying Y T = (X T K) +. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 7 / 34

15 8/34 3. APPLICATIONS OF BSDE : FINANCE & PDE Denoe by R : he amoun ha he wrier invess in he sock Y R : he remaining amoun which is invesed in he bond R deermines a sraegy of he invesmen which is called a porfolio. By seing Z = σr, we obain he following BSDE dx = bx d + σx db dy = (ry + b r σ Z) d + Z db, [, T ], }{{} f (,Y,Z ) X = x, Y T = (X T K) }{{ +. } ξ Pardoux & Peng resul = here exis a unique soluion (Y, Z ). The opion price a ime = is given by Y, and he porfolio is given by R = Z σ. (9) Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 8 / 34

16 9/34 3. APPLICATIONS OF BSDE : FINANCE & PDE Le u be he soluion of he following sysem of semi-linear parabolic PDE s: { u (, x) Tr(σσ u)(, x) + b u(, x) + f (, x, u(, x), uσ(, x)) = u(t, x) = g(x). (1) Inroducing {(Y s,x, Z s,x ) ; s T } he adaped soluion of he backward sochasic differenial equaion { dy = f (, X s,x, Y, Z )ds Z db Y T = g(x,x T ), (11) where (X s,x ) denoes he soluion of he following sochasic differenial equaion { dx = b(, X )d + σ(, X )db X s = x. (12) Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 9 / 34

17 1/34 3. APPLICATIONS OF BSDE : FINANCE & PDE Then we have : u is a classical soluion of PDE (1) = (Y s,x is a soluion he BSDE (11). = u(, X s,x ), Z s,x = u(, X s,x )σ(s, X s,x )) There exiss a soluion o he BSDE (11)= u(, x) = Y,x, is a viscosiy soluion of PDE (1). This formula is a generalizaion of Feynman-Kac formula. Suppose ha f (, x, y, z) = c(, x)y + h(, x), we obain Y,x [ =IE g(x,x 1 ) exp 1 + ( 1 h(s, Xs,x ) exp which is he classical Feynman-Kac formula. ) c(r, Xr,x )dr ( s ) ] c(r, Xr,x )dr ds, Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 1 / 34

18 11/34 4. BSDEs wih locally Lipschiz coefficien Consider he following assumpions: (A1) f is coninuous in (y, z) for almos all (, ω). (A2) There exis K > and α 1 such ha f (, ω, y, z) K(1+ y α + z α ). (A3) For each N >, here exiss L N such ha: f (, y, z) f (, y, z ) L N ( y y + z z ) y, y, z, z N. (A4) ξ L 2 (Ω, F T ; IR d ) Theorem : Bahlali 22 Assume moreover ha here exiss a posiive consan L such ha L N = L + log N hen here exiss a unique soluion for he BSDE (1). Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 11 / 34

19 11/34 4. BSDEs wih locally Lipschiz coefficien Consider he following assumpions: (A1) f is coninuous in (y, z) for almos all (, ω). (A2) There exis K > and α 1 such ha f (, ω, y, z) K(1+ y α + z α ). (A3) For each N >, here exiss L N such ha: f (, y, z) f (, y, z ) L N ( y y + z z ) y, y, z, z N. (A4) ξ L 2 (Ω, F T ; IR d ) Theorem : Bahlali 22 Assume moreover ha here exiss a posiive consan L such ha L N = L + log N hen here exiss a unique soluion for he BSDE (1). Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 11 / 34

20 12/34 5. BSDEs wih locally monoone coefficien Quesion Quesion Le f (y) := y log y. Suppose ha ξ L 2 (F T ) or ξ L p (F T ), p > 1 and consider he following BSDE wih logarihmic nonlineariy Does his equaion has a unique soluion? Y = ξ Y s log Y s ds Z sdw s. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 12 / 34

21 13/34 5. BSDEs wih locally monoone coefficien-assumpions Consider he following assumpions: (H1) f is coninuous in (y, z) for almos all (, ω), (H2) There exis M >, γ < 1 2 and η L1 ([, T ] Ω) such ha, y, f (, ω, y, z) η + M y 2 + γ z 2 P a.s., a.e. [, T ]. (H3) Almos quadraic growh : M 1 >, α < 2, α > 1 and η L α ([, T ] Ω) s. : f (, ω, y, z) η + M 1 ( y α + z α ). Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 13 / 34

22 14/34 5. BSDEs wih locally monoone coefficien (H4) There exiss a real valued sequence (A N ) N>1 and consans M > 1, r > 1 such ha: i) N > 1, 1 < A N N r. ii) lim N A N =. iii) Locally monoone condiion : For every N IN, y, y, z, z such ha y, y, z, z N, we have y y, f (, y, z) f (, y, z ) M y y 2 loga N + M y y z z log A N + MA 1 N. Theorem : Bahlali-Essaky-Hassani-Pardoux, 22 Le ξ be a square inegrable random variable. Assume ha (H1) (H4) are saisfied. Then he BSDE has a unique soluion. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 14 / 34

23 15/34 1. L p -soluions o BSDEs wih super-moivaion Le s menion some consideraions which have moivaed he presen work. The growh condiions on he nonlineariy consiue a criical case. Indeed, i is known ha for any ε >, he soluions of he ordinary differenial equaion X = x + X 1+ε s ds explode a a finie ime. The logarihmic nonlineariies appear in some PDEs arising in physics. In erms of coninuous-sae branching processes, he logarihmic nonlineariy u log u corresponds o he Neveu branching mechanism. This process was inroduced by Neveu. For insance, he super-process wih Neveu s branching mechanism is relaed o he Cauchy problem, { u u + u log u = on (, ) IRd (13) u( + ) = ϕ > Hence, our resul can be seen as an alernaive approach o he PDEs. I is worh noing ha our condiion on he coefficien f is new even for he classical Iô s forward SDEs. s s X s = x + X r log X r dr + X r log X r dw r, s T. (14) For insance, he problem o esablish he exisence of a pahwise unique soluion o he following équaion (14) sill remains open. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 15 / 34

24 16/34 5. BSDEs wih locally monoone coefficien Example Example Le f (y) := y log y hen for all ξ L 2 (F T ) he following BSDE has a unique soluion Y = ξ Y s log Y s ds Z sdw s. Indeed, f saisfies (H.1)-(H.3) since y, f (y) 1 and f (y) ε y 1+ε for all ε >. (H.4) is saisfied for every N > e and A N = N. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 16 / 34

25 17/34 5. BSDEs wih locally monoone coefficien Idea of he proof We define a family of semi-norms ( ρ n(f ) ) n IN by, ρ n(f ) = IE sup f (s, y, z) ds. y, z n We Approximae f by a sequence (f n) n>1 of Lipschiz funcions : Lemma Le f be a process which saisfies (H.1) (H.3). T hen here exiss a sequence of processes (f n) such ha, (a) For each n, f n is bounded and globally Lipschiz in (y, z) a.e. and P-a.s.ω. There exiss M >, such ha: (b) sup n f n(, ω, y, z) η + M + M 1 ( y α + z α ). P-a.s., a.e. [, T ]. (c) sup < y, f n(, ω, y, z) > η + M + M y 2 + γ z 2 n (d) For every N, ρ N (f n f ) as n. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 17 / 34

26 18/34 5. BSDEs wih locally monoone coefficien Key seps of he proof We consider he following BSDE Y n = ξ + f n(s, Ys n, Z s n )ds Zs n dws, T. Lemma There exis a universal consan l such ha a) IE e 2Ms Z fn s 2 ds [ ] 1 e 2MT IE ξ 2 +2IE e 2Ms (η + M )ds = K 1 1 2γ b) IE sup (e 2M Y fn 2 ) lk 1 = K 2 T c) IE e 2Ms f n(s, Ys fn s fn ds ] 4 [IE α 1 e 2Ms ((η + M ) α + 4)ds + M1 α K 1 + TM1 α K 2 = K 3 d) IE e 2Ms f (s, Ys fn s fn ds K 3, where α = min(α, 2 α ). Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 18 / 34

27 19/34 Hence he following convergences hold rue Y n Y, weakly sar in L 2 (Ω, L [, T ]) Z n Z, weakly in L 2 (Ω [, T ]) f n(., Y n, Z n ) Γ. weakly in L α (Ω [, T ]), Moreover Y = ξ + Γ sds Z sdw s, [, T ]. We Apply Iô s formula o ( Y n Y m 2 + ε) p for some < p < 1, insead of Y n Y m 2 : lim n + ( IE sup T ) Y n Y β + IE Zs n Zs ds =, 1 < β < 2. We Idenify Γ s by proving ha : lim E n f n(s, Ys n, Z s n ) f (s, Ys, Zs) ds =. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 19 / 34

28 2/34 6. L p -soluions o BSDEs wih locally monoone coefficien-definiion Le p > 1 is an arbirary fixed real number and all he considered processes are (F )-predicable. Definiion A soluion of equaion (8 is an (F )-adaped and IR d+dr -valued process (Y, Z) such ha IE sup Y p + IE T [ Z s 2 ds ] p 2 T + IE f (s, Y s, Z s) ds < + and saisfies Y = ξ + f (s, Y s, Z s)ds Z sdb s, T. (15) l Hassan Essaky Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 2 / 34

29 21/34 6. L p -soluions o BSDEs wih locally monoone coefficien-assumpions We consider he following assumpions on (ξ, f ): (H.) There are M L (Ω; L 1 ([, T ]; IR +)), K L (Ω; L 2 ([, T ]; IR +)) and γ ], such ha: IE ξ p e p 2 λ sds (H.1) f is coninuous in (y, z) for almos all (, ω). <, where λ s := 2M s + K 2 s 2γ 1 (p 1) [ 2 (H.2) s p 2 λ r dr There are η and f L (Ω [, T ]; IR +) saisfying IE e η sds < s p 1 λ 2 r dr and IE e fs ds <, where λ is defined in assumpion (H.), such ha: y, f (, y, z) η + f y + M y 2 + K y z. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 21 / 34

30 22/34 6. L p -soluions o BSDEs wih locally monoone coefficien-assumpions { There are η L q (Ω [, T ]; IR +)) (for some q > 1) and α ]1, p[, α ]1, p 2[ (H.3) (H.4) such ha: f (, ω, y, z) η + y α + z α. There are v L q (Ω [, T ]; IR +)) (for some q > ) and K IR + such ha for every N IN and every y, y z, z saisfying y, y, z, z N 1 v (ω) N y y, f (, ω, y, z) f (, ω, y, z ) K log A N y y 2 + K log A N y y z z +K log A N A N where A N is a increasing sequence and saisfies A N > 1, lim N A N = and A N N µ for some µ >. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 22 / 34

31 23/34 6. L p -soluions o BSDEs wih locally monoone coefficien-exisence and Uniqueness Theorem : Bahlali-Essaky-Hassani If (H.)-(H.4) hold hen (8) has a unique soluion (Y, Z). Moreover we have IE sup { C Y p p [ 2 e λs ds + IE IE ξ p p 2 e e s λr dr Z s 2 ds s ( λ s ds + IE e λr dr η sds for some consan C depending only on p and γ. ) p 2 ] p 2 + IE ( e 1 2 s λr dr f s ds ) p }. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 23 / 34

32 24/34 6. L p -soluions o BSDEs wih locally monoone coefficien-examples Example 1 Le f (y) := y log y hen for all ξ L p (F T ) he following BSDE has a unique soluion Y = ξ Y s log Y s ds Z sdw s. Indeed, f saisfies (H.1)-(H.3) since y, f (y) 1 and f (y) ε y 1+ε for all ε >. (H.4) is saisfied for every N > e wih v s = and A N = N. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 24 / 34

33 25/34 6. L p -soluions o BSDEs wih locally monoone coefficien-examples Le g(y) := y log Example 2 y 1+ y and h C(IRdr ; IR +) C 1 (IR dr {}; IR +) be such ha : h(z) = { z log z if z < 1 ε z log z if z > 1 + ε, where ε ], 1[. Finally, we pu f (y, z) := g(y)h(z). Then for every ξ L p (F T ) he following BSDE has a unique soluion Y = ξ + f (Y s, Z s)ds Z sdw s. (H.4) is saisfied for every N > e wih v s = and A N = N Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 25 / 34

34 26/34 6. L p -soluions o BSDEs wih locally monoone coefficien-examples Example 3 Le (X ) T be an (F ) adaped and IR k valued process saisfying : X = X + b(s, X s)ds + σ(s, X s)dw s, where X IR k and σ, b : [, T ] IR k IR kr IR k are measurable funcions such ha σ(s, x) c and b(s, x) c(1 + x ), for some consan c. Consider he following BSDE Y = g(x T ) + ( X s q Y s Y s log Y s )ds Z sdw s. where q ], 2[ and g is a measurable funcion saisfying g(x) c exp c x q, for some consans c >, q [, 2[. The previous BSDE has a unique soluion (Y, Z) such ha IE sup Y p +IE [ Z s 2 ds ] p 2 C exp (C X 2 ). (H.4) is saisfied wih v s = exp X s q and A N = N. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 26 / 34

35 27/34 6. L p -soluions o BSDEs wih locally monoone coefficien-examples Example 4 Le (ξ, f ) saisfying (H.)-(H.3) and (H.4) There are a posiive process C saisfying IE e q C s ds < and K IR + such ha: ) y y, f (, ω, y, z) f (, ω, y, z ) K y y (C 2 (ω)+ log( y y ) +K y y z z C (ω)+ log z z. Then he following BSDE has a unique soluion Y = ξ + f (s, Y s, Z s)ds Z sdw s. (H.4) is saisfied wih v s = exp (C s) and A N = N. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 27 / 34

36 28/34 6. L p -soluions o BSDEs wih locally monoone coefficien-idea of he proof We Approximae f by a sequence (f n) n>1 of Lipschiz funcions : f n(, y, z) = (c 1 e) 2 1 {Λ n} ψ(n 2 y 2 )ψ(n 2 z 2 ) m (d+dr) IR d IR dr f (, y u, z v)π d i=1 ψ(mu i )Π d i=1 Πr j=1 ψ(mv ij)dudv, wih m := n2p and Λ := η + η h + f + M + K + 1 where h h is a predicable process such ha < h 1. We consider he following BSDE Y n = ξ1 { ξ n} + f n(s, Ys n, Z s n )ds Zs n dws, T. We Apply Iô s formula o ({ Y n Y m 2 + ε}( 1 ε )2C ) β 2 for some 1 < β < p 2, insead of Y n Y m 2 we have: For every p < p, (Y n, Z n ) (Y, Z) srongly in L p (Ω; C([, T ]; IR d )) L p (Ω; L 2 ([, T ]; IR dr )). For every ˆβ < 2 α p α p α q lim IE n f n(s, Y n s, Z n s ) f (s, Ys, Zs) ˆβ ds =. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 28 / 34

37 29/34 1. L p -soluions o BSDEs wih super-linear growh coefficien-applicaion o PDEs { Consider he following sysem of semilinear PDE (P (g,f ) ) u(, x) + Lu(, x) + F (, x, u(, x), σ u(, x)) = ], T [, x IR k u(t, x) = g(x) x IR k where L := 1 (σσ ) ij ij b i i, σ Cb 3(IRk, IR kr ), b Cb 2(IRk, IR k ). Le H 1+ := δ,β>1 i,j i { v C([, T ]; L β (IR k, e δ x dx; IR d )) : IR k σ v(s, x) β e δ x dxds < }. Definiion A (weak) soluion of (P (g,f ) ) is a funcion u H 1+ such ha for every ϕ C c 1 T < u(s), ϕ(s) > ds+ < u(), ϕ() > s =< g, ϕ(t ) > + where < f (s), h(s) >= IR < F (s,., u(s), σ T u(s)), ϕ(s) > ds + k f (s, x)h(s, x)dx. < Lu(s), ϕ(s) > ds, Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 29 / 34

38 3/34 1. L p -soluions o BSDEs wih super-linear growh coefficien-applicaion o PDE (A.) g(x) L p (IR k, e δ x dx; IR d ). (A.1) F (, x,.,.) is coninuous a.e. (, x) There are η L p 2 1 ([, T ] IR k, e δ x ddx; IR +)), (A.2) f L p ([, T ] IR k, e δ x ddx; IR +)), and M, M IR + such ha y, F (, x, y, z) η (, x) + f (, x) y + (M + M x ) y 2 + M + M x y z. (A.3) There are η L q ([, T ] IR k, e δ x ddx; IR +)) (for some q > 1), α ]1, p[ and α ]1, p 2[ such ha F (, x, y, z) η (, x) + y α + z α. (A.4) There are K, r IR + such ha N IN and every e r x, y, y, z, z N, y y ; F (, ( x, y, z) F (, ) x, y, z ) 1 K log N N + y y 2 + K log N y y z z. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 3 / 34

39 31/34 1. Exisence and uniqueness of soluions o PDE Consider he diffusion process wih infiniesimal operaor L s s Xs,x = x + b(xr,x )dr + σ(xr,x )dw r, s T Theorem : Bahlali-Essaky-Hassani Under assumpion (A.)-(A.4) we have 1) The PDE (P (g,f ) ) has a unique soluion u on [, T ] 2) For all [, T ] here exiss D IR k such ha i) D c 1 dx = ii) For all [, T ] and all x D (E (ξ,x,f,x ) ) has a unique soluion (Y,x, Z,x ) on [, T ] where ξ,x := g(x,x T ) and f,x (s, y, z) := 1 {s>} F (s, Xs,x, y, z) 3) For all [, T ] ( u(s, X,x s ), σ u(s, X,x s ) ) = ( Y,x s ), Zs,x a.e.(s, x, ω) Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 31 / 34

40 32/34 1. Exisence and uniqueness of soluions o PDE-Idea of he proof We Approximae F by a sequence (F n) n>1 of Lipschiz funcions : F n(, x, y, z) = (n 2p e x ) (d+dr) (c 1 e) 2 1 {η (,x)+η (,x)+f (,x)+ x n} ψ(n 2 y 2 )ψ(n 2 z 2 ) IR d IR dr F (, x, y u, z v)πd i=1 ψ(n2p e x u i )Π d i=1 Πr j=1 ψ(n2p e x v ij )dudv, We consider (Y,x,n, Z,x,n ) be he unique soluion of BSDE (1) avec ξ,x n and fn,x (s, y, z) := 1 {s>} F n(s, Xs,x, y, z), wih g n(x) := g(x)1 { g(x) n}. There exiss a unique soluion u n of PDE (P (gn,fn) ) { u n (, x) := g n(x,x T ) + Lu n (, x) + F n(, x, u n (, x), σ u n (, x)) = ], T [, x IR k u n (T, x) = g n(x) x IR k such ha for all u n (s, Xs,x ) = Y,x,n s and σ u n (s, Xs,x ) = Z,x,n s We have he following convergence : lim sup u n (, x) u m (, x) p e δ x dx = n,m T IR k T lim n,m IR k a.e (s, ω, x). σ u n (, x) σ u m (, x) p 2 e δ x ddx =. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 32 / 34

41 33/34 1. Exisence and uniqueness of soluions o PDE-Idea of he proof For uniqueness we prove ha he sysem of semilinear PDEs { u(, x) + Lu(, x) + f (, x, u(, x), u(, x)) =, ], T [, x IR k u(t, x) = g(x), x IR k has a unique soluion if and only if is he unique soluion of he linear sysem { u(, x) + Lu(, x) =, ], T [, x IR k u(t, x) =, x IR k Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 33 / 34

42 34/34 For Furher Reading K. Bahlali Exisence and uniqueness of soluions for BSDEs wih locally Lipschiz coefficien. Elecron. Comm. Probab., 7, , 22. K. Bahlali, E.H. Essaky, M. Hassani, E. Pardoux Exisence, uniqueness and sabiliy of backward sochasic differenial equaions wih locally monoone coefficien. C. R. Acad. Sci., Paris 335, no. 9, , 22. K. Bahlali, E.H. Essaky, M. Hassani, E. Pardoux p-inegrable soluions o mulidimensional BSDEs and degenerae sysems of PDEs wih logarihmic nonlineariies. Preprin. Mulidisciplinary Faculy (Cadi Exisence-uniqueness Ayyad Universiy of soluion Mulidisciplinary for BSDE Faculy Safi, Morocco ITN Rosco 34 / 34