Uniqueness of solutions to quadratic BSDEs. BSDEs with convex generators and unbounded terminal conditions

Transcription

1 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs On he uniqueness of soluions o quadraic BSDEs wih convex generaors and unbounded erminal condiions IRMAR, Universié Rennes 1 Châeau de Mons Uniqueness of soluions o quadraic BSDEs

2 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs 1 Recalls and basic resuls on BSDEs Wha is a BSDE? Wha we know abou quadraic BSDEs 2 Uniqueness resul Framework and ools skech of he proof Consrucion of he conrol problem 3 Links wih PDEs Uniqueness of soluions o quadraic BSDEs

3 Definiion Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Wha is a BSDE? Wha we know abou quadraic BSDEs Le (Ω, F, P) be a probabiliy space, (W ) R + be a Brownian moion in R d, (F ) R + be his augmened naural filraion, T be a nonnegaive real number, ξ a real F T -measurable random variable, g : [0, T ] Ω R R 1 d R. Y = ξ g(r, Y r, Z r )dr + Z r dw r, 0 T. (1.1) Uniqueness of soluions o quadraic BSDEs

4 Definiion Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Wha is a BSDE? Wha we know abou quadraic BSDEs Le (Ω, F, P) be a probabiliy space, (W ) R + be a Brownian moion in R d, (F ) R + be his augmened naural filraion, T be a nonnegaive real number, ξ a real F T -measurable random variable, g : [0, T ] Ω R R 1 d R. Y = ξ g(r, Y r, Z r )dr + Z r dw r, 0 T. (1.1) Definiion A soluion o (1.1) is a pair of processes (Y, Z ) 0 T such ha : 1 (Y, Z ) is a predicable process wih values in R R 1 d, 2 P a.s. Y is coninuous and 0 g(r, Y r, Z r ) + Z r 2 dr < 3 (Y, Z ) verifies (1.1). Uniqueness of soluions o quadraic BSDEs

5 Simple example Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Wha is a BSDE? Wha we know abou quadraic BSDEs We ake g = 0. 1 A naural idea is o consider Y := E[ξ F ]. 2 The maringale represenaion heorem gives us Y = E[ξ] + 0 Z sdw s. 3 By a simple calculus we obain Y = ξ Z s dw s. Uniqueness of soluions o quadraic BSDEs

6 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Exisence and uniciy resuls Wha is a BSDE? Wha we know abou quadraic BSDEs Exisence and uniqueness of BSDEs when g is Lipschiz wih respec o y and z : E. Pardoux e S. Peng (1990). Uniqueness of soluions o quadraic BSDEs

7 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Exisence and uniciy resuls Wha is a BSDE? Wha we know abou quadraic BSDEs Exisence and uniqueness of BSDEs when g is Lipschiz wih respec o y and z : E. Pardoux e S. Peng (1990). Numerous works weak assumpions on g. Uniqueness of soluions o quadraic BSDEs

8 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Exisence and uniciy resuls Wha is a BSDE? Wha we know abou quadraic BSDEs Exisence and uniqueness of BSDEs when g is Lipschiz wih respec o y and z : E. Pardoux e S. Peng (1990). Numerous works weak assumpions on g. Exisence and uniqueness of soluions o quadraic BSDEs wih respec o z wih ξ bounded : M. Kobylanski (2000), J.-P. Lepelier e J. San Marín (1998). Uniqueness of soluions o quadraic BSDEs

9 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Exisence and uniciy resuls Wha is a BSDE? Wha we know abou quadraic BSDEs Exisence and uniqueness of BSDEs when g is Lipschiz wih respec o y and z : E. Pardoux e S. Peng (1990). Numerous works weak assumpions on g. Exisence and uniqueness of soluions o quadraic BSDEs wih respec o z wih ξ bounded : M. Kobylanski (2000), J.-P. Lepelier e J. San Marín (1998). Exisence of soluions o quadraic BSDEs wih ξ unbounded : P. Briand e Y. Hu (2006). Uniqueness of soluions o quadraic BSDEs

10 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Exisence and uniciy resuls Wha is a BSDE? Wha we know abou quadraic BSDEs Exisence and uniqueness of BSDEs when g is Lipschiz wih respec o y and z : E. Pardoux e S. Peng (1990). Numerous works weak assumpions on g. Exisence and uniqueness of soluions o quadraic BSDEs wih respec o z wih ξ bounded : M. Kobylanski (2000), J.-P. Lepelier e J. San Marín (1998). Exisence of soluions o quadraic BSDEs wih ξ unbounded : P. Briand e Y. Hu (2006). Uniqueness of soluions o quadraic BSDEs wih g a convex funcion wih respec o z and ξ unbounded : P. Briand e Y. Hu (2008). Uniqueness of soluions o quadraic BSDEs

11 framework Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Framework and ools skech of he proof Consrucion of he conrol problem Assumpions : There exis hree consans β 0, γ > 0 and r 0 ogeher wih wo progressively measurable nonnegaive sochasic processes (ᾱ ) 0 T and (α ) 0 T such ha, P-a.s., 1 z g(, y, z) is a convex funcion (, y) [0, T ] R ; 2 (, z) [0, T ] R 1 d, g(, y, z) g(, y, z) β y y, (y, y ) R 2 ; 3 growh condiion : (, y, z) [0, T ] R R 1 d, α r( y + z ) g(, y, z) ᾱ + β y + γ 2 z 2. Uniqueness of soluions o quadraic BSDEs

12 Exisence resul Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Framework and ools skech of he proof Consrucion of he conrol problem Theorem If here exiss p > 1 such ha [ ( (γe βt E exp ξ + )) ( ) p ] T ᾱ d + (ξ + ) p + α d < hen he BSDE (1.1) has a soluion (Y, Z ). Uniqueness of soluions o quadraic BSDEs

13 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Fenchel-Legendre ransform Framework and ools skech of he proof Consrucion of he conrol problem Since g(, y,.) is a convex funcion, we can define he Fenchel-Legendre ransform of g : f (, y, q) = sup (zq g(, y, z)), [0, T ], q R d, y R. z f is a funcion wih value in R {+ }. Proposiion (, y, y, q) [0, T ] R R R d such ha f (, y, q) < +, f (, y, q) < + e f (, y, z) f (, y, z) β y y. f is a convex funcion wih respec o q, The Fenchel-Legendre ransform of f is g. Uniqueness of soluions o quadraic BSDEs

14 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Framework and ools skech of he proof Consrucion of he conrol problem skech of he proof for he uniqueness resul Wha happened when g does no depend on y? Y = ξ g(s, Z s )ds + Z s dw s. We have g(s, Z s ) = sup qs (Z s q s f (s, q s )) = Z s q s f (s, q s). Uniqueness of soluions o quadraic BSDEs

15 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Framework and ools skech of he proof Consrucion of he conrol problem skech of he proof for he uniqueness resul Wha happened when g does no depend on y? Y = ξ g(s, Z s )ds + Z s dw s. We have g(s, Z s ) = sup qs (Z s q s f (s, q s )) = Z s q s f (s, q s). Y = ξ + ξ + f (s, qs)ds + f (s, q s )ds + Z s (dw s q sds) (2.1) Z s (dw s q s ds). (2.2) Uniqueness of soluions o quadraic BSDEs

16 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Framework and ools skech of he proof Consrucion of he conrol problem skech of he proof for he uniqueness resul Wha happened when g does no depend on y? Y = ξ g(s, Z s )ds + Z s dw s. We have g(s, Z s ) = sup qs (Z s q s f (s, q s )) = Z s q s f (s, q s). Finally, Y = ξ + ξ + Y = ess inf q A EQ f (s, qs)ds + f (s, q s )ds + [ ξ + Z s (dw s q sds) (2.1) Z s (dw s q s ds). (2.2) ] f (s, q s )ds F. Uniqueness of soluions o quadraic BSDEs

17 Quesions Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Framework and ools skech of he proof Consrucion of he conrol problem Are we allowed o apply Girsanov? Which admissible conrol se A could we choice? Wha happened when f depends on y? Uniqueness of soluions o quadraic BSDEs

18 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Framework and ools skech of he proof Consrucion of he conrol problem Consrucion of he conrol problem (1/4) A := { (q s ) s [0,T ], (M ) [0,T ] is a maringale, E Q [ ξ q s 2 ds < + P a.s., f (s, 0, q s ) ds ( wih M := exp q s dw s ] < +, 0 ) q s 2 ds and dq } dp := M T Uniqueness of soluions o quadraic BSDEs

19 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Framework and ools skech of he proof Consrucion of he conrol problem Consrucion of he conrol problem (1/4) A := { (q s ) s [0,T ], (M ) [0,T ] is a maringale, E Q [ ξ q s 2 ds < + P a.s., f (s, 0, q s ) ds ( wih M := exp q s dw s 1 2 There exiss a soluion o he BSDE Y q = ξ + wih dw q := dw q d. 0 f (s, Y q s, q s )ds + ] < +, 0 ) q s 2 ds Z q s dw q s, 0 T. Uniqueness of soluions o quadraic BSDEs and dq } dp := M T

20 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Framework and ools skech of he proof Consrucion of he conrol problem Consrucion of he conrol problem (2/4) We have Y ess inf q A Y q. We mus show ha q A. If T is small enough and if here exis some given exponenial momens for sup 0 T Y + 0 ᾱsds and sup 0 T Y + hen we are able o show ha MT is a maringale. In he general case we divide [0, T ] ino subses. Thus, for N big enough we define i := it N wih i {0,..., N}. Uniqueness of soluions o quadraic BSDEs

21 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Framework and ools skech of he proof Consrucion of he conrol problem Consrucion of he conrol problem (3/4) A i, i+1 (η) := { (q s ) s [i, i+1 ], i+1 i+1 (M i ) [i, i+1 ] is a maringale, E [ η Qi + wih M i := exp ( i i q s dw s 1 2 There exiss a soluion o he BSDE Y η,q = η + i+1 f (s, Y η,q s, q s )ds + q s 2 ds < + P a.s., i i+1 i q s 2 ds f (s, 0, q s ) ds ) ] < +, } and dqi dp := Mi i+1. Z η,q s dw q s, i i+1. Uniqueness of soluions o quadraic BSDEs

22 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Framework and ools skech of he proof Consrucion of he conrol problem Consrucion of he conrol problem (4/4) { A := (q s ) s [0,T ], q [N 1,T ] A N 1,T (ξ), ( ) i {N 2,..., 0}, q [i, i+1 ] A i, i+1 Y q i+1 wih Y q i+1 := Y Y q },q [i+1, i+2 i+2 ] i+1 and Y q T := ξ. We can define our cos funcional i {N 1,..., 0}, [ i, i+1 ], Y q := Y Y q,q [i, i+1 i+1 ]. Uniqueness of soluions o quadraic BSDEs

23 Resul Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Framework and ools skech of he proof Consrucion of he conrol problem Theorem We suppose ha here exiss a soluion (Y, Z ) of he BSDE (1.1) verifying p > γ, ε > 0, [ ( ( E exp p sup Y + ᾱ s ds) ) ( )] + exp ε sup Y + < +, 0 T 0 0 T Then we have Y = ess inf q A Y q, and here exiss q A such ha Y = Y q. Moreover, his implies ha he soluion (Y, Z ) is unique among soluions verifying such assumpion. Uniqueness of soluions o quadraic BSDEs

24 Remarks : Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Framework and ools skech of he proof Consrucion of he conrol problem 1 To obain he exisence of a soluion (Y, Z ) ha verifies such assumpion i is sufficien o suppose ha ξ + 0 ᾱd have an exponenial momen of order qe β avec q > γ and ξ α d have an exponenial momen of order ε > 0. 2 When g does no depend on y, we do no have o divide [0, T ]. We have [ ] T Y = ess inf q A 0,T (ξ) EQ ξ + f (s, q s )ds F, [0, T ]. Uniqueness of soluions o quadraic BSDEs

25 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Link wih PDEs Le us consider he following semi-linear PDE u(, x) + Lu(, x) g(, x, u(, x), σ x u(, x)) = 0, u(t,.) = h, (3.1) wih L is he infiniesimal generaor of he diffusion X,x s = x+ s and he BSDE Y,x = h(x,x T ) b(r, Xr,x )dr+ g(s, X,x s s σ(r)dw r,, Y,x s s T, and X,x s = x, s, (3.2), Zs,x )ds Zs,x dw s, 0 T, (3.3) Uniqueness of soluions o quadraic BSDEs

26 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Link wih PDEs Le us consider he following semi-linear PDE u(, x) + Lu(, x) g(, x, u(, x), σ x u(, x)) = 0, u(t,.) = h, (3.1) wih L is he infiniesimal generaor of he diffusion X,x s = x+ s and he BSDE Y,x = h(x,x T ) b(r, Xr,x )dr+ g(s, X,x s s σ(r)dw r,, Y,x s s T, and X,x s = x, s, (3.2), Zs,x )ds Zs,x dw s, 0 T, (3.3) The nonlinear Feynman-Kac formula consiss in proving ha he funcion defined by he formula (, x) [0, T ] R d, is a viscosiy soluion o he PDE (3.1). u(, x) := Y,x (3.4) Uniqueness of soluions o quadraic BSDEs

27 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Exponenial Momens We suppose σ coninuous and b K-Lipschiz in x. We have Lemma [ [ [ 1 λ 0, 2e 2KT σ 2 T, C 0, E sup 0 T e λ X 0,x 0 2 ] Ce C x 0 2. Uniqueness of soluions o quadraic BSDEs

28 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Assumpions We suppose ha g : [0, T ] R d R R d R and h : R d R are coninuous and ha here exis five consans r 0, β 0, γ 0, α 0 e α 0 such ha : 1 g(, x, y, z) g(, x, y, z) β y y ; 2 z g(, x, y, z) is a convex funcion on R 1 d ; 3 r(1 + x 2 + y + z ) g(, x, y, z) r + α x 2 + β y + γ 2 z 2, r α x 2 h(x) r(1 + x 2 ); 4 g(, x, y, z) g(, x, y, z) r(1 + x + x ) x x, h(x) h(x ) r(1 + x + x ) x x ; 5 α + T α < 1 2γe 3βT σ 2 T. Uniqueness of soluions o quadraic BSDEs

29 Resuls Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Proposiion The funcion u defined by (3.4) is coninuous on [0, T ] R d and saisfies (, x) [0, T ] R d, u(, x) C(1 + x 2 ). Moreover, u is a viscosiy soluion o he PDE (3.1). Uniqueness of soluions o quadraic BSDEs

30 Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs Bibliographie Magdalena Kobylanski. Backward sochasic differenial equaions and parial differenial equaions wih quadraic growh. Ann. Probab., 28(2) : , Philippe Briand and Ying Hu. BSDE wih quadraic growh and unbounded erminal value. Probab. Theory Relaed Fields, 136(4) : , Philippe Briand and Ying Hu. Quadraic BSDEs wih convex generaors and unbounded erminal condiions. Probab. Theory Relaed Fields, 141(3-4) : , Ph. Briand, B. Delyon, Y. Hu, E. Pardoux, and L. Soica. L p soluions of backward sochasic differenial equaions. Sochasic Process. Appl., 108(1) : , Uniqueness of soluions o quadraic BSDEs