Dual Representation as Stochastic Differential Games of Backward Stochastic Differential Equations and Dynamic Evaluations

Transcription

1 arxiv:mah/ v1 [mah.pr] 15 Feb 2006 Dual Represenaion as Sochasic Differenial Games of Backward Sochasic Differenial Equaions and Dynamic Evaluaions Shanjian Tang Absrac In his Noe, assuming ha he generaor is uniform Lipschiz in he unknown variables, we relae he soluion of a one dimensional backward sochasic differenial equaion wih he value process of a sochasic differenial game. Under a dominaion condiion, an F- consisen evaluaions is also relaed o a sochasic differenial game. This relaion comes ou of a min-max represenaion for uniform Lipschiz funcions as affine funcions. The exension o refleced backward sochasic differenial equaions is also included. 1 Inroducion Le (Ø, F,P) be a probabiliy space, and {B s ;s 0} a d-dimensional Brownian moion defined on (Ø, F,P). Le F be he σ-algebra generaed by {B s ;0 s } and he oaliy of P-null ses in F, L 2 (F ) he se of all F -measurable random variables X such ha E X 2 <, and L 2 F (0,T) he se of F -adaped processes ϕ such ha E T 0 ϕ 2 d <. Denoe by T he se of all F s -sopping imes aking values in [,T]. This work is parially suppored by he NSFC under grans (disinguished youh foundaion) and (key projec), and he Science Foundaion of Chinese Minisry of Educaion under gran Deparmen of Finance and Conrol Sciences, School of Mahemaical Sciences, Fudan Universiy, Shanghai , China, & Key Laboraory of Mahemaics for Nonlinear Sciences (Fudan Universiy), Minisry of Educaion. sjang@fudan.edu.cn.

2 Consider he following one dimensional backward sochasic differenial equaion (BSDE): { dys = f(s,y s,z s )ds + z s,db s, 0 s T; y T = ξ L 2 (1) (F T ). I is known ha when he generaor is convex or concave wih respec o he unknown variables, BSDE (1) is relaed wih a sochasic conrol problem. More precisely, assume ha f is concave in he las wo variables. Consider he Fenchel-Legender ransformaion: F(ø,,β 1,β 2 ) := sup[f(ø,,y,z) β 1 y β 2,z ] (2) (y,z) for any (ø,,β 1,β 2 ) Ø [0,T] lr lr d. Define D F (ø) = {(β 1,β 2 ) lr lr d : F(ø,,β 1,β 2 ) < }. (3) Then he se D F is a.s. bounded. I follows from well-known resuls (see, e.g., [4]) ha f(ø,,y,z) = inf (β 1,β 2 ) D F (ø) [F(ø,,β 1,β 2 ) + β 1 y + β 2,z ], (4) and he infimum is achieved. Le us now denoe by A he se of bounded progressively measurable lr lr d valued processes {β 1 (),β 2 ()) : 0 T } such ha E T 0 F(,β 1 (),β 2 ()) 2 d <. (5) To each (β 1,β 2 ) A, we associae he unique adaped soluion (Y β 1,β 2,Z β 1,β 2 ) of BSDE (1) wih he coefficien f being replaced wih he affine one f β 1,β 2 (,y,z) := F(,β 1 (),β 2 ()) + β 1 ()y + β 2 (),z. In [4, pages 35 37], he soluion y of BSDE (1) is inerpreed as he value process of a conrol problem. Tha is, where y = essinf (β 1,β 2 ) A E[Φ(,β 1,β 2 ) F ] (6) T Φ(,β 1,β 2 ) := β 1,β 2,T ξ + β 1,β 2,s F(s,β 1 (s),β 2 (s))ds (7) 2

3 and for each [0,T], { β 1,β 2,s : s T } is he unique soluion of he following sochasic differenial equaion (SDE): d,s =,s [β 1 (s)ds + β 2 (s),db s ], s [,T];, = 1. (8) The purpose of his Noe is o obain a similar dual represenaion for he soluion y of BSDE (1) under he Lipschiz assumpion on he generaor, insead of he convexiy assumpion on he generaor f. Assume hroughou he res of he Noe ha here is a consan C > 0 such ha (B1) f(,y,z) L 2 F (0,T) for any pair (y,z) lr lrd ; (B2) f(,y 1,z 1 ) f(,y 2,z 2 ) C( y 1 y 2 + z 1 z 2 ) for any [0,T] and (y 1,z 1 ),(y 2,z 2 ) lr lr d. Then, for any X L 2 (F ), here is unique adaped soluion {(Y s,z s );0 s } of BSDE (1) wih he erminal condiion: Y = X. Define E f s, [X] := Y s for any s [0,]. The res of his Noe is organized as follows. In secion 2, we give a Min-Max represenaion of a Lipschiz funcion in erms of affine funcions, which is he basis of he Noe. In Secion 3, we presen he dual formula for he soluion of one dimensional BSDE (1). In Secion 4, he formula obained in Secion 3 is applied o he dynamical evaluaion and a dual formula is herefore derived for an F -consisen evaluaion. Finally in Secion 5, a dual formula is also obained for one dimensional refleced backward sochasic differenial equaions (RBSDEs) (20). 2 Min-max represenaion of a Lipschiz funcion as affine funcions The following represenaion is due o Evans and Souganidis [2, pages ]. Lemma 2.1. Le f : [0,T] Ω lr n lr be a Lipschiz funcion. Tha is, here is a consan C > 0 such ha f(,x 1 ) f(,x 2 ) C x 1 x 2, x 1,x 2 lr n. (10) Then for each [0,T] and x lr n, f(,x) = max min z lr n y O n(0,1) 3 (9) {C y,x + F(,y,z)} (11)

4 where F(,y,z) := f(,z) C y,z for y,z lr n and O n (0,1) is he closed uni ball in lr n. Proof. In view of he assumpion (10), we have for any x lr n f(,x) = max z lr n{f(,z) C x z } = max min {f(,z) + C y,x z }. z lr n y O n(0,1) Remark 1. See Fleming [5, pages ] or Evans [1] for oher, more complicaed ways of wriing a nonlinear funcion as he max-min (or min-max) of affine mappings. 3 Backward sochasic differenial equaions and relaed sochasic differenial games Denoe (L 2 F (0,T))d+1 by L 2 F (0,T;lRd+1 ), and by V d+1 he subse of L 2 F (0,T;lRd+1 ) whose elemen akes values in he closed uni ball O d+1 (0,1). Define he funcion F : Ø [0,T] lr d+1 lr d+1 lr as follows: F(ø,s,β 1,β 2,α 1,α 2 ) = f(ø,s,α 1,α 2 ) Cβ 1 α 1 C β 2,α 2 (12) for any (ø,s,β 1,β 2,α 1,α 2 ) Ø [0,T] lr d+1 lr d+1. Then, in view of Lemma 2.1, we have for any (ø,s,β 1,β 2,α 1,α 2 ) Ø [0,T] lr d+1 lr d+1, f(ø,,y,z) = max min α lr d+1 β O d+1 (0,1) [F(ø,,β,α) + Cβ 1 y + C β 2,z ]. (13) Given α L 2 F (0,T;lRd+1 ) and β V d+1, consider he relaed BSDE: dy s = [Cβ 1 (s)y s + C β 2 (s),z s +F(s,β 1 (s),β 2 (s),α 1 (s),α 2 (s))]ds + Z s,db s ; Y T = ξ L 2 (F T ). (14) The soluion is denoed by (Y α,β,z α,β ) when i is necessary o emphasize he dependence on (α,β) wih α = (α 1,α 2 ) and β = (β 1,β 2 ). 4

5 Inroduce he following sochasic differenial equaion (SDE): dγ,s = Γ,s [Cβ 1 (s)ds + C β 2 (s),db s ], s [,T]; Γ, = 1. (15) Is soluion is denoed by Γ β,s, s T o indicae he dependence on β = (β 1,β 2 ). We have Y α,β [ T ] = E Γ β,s F(s,β 1(s),β 2 (s),α 1 (s),α 2 (s))ds + Γ β,t ξ F for any [0,T]. (16) Theorem 3.1. Assume ha he funcion f saisfies (9). Le (y, z) be he adaped soluion of BSDE (1) and {Γ β,s ; s T } he soluion of SDE (15). Then we have for any [0,T], essinf E α L 2 F (0,T;lRd+1 ) β V d+1 y = esssup [ T Γ β,s F(s,β 1(s),β 2 (s),α 1 (s),α 2 (s))ds +Γ β,t ξ F ]. (17) 4 An F -consisen evaluaions and is dual represenaion as a sochasic differenial game Définiion 4.1. A sysem of operaors E s, : L 2 (F ) L 2 (F s ),0 s T is called an F -consisen evaluaion defined on [0,T] if i saisfies he following four properies: for any 0 s T and any X 1,X 2 L 2 (F ), (A1) E s, [X 1 ] E s, [X 2 ], a.s., if X 1 X 2, a.s. ; (A2) E, [X 1 ] = X 1, a.s. ; (A3) E r,s [E s, [X 1 ]] = E r, [X 1 ], a.s. ; (A4) χ A E s, [X 1 ] = χ A E s, [χ A X 1 ], a.s. for any A F s. In view of Peng [7, Corollary 4.2, page 588], he following is an immediae consequence of Theorem??. 5

6 Theorem 4.1. Le {E s, } 0 s T denoe an F -consisen evaluaion defined on [0,T]. Assume ha here is a funcion g µ (,y,z) := µ( y + z ),(,y,z) [0,T] lr lr d for some µ > 0 such ha he F - in he following sense: for any s, [0,T] such ha s and for any X 1,X 2 L 2 (F ), we have consisen evaluaion {E s, } 0 s T is dominaed by E gµ s, E s, [X 1 ] E s, [X 2 ] E gµ s, [X 1 X 2 ], a.s.. (18) Furhermore, assume ha here is g 0 L 2 F (0,T) such ha E gµ+g 0 s, [0] E s, [0] E gµ+g 0 s, [0]. Then here is a funcion f : Ω [0,T] lr d+1 lr which saisfies (9), such ha ] E s, [ξ] = esssup essinf [Γ βs, ξ + Γ β s,r F(r,α(r),β(r))dr F s. E α L 2 F (0,T;lRd+1 ) β V d+1 Here {Γ β,s ; s T } is he soluion of SDE (15) and he funcion F : Ω [0,T] lr d+1 lr d+1 lr is given by (12). 5 Refleced backward sochasic differenial equaions and relaed mixed sochasic differenial games We make he following assumpion. (B3) The obsacle {S,0 T } is a coninuous progressively measurable real-valued process saisfying s E sup (S + )2 <, S T ξ,a.s.. (19) 0 T Consider he following RBSDE: dy = f(,y,z )d da + z,db ; y T = ξ L 2 (F T ); y S,a.s. [0,T]; T 0 (y S )da = 0. (20) In view of [3, Theorem 5.2, page 718], i has a unique soluion (y,z,a). 6

7 Given α L 2 F (0,T;lRd+1 ) and β V d+1, idenically as in Secion 3, consider he funcion F given by (12) and he relaed RBSDE: dy s = [Cβ 1 (s)y s + C β 2 (s),z s +F(s,β 1 (s),β 2 (s),α 1 (s),α 2 (s))]ds da s + Z s,db s ; Y T = ξ L 2 (F T ); y s S s,a.s. s [0,T]; T 0 (Y s S s )da s = 0. (21) The unique soluion is denoed by (Y α,β,z α,β,a α,β ). We have for any [0,T], Y α,β = esssupe τ T [ τ Γ β,s F(s,β 1(s),β 2 (s),α 1 (s),α 2 (s))ds +Γ β,τ S τχ {τ<t } + Γ β,τ ξχ {τ=t } F ]. (22) Theorem 5.1. Assume ha he funcion f saisfies (9) and he obsacle {S,0 T } saisfies assumpion (B3). Le (y,z,a) be he adaped soluion of RBSDE (20) and {Γ β,s ; s T } he soluion of SDE (15). Then we have for any [0,T], [ τ y = esssup essinf E Γ β,s F(s,β 1(s),β 2 (s),α 1 (s),α 2 (s))ds α L 2 F (0,T;lRd+1 ),τ T β V d+1 ] +Γ β,τ S τχ {τ<t } + Γ β,τ ξχ {τ=t } F. References (23) [1] L. C. Evans, Some Min-Max mehods for he Hamilon-Jacobi equaions, Indiana Univerisy Mahemaics Journal, 33 (1984), [2] L. C. Evans and P. E. Souganidis, Differenial games and represenaion formulas for soluions of Hamilon-Jacobi-Isaacs equaions, Indiana Univerisy Mahemaics Journal, 33 (1984), [3] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, and M. C. Quenez Refleced soluion of backward SDE s, and relaed obsacle problems for PDE s, Annals of Probabiliy, 25 (1997),

8 [4] N. El Karoui, S. Peng, and M. C. Quenez Backward sochasic differenial equaions in finance, Mah. Finance, 7 (1997), [5] W. Fleming, The Cauchy problem for degenerae parabolic equaions, J. Mah. Mech., 13 (1964), [6] E. Pardoux and S. Peng, Adaped soluion of a backward sochasic differenial equaion, Sysems Conrol Leers, 14 (1990), [7] S. Peng, Dynamical evaluaion, C. R. Acad. Sci. Paris, Ser. I 339 (2004),