1 An Inroducion o Backward Sochasic Differenial Equaions (BSDEs) PIMS Summer School 2016 in Mahemaical Finance June 25, 2016 Chrisoph Frei cfrei@ualbera.ca
This inroducion is based on Touzi [14], Bouchard [1] and Rukowski [12] as well as papers given in he reference secion. Throughou his inroducion, we consider a d- dimensional Brownian moion W on a complee probabiliy space (Ω, F, P ) over a finie horizon [0, T ] for a fixed T > 0. We denoe by (F ) 0 T he augmened filraion generaed by W. 2
3 1 Moivaion Consider d = 1 and a model of a financial marke wih We as- one risk-free asse wih ineres rae r. sume ha r is bounded and predicable, one risky asse whose prices (S ) 0 T are given by S 0 > 0 and ds S = µ d + σ dw, where µ is bounded and predicable, and σ is posiive, bounded away from zero and predicable, an invesor, whose amoun of money invesed in S a ime is denoed by π, and his/her oal wealh is denoed by Y, wealh process dynamics dy = π S ds + r (Y π ) d = (π µ π r + r Y ) d + π σ dw, assuming ha he sraegy is self-financing,
a European opion wih payoff ξ a ime T, where ξ is a square-inegrable, F T -measurable random variable. Assuming ha he invesor wans o replicae he opion payoff, we wan o solve dy = (π µ π r + r Y ) d + π σ dw, (1) Y T = ξ, 4 (2) which is a sochasic differenial equaion for Y wih erminal condiion Y T = ξ. In his case (see Secion 4 below), we can solve explicily he problem wih Y given by [ Y = E Q e T r s ds ξ F ], where he probabiliy measure Q is defined by dq dp = exp ( 0 µ s r s σ s dw s 1 2 0 (µ s r s ) 2 We will see ha he reason why we can solve explicily his problem is ha he erm π (µ r ) + r Y in (1) is affine in π and Y. σ 2 s ds ).
However, he problem has no more an explici soluion if he ineres raes for borrowing and lending are differen, say, r for lending and r for borrowing. The wealh process dynamics hen is dy = ( π µ + r (Y π ) + r (Y π ) ) d Y T = ξ, + π σ dw, which has no more a d-erm affine in π and Y. 5 2 Definiion of BSDEs Definiion: An n-dimensional BSDE is of he form dy = f (Y, Z ) d + Z dw for [0, T ], Y T = ξ, (3) where given are he generaor (also called driver) f, which is a mapping f : [0, T ] Ω R n R n d R n, which saisfied appropriae measurabiliy condiions, namely, for every fixed (y, z) R n R n d, he process (f (y, z)) 0 T is predicable;
he erminal condiion ξ, which is an F T -measurable, square-inegrable random variable wih values in R n. A soluion o (3) consiss of (Y, Z) for an R n -valued, adaped process (Y ) 0 T an R n d -valued, predicable process (Z ) 0 T saisfying (3). 6 Remark: We can wrie (3) equivalenly as Y = ξ + f s (Y s, Z s ) ds Z s dw s. (4) 3 BSDEs wih zero generaor In he case of f 0, (4) reduces o Y = ξ Z s dw s. (5)
To find a soluion (Y, Z), recall he maringale represenaion heorem, which says ha every F T -measurable, square-inegrable ξ can be wrien as ξ = E[ξ] + β s dw s 0 for a unique predicable, square-inegrable process β. By seing Y = E[ξ F ] and Z = β, we obain a soluion o (5), which is unique in he class of squareinegrable soluions. 7 4 BSDEs wih affine generaor We now consider he case n = 1 and where f (y, z) = a + b y + c z, a is an R-valued, predicable process such ha E [ 0 a d ] <, b is an R-valued, bounded, predicable process, c is an R d -valued, bounded, predicable process.
In his case, we can reduce he corresponding BSDE o a problem of a BSDE wih zero generaor. 8 1. To eliminae he erm c z in he generaor, we apply Girsanov s heorem. Recall ha under he measure Q given by dq dp = exp ( 0 c dw 1 2 he process (B ) 0 T defined by 0 c 2 d B = W c s ds, [0, T ] 0 is a Brownian moion. We can rewrie as dy = (a + b Y + c Z ) d + Z dw dy = (a + b Y ) d + Z db and consider his BSDE under Q. ),
2. To eliminae he erm b y in he generaor, we use he ransformaion Y = Y e so ha by Iô s formula 0 b s ds dy = e 0 b s ds dy + b Y e 0 b s ds d = a e 0 b s ds d + Z e 0 b s ds db, (6) wih Y T = ξ := e 0 b s ds ξ, 3. To eliminae he remaining erm in he generaor, we wrie (6) as ( ) u d Y + a ue 0 b s ds du = Z e 0 b s ds 0 }{{} db. =:Z } {{ } =:Y wih Y T = ξ = e 0 b s ds ξ + 0 a u e u 0 b s ds du, 4. By Secion 3, we know ha Y = E Q [ξ F ] = E [e Q 0 b s ds u ] ξ + a ue 0 b s ds du F, 0 9
10 hence Y = E [e Q 0 b s ds ξ + Y = E [e Q b s ds ξ + u ] a u e 0 b s ds du F, a u e u b s ds du F ]. 5 BSDEs wih Lipschiz-coninuous generaor Theorem 1 (Pardou and Peng [10]) Assume ha E[ ξ 2 ] <, E [ 0 f (0, 0) 2 d ] < and f (y, z) f (y, z ) C( y y + z z ) (7) for a consan C and all y, y R n, z, z R n d. Then he BSDE (3) has a unique square-inegrable soluion.
Remarks: 11 The proof is based on fixed poin argumen, working in a combined space for (Y, Z) wih norm [ E 0 eα ( Y 2 + Z 2 ) d for a suiably chosen consan α. A sequence (Y (n), Z (n) ) can be defined by Y (n) ( = ξ + f s Y s (n 1), Z s (n 1) ) ds Z (n) s dw s using he maringale represenaion for he random variable ξ + ( T (n 1) 0 f s Y s, Z s (n 1) ) ds. Theorem 1 holds for general n (mulidimensional Y ), bu in applicaions from mahemaical finance, he Lipschiz coninuiy for he generaor is ofen oo resricive. ] Having esablished exisence and uniqueness of BSDE soluions, we can compare hem for differen erminal condiions and generaors.
12 Theorem 2 (Peng [11] and El Karoui e al. [4]) a Assume n = 1. Consider wo square-inegrable soluions (Y i, Z i ) o BSDEs wih erminal condiions ξ i and generaor f i such ha E [ 0 f i (0, 0) 2 d ] <, E[ ξ i 2 ] < and (7) is saisfied for f i for i = 1, 2. Assume furher ha ξ 1 ξ 2, f 1 (Y 2, Z 2 ) f 2 (Y 2, Z 2 ). Then Y 1 Y 2 for all [0, T ]. While he Brownian moion W can be mulidimensional in Theorem 2, we need ha Y is one-dimensional. Under addiional condiions, comparison resuls for mulidimensional Y are available, see for example, Hu and Peng [7] or Cohen e al. [2], bu hey only hold under addiional condiions on he generaors. This is one of he reasons why he exisence and uniqueness resuls can be exended o he quadraic generaors only in he case of n = 1; see nex secion.
6 BSDEs wih quadraic generaor 13 Le us sar wih an example of a quadraic BSDE for n = 1 and d = 1 by considering dy = 1 2 Z 2 d + Z dw. Recall ha he sochasic exponenial X = exp ( 0 Z s dw s 0 1 2 Z s 2 ds is a maringale if Z s dw s saisfies sufficien inegrabiliy condiions, for example, if i is a BMO (bounded mean oscillaion) maringale or if he Novikov condiion E [ exp ( is saisfied. Noe ha 0 )] 1 2 Z s 2 ds < d ( ln(x ) ) = 1 2 Z 2 d + Z dw. )
14 Therefore, we use an exponenial ransformaion d ( e Y ) = e Y dy + 1 2 ey d Y = e Y 1 2 Z 2 d + e Y Z dw + 1 2 ey Z 2 d = e Y Z dw, which yields so ha e ξ e Y = d ( e Y s ) = e Ys Z s dw s, Y = ln ( E [ e ξ F ]), assuming ha e Y sz s dw s is a rue maringale (and no merely a local maringale). The example shows ha BSDEs wih quadraic generaors are sill meaningful, bu square inegrabiliy of he erminal condiion may no be enough because e ξ appears in he soluion of he above example.
Theorem 3 (Simplified version from Kobylanski [9]) Assume n = 1, ha ξ is bounded and f (y, z) c + c z 2, f (y, z) z c + c z f (y, z) c + c z 2 y 15 for a consan c and all y R, z R d. Then he BSDE (3) has a unique soluion (Y, Z) wih bounded Y and square-inegrable Z. Remarks: A comparison heorem similarly o Theorem 2 holds for BSDEs wih quadraic generaors and n = 1. Theorem 3 canno be generalized o BSDEs wih generaors of super-quadraic growh in z. Counerexamples for such a case can be found in Delbaen e al. [3].
16 Example (from Frei and dos Reis [6]): We ake d = 1 (dimension of W ) and consider he wo-dimensional (n = 2) BSDE dy 1 = Z 1 dw, YT 1 = ξ, (8) = ( Z 1 2 + 12 ) Z2 2 d + Z 2 dw, YT 2 = 0, dy 2 for a given bounded random variable ξ. (9) For some bounded ξ, he BSDE (8), (9) has no squareinegrable soluion. Main ideas o show his: assume ha a square-inegrable soluion (Y, Z) exiss; (8) deermines Y 1 and Z 1 uniquely; using his Z 1 in (9) gives E [ exp ( = exp ( Y 2 0 Z1 2 d 0 exp ( Y0 2 ) ; ) E [ exp )] ( 0 Z2 dw 1 2 0 Z2 2 d )]
17 i is possible o consruc a process β such ha β dw is bounded bu [ E exp ( 0 β 2 d )] =, hen se ξ = 0 β dw. Commens: The quesions abou exisence of soluions o mulidimensional, quadraic BSDEs are relaed o ineresed economic quesions abou he exisence of a Nash equilibrium in a model of a financial marke where differen invesors ake he relaive performance compared o each oher ino accoun (see [6]). The sudy of mulidimensional, quadraic BSDEs is a very recen research opic. There are exisence resuls available for paricular cases, for example, if
18 he erminal condiion is small enough (see Tevzadze [13]), he ime horizon [0, T ] is shor (see Frei [5]), he generaor has a paricularly quadraic diagonal srucure (see Hu and Tang [8]), under some Markovian assumpions on he generaor and erminal condiion (see Xing and Žiković [15]). References [1] B. Bouchard: BSDEs: Main Exisence and Sabiliy Resuls, Lecure Noes o lecures given a he London School of Economics, 2015, available a www.ceremade.dauphine.fr/ bouchard [2] S. Cohen, R. Ellio and C. Pearce: A General Comparison Theorem for Backward Sochasic Differenial Equaions. Advances in Applied Probabiliy, 42:878-898, 2010 [3] F. Delbaen, Y. Hu, and X. Bao: Backward SDEs wih Superquadraic Growh. Probabiliy Theory and Relaed Fields 150:145 192, 2011
19 [4] N. El Karoui, S. Peng and M. Quenez: Backward Sochasic Differenial Equaions in Finance. Mahemaical Finance 7:1 71, 1997 [5] C. Frei: Spliing Mulidimensional BSDEs and Finding Local Equilibria. Sochasic Processes and heir Applicaions 124:2654 2671, 2014 [6] C. Frei and G. dos Reis: A Financial Marke wih Ineracing Invesors: Does an Equilibrium Exis?, Mahemaics and Financial Economics 4:61 182, 2011 [7] Y. Hu and S. Peng: On he Comparison Theorem for Mulidimensional BSDEs. Compes Rendus Mahemaique 343:135 140, 2006 [8] Y. Hu and S. Tang: Muli-dimensional Backward Sochasic Differenial Equaions of Diagonally Quadraic Generaors. Sochasic Processes and heir Applicaions 126: 1066 1086, 2016 [9] M. Kobylanski: Backward Sochasic Differenial Equaions and Parial Differenial Equaions wih Quadraic Growh. Annals of Probabiliy 28: 558 602, 2000
20 [10] E. Pardoux and S. Peng: Adaped Soluion of a Backward Sochasic Differenial Equaion. Sysems and Conrol Leers 14:55 61, 1990 [11] S. Peng: Sochasic Hamilon-Jacobi-Bellman Equaions, SIAM Journal of Conrol and Opimizaion 30:284 304, 1992 [12] M. Rukowski: Backward Sochasic Differenial Equaions wih Applicaions, Lecure Noes o lecures given a he Universiy of Sydney, 2015 [13] R. Tevzadze: Solvabiliy of Backward Sochasic Differenial Equaions wih Quadraic Growh. Sochasic Processes and heir Applicaions 118:503 515, 2008 [14] N. Touzi: Opimal Sochasic Conrol, Sochasic Targe Problems, and Backward SDEs, Lecure Noes o lecures given a he Fields Insiue, 2010, available a www.cmap.polyechnique.fr/ ouzi/ [15] H. Xing and G. Žiković: A Class of Globally Solvable Markovian Quadraic BSDE Sysems and Applicaions, 2016, available a arxiv:1603.00217