10384 19020071152070 UDC The Establishment of Fractional BSDE : : : 2010 4 : 2010 5 : 2010 6 2010 4
1. 2.
1973 1990 Pardoux. Duffie Epstein ( ) Feynman-Kac Navier-Stokes El Karoui Quenez ( )... - i
Abstract Backward stochastic differential equation is a hot new field currently. As for BSDE study, the linear case began in 1973, but the basic framework of the general nonlinear case was provided by Shige Peng and Pardoux in 1990 and proved its existence and uniqueness of solution. The study history of BSDE theory is rather short but the progress is very rapid, because the theory itself, in addition to its unique system and interesting nature, has important application prospects: Duffie and Epstein found it can be used to describe the consumer preferences in uncertain economic environment (i.e. utility function theory); Shige Peng obtained nonlinear Feynman-Kac formula by virtue of BSDE, which can be used to handle the well-known important nonlinear partial differential equations such as reaction-diffusion equation and Navier-Stokes equation; ElKaroui and Quenez found that the theoretical prices of many important derivative securities (such as options and futures) in financial markets can be derived by means of BSDE. Nowadays, BSDE has a new development direction: the standard Brownian motion drive is changed into fractional Brownian motion drive. Empirical analysis demonstrates that the BSDE driven by fbm is more realistic, and many ii
domestic and foreign scholars have already engaged in the study of stochastic analysis theory and applications of fbm. These thesis sums up the study achievement of fbm and its stochastic analysis theory, and summarizes the establishment of the FrBSDE for better understanding and further development of the theory. Keywords: fractional Brownian motion, backward stochastic differential equation, quasi-conditional expectation, backward stochastic differential equation driven by fractional Brownian motion. iii
i Abstract ii iv 1 1.1 BSDE.......................... 1 1.2 FrBSDE..................... 3 1.3............................... 5 7 2.1......................... 7 2.2...................... 7 2.3......................... 9 2.3.1........................... 9 2.3.2.......................... 10 2.3.3 Hölder......................... 10 iv
2.3.4........................... 11 2.3.5 H 1 2 BH................. 11 13 3.1 Girsanov............................. 13 3.2 Malliavin φ -....................... 14 3.3 Itô.............................. 16 3.4 Itô................................ 18 3.5............................. 21-23 4.1 -.......................... 23 4.2 -.......................... 25 32 5.1............................ 32 5.2 FrBSDE............................. 40 42 46 v
Contents 1 Introduction 1 1.1 Development history of BSDE.................... 1 1.2 Research significance and progress of FrBSDE........... 3 1.3 Framework of paper.......................... 5 2 Fractional Brownian motion 7 2.1 Definition of fbm........................... 7 2.2 Integral representation of fbm.................... 7 2.3 Proposition of fbm.......................... 9 2.3.1 Self-similarity......................... 9 2.3.2 Long-range dependence.................... 10 2.3.3 Hölder continuity....................... 10 2.3.4 Path differentiability..................... 11 2.3.5 The fbm is not a semimartingale for H 1........ 11 2 3 Stochastic calculus for fbm 13 3.1 Girsanov translation......................... 13 vi
3.2 Malliavin derivative and φ -derivative................ 14 3.3 Itô stochastic integration....................... 16 3.4 Itô formula.............................. 18 3.5 Integration by parts formula..................... 21 4 Quasi-conditional expectation 23 4.1 Definition of quasi-conditional expectation............. 23 4.2 Proposition of quasi-conditional expectation............ 25 5 Fractional backward stochastic differential equation 32 5.1 Existence and uniqueness of solutions................ 32 5.2 Linear FrBSDE............................ 40 vii
1.1 BSDE (BSDE) Pardoux. (SDE). { dx(t) = b(x(t), t)dt + B(X(t), t)dw (t), X(0) = X 0. SDE [1]... 1973 J.M.Bismut BSDE [2]. 1990 Pardoux BSDE
[3] { dy(t) = f (t, y(t), z(t)) dt + z(t)dw (t), y(t ) = ξ, BSDE. 1992 Duffie Epstein { dy(t) = g (y(t), z(t)) dt z(t)dw (t), y(t ) = 0. g z ( riskaversion )... BSDE ( ) ( Lipschitz ) BSDE [4] [5] [6] [7] [8]. Jensen [9] [10] [11] [12] BSDE. 1994 Elkaroai 2
BSDE. Black-Scholes BSDE BSDE BSDE ([13],[14],[15] ) 1995 BSDE g- ([16],[17],[18] ). g- Feynman-Kac ([19],[20],[21] ) 10 ([22],[23],[24] ) BSDE (FBSDE ) (RSBSDE). 1.2 FrBSDE Black-Scholes 3
ds(t) = µs(t)dt + σs(t)db(t), B(t). Black-Scholes. ( ) ; Hurst 1 2.. E.E.Peter 1972 1 1990 12 Hurst H Ĥ = 0.642 S&P tick Ĥ 0.6 [25] ; I.Simonsen K.Sneppen Hurst 0.4 [26]. 4
(FrBSDE) BSDE... Bernt Øksendal Christian Bender Robert J.Elliott J.Van der Hoek ([27],[28],[29],[30] ) Itô Girsanov., C.Bender FrBSDE [31]; FrBSDE FrBSDE - FrBSDE [32] FrBSDE. 1.3.. fbm Hölder. 5
Girsanov Itô. - FrBSDE. FrBSDE. 6
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