Forward-Backward Stochastic Differential Equations and Their Applications

by Ma, Jin; Yong, Jiongmin

• ISBN13: 9783540659600
• ISBN10: 3540659609
• Format: Paperback
• Copyright: 1999-06-01
• Publisher: Springer Verlag

Summary

This volume is a survey/monograph on the recently developed theory of forward-backward stochastic differential equations (FBSDEs). Basic techniques such as the method of optimal control, the "Four Step Scheme", and the method of continuation are presented in full. Related topics such as backward stochastic PDEs and many applications of FBSDEs are also discussed in detail. The volume is suitable for readers with basic knowledge of stochastic differential equations, and some exposure to the stochastic control theory and PDEs. It can be used for researchers and/or senior graduate students in the areas of probability, control theory, mathematical finance, and other related fields.

Table of Contents

Preface	vii
Introduction	1	(24)
Some Examples	1	(7)
A first glance	1	(2)
A stochastic optimal control problem	3	(1)
Stochastic differential utility	4	(3)
Option pricing and contingent claim valuation	7	(1)
Definitions and Notations	8	(2)
Some Nonsolvable FBSDEs	10	(4)
Well-posedness of BSDEs	14	(5)
Solvability of FBSDEs in Small Time Durations	19	(3)
Comparison Theorems for BSDEs and FBSDEs	22	(3)
Linear Equations	25	(26)
Compatible Conditions for Solvability	25	(5)
Some Reductions	30	(3)
Solvability of Linear FBSDEs	33	(12)
Necessary conditions	34	(5)
Criteria for solvability	39	(6)
A Riccati Type Equation	45	(4)
Some Extensions	49	(2)
Method of Optimal Control	51	(29)
Solvability and the Associated Optimal Control Problem	51	(6)
An optimal control problem	51	(3)
Approximate Solvability	54	(3)
Dynamic Programming Method and the HJB Equation	57	(3)
The Value Function	60	(9)
Continuity and semi-concavity	60	(4)
Approximation of the value function	64	(5)
A Class of Approximately Solvable FBSDEs	69	(6)
Construction of Approximate Adapted Solutions	75	(5)
Four Step Scheme	80	(23)
A Heuristic Derivation of Four Step Scheme	80	(4)
Non-Degenerate Case---Several Solvable Classes	84	(5)
A general case	84	(2)
The case when h has linear growth in z	86	(2)
The case when m = 1	88	(1)
Infinite Horizon Case	89	(14)
The nodal solution	89	(3)
Uniqueness of nodal solutions	92	(6)
The limit of finite duration problems	98	(5)
Linear, Degenerate Backward Stochastic Partial Differential Equations	103	(34)
Formulation of the Problem	103	(3)
Well-posedness of Linear BSPDEs	106	(5)
Uniqueness of Adapted Solutions	111	(7)
Uniqueness of adapted weak solutions	111	(2)
An Ito formula	113	(5)
Existence of Adapted Solutions	118	(8)
A Proof of the Fundamental Lemma	126	(4)
Comparison Theorems	130	(7)
The Method of Continuation	137	(32)
The Bridge	137	(3)
Method of Continuation	140	(8)
The solvability of FBSDEs linked by bridges	140	(3)
A priori estimate	143	(5)
Some Solvable FBSDEs	148	(6)
A trivial FBSDE	148	(1)
Decoupled FBSDEs	149	(2)
FBSDEs with monotonicity conditions	151	(3)
Properties of Bridges	154	(4)
Construction of Bridges	158	(11)
A general consideration	158	(3)
A one dimensional case	161	(8)
FBSDEs with Reflections	169	(24)
Forward SDEs with Reflections	169	(2)
Backward SDEs with Reflections	171	(10)
Reflected Forward-Backward SDEs	181	(12)
A priori estimates	182	(4)
Existence and uniqueness of the adapted solutions	186	(4)
A continuous dependence result	190	(3)
Applications of FBSDEs	193	(42)
An Integral Representation Formula	193	(4)
A Nonlinear Feynman-Kac Formula	197	(4)
Black's Consol Rate Conjecture	201	(6)
Hedging Options for a Large Investor	207	(19)
Hedging without constraint	210	(9)
Hedging with constraint	219	(7)
A Stochastic Black-Scholes Formula	226	(6)
Stochastic Black-Scholes formula	227	(2)
The convexity of the European contingent claims	229	(2)
The robustness of Black-Scholes formula	231	(1)
An American Game Option	232	(3)
Numerical Methods for FBSDEs	235	(22)
Formulation of the Problem	235	(2)
Numerical Approximation of the Quasilinear PDEs	237	(13)
A special case	237	(1)
Numerical scheme	238	(2)
Error analysis	240	(4)
The approximating solutions {u(n)}∞n=1	244	(1)
General case	245	(2)
Numerical scheme	247	(1)
Error analysis	248	(2)
Numerical Approximation of the Forward SDE	250	(7)
Comments and Remarks	257	(2)
References	259	(10)
Index	269