## Forward-Backward Stochastic Differential Equations and Their Applications, Issue 1702This book is intended to give an introduction to the theory of forwa- backward stochastic di erential equations (FBSDEs, for short) which has received strong attention in recent years because of its interesting structure and its usefulness in various applied elds. The motivation for studying FBSDEs comes originally from stochastic optimal control theory, that is, the adjoint equation in the Pontryagin-type maximum principle. The earliest version of such an FBSDE was introduced by Bismut [1] in 1973, with a decoupled form, namely, a system of a usual (forward)stochastic di erential equation and a (linear) backwardstochastic dieren tial equation (BSDE, for short). In 1983, Bensoussan [1] proved the well-posedness of general linear BSDEs by using martingale representation theorem. The r st well-posedness result for nonlinear BSDEs was proved in 1990 by Pardoux{Peng [1], while studying the general Pontryagin-type maximum principle for stochastic optimal controls. A little later, Peng [4] discovered that the adapted solution of a BSDE could be used as a pr- abilistic interpretation of the solutions to some semilinear or quasilinear parabolic partial dieren tial equations (PDE, for short), in the spirit of the well-known Feynman-Kac formula. After this, extensive study of BSDEs was initiated, and potential for its application was found in applied and t- oretical areas such as stochastic control, mathematical n ance, dieren tial geometry, to mention a few. The study of (strongly) coupled FBSDEs started in early 90s. In his Ph. |

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### Contents

II | 1 |

III | 3 |

IV | 4 |

V | 7 |

VI | 8 |

VII | 10 |

VIII | 14 |

IX | 19 |

XLIV | 140 |

XLVI | 143 |

XLVII | 148 |

XLIX | 149 |

L | 151 |

LI | 154 |

LII | 158 |

LIV | 161 |

X | 22 |

XI | 25 |

XII | 30 |

XIII | 33 |

XIV | 34 |

XV | 39 |

XVI | 45 |

XVII | 49 |

XVIII | 51 |

XIX | 54 |

XX | 57 |

XXI | 60 |

XXIII | 64 |

XXIV | 69 |

XXV | 75 |

XXVI | 80 |

XXVII | 84 |

XXIX | 86 |

XXX | 88 |

XXXI | 89 |

XXXIII | 92 |

XXXIV | 98 |

XXXV | 103 |

XXXVI | 106 |

XXXVII | 111 |

XXXIX | 113 |

XL | 118 |

XLI | 126 |

XLII | 130 |

XLIII | 137 |

### Other editions - View all

Forward-Backward Stochastic Differential Equations and their Applications Jin Ma,Jiongmin Yong Limited preview - 2007 |

Forward-Backward Stochastic Differential Equations and Their ..., Issue 1702 Jin Ma,J.-M. Morel,Jiongmin Yong No preview available - 1999 |

### Common terms and phrases

adapted weak solution admits a unique admits an adapted Applying Ito's formula approximately solvable assume assumption Black-Scholes formula boundary value problem boundedness Brownian motion BSDE BSPDE Cetraro Chapter coeﬃcients comparison theorems condition Consequently consider the following continuous convex define deﬁned definition denote depending deterministic differential dX(t dY(t Euler transformation exists a constant exists a unique Feynman-Kac formula ﬁrst following FBSDE following result Four Step Scheme Gronwall's inequality h and g Hence implies Jt Jt Lemma linear FBSDE Lipschitz constant Lipschitz continuous Martina Franca Martingale Representation Theorem nodal solution obtain optimal control optimal control problem parabolic Proposition prove quasilinear satisﬁes satisfies the following solution of 1.1 solvable solve square integrable stochastic diﬀerential equations Suppose Theorem 2.1 Theory uniformly bounded uniformly Lipschitz unique adapted solution uniquely solvable value function viscosity solution well-posedness