Article

# On Girsanov's transform for backward stochastic differential equations

11/2010;

Source: arXiv

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**ABSTRACT:**This book uses the so-called Malliavin calculus and large deviation techniques to study the asymptotics as t↓↓0 of the conditional probabilities of bridges associated with certain hypoelliptic diffusions. In particular, stochatic differential equations of the form dx=∑ i=1 m X i (x)dw i ;x(0)=x 0 are studied, where X 1 ,···,X m are smooth vector fields as on a manifold M. Let be x s =ψ s (w). Associated to the latter differential equation is the second order operator L=1/2∑ i=1 m X i 2 . The law of ψ 1 (w) is given by the kernel e L (x 0 ,·). Under certain standard assumptions of Hörmander, one considers the asymptotics of the law p t (x) of ψ 1 (tdw) as t↓↓0· In the elliptic case, rather complete results are obtained. In case ”hypoelliptic” is substituted for ”elliptic”, some basic machinery is developed and some conjectures are formulated. The special case of the Hessenberg group is considered in some detail. The book assumes a fair amount of sophistication on the part of the reader - some background in probability, Riemannian geometry, and the basic language of differential equations is advised. However the author is fairly kind to his readers. He works out special cases, writes out most details, and gives copious references. The author is able to keep his book fairly brief and yet provide an introduction to an active area of research. - [Show abstract] [Hide abstract]

**ABSTRACT:**The book consists of 4 chapters, and an index. There is a bibliography at the end of each chapter. The book deals with quasilinear hyperbolic systems in one and several space variables. These systems are important in fluid mechanics, combustion theory and elsewhere. In applications these systems can be often written as systems of conservation laws. In chapter 1, introduction, some physical examples are given (gas dynamics, combustion theory, nonlinear wave equation, etc.), and weakly nonlinear asymptotics of solutions of a class of systems of conservation laws are discussed. In chapter 2, smooth solutions and the equations of incompressible fluid flow and the existence of smooth solutions for a general system of conservation laws with smooth initial data is discussed. The smooth solution may not exist because of the blow-up or because of the formation of shock waves. Compressible and incompressible fluid flows and combustion equations at low Mach’s number are studied. In chapter 3, the formation of shock waves in smooth solutions and a number of the results concerning the breakdown of smooth solutions are discussed (shock formation for scalar conservation laws in several space variables, for 2×2 strictly hyperbolic systems, for some m×m systems, and for a quasilinear wave equation). In chapter 4, the existence and stability of shock fronts in several space variables, the discontinuous weak solutions of systems of conservation laws in several space variables are discussed. - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper we investigate the nature of the adapted solutions to a class of forward-backward stochastic differential equations (SDEs for short) in which the forward equation is non-degenerate. We prove that in this case the adapted solution can always be sought in an ordinary sense over an arbitrarily prescribed time duration, via a direct Four Step Scheme. Using this scheme, we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic PDE system. Moreover the uniqueness of the adapted solutions (over an arbitrary time duration), as well as the continuous dependence of the solutions on the parameters, can all be proved within this unified framework. Some special cases are studied separately. In particular, we derive a new form of the integral representation of the Clark-Haussmann-Ocone type for functionals (or functions) of diffusions, in which the conditional expectation is no longer needed.Probability Theory and Related Fields 08/1994; 98(3):339-359. · 1.39 Impact Factor

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