On Girsanov's transform for backward stochastic differential equations

Source: arXiv

ABSTRACT By using a simple observation that the density processes appearing in Ito's martingale representation theorem are invariant under the change of measures, we establish a non-linear version of the Cameron-Martin formula for solutions of a class of systems of quasi-linear parabolic equations with non-linear terms of quadratic growth. We also construct a local stochastic flow and establish a Bismut type formula for such system of quasi-linear PDEs. Gradient estimates are obtained in terms of the probability representation of the solution. Another interesting aspect indicated in the paper is the connection between the non-linear Cameron-Martin formula and a class of forward-backward stochastic differential equations(FBSDEs). Comment: 21 pages

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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.
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    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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