[For the entire collection see Zbl 0724.00018.]
This is a survey of some recent results on the absolute continuity of the image of the Wiener measure \(P\) by anticipating Girsanov transformations. More precisely, for a process \((u_ t; 0 \leq t \leq 1)\), let the transformation \(T\) on \(C_ 0 ([0,1])\) be defined by \(T(\omega)_ t=\omega_ t+\int^ t_ 0 u_ s (\omega) ds\). The author gives the non-adapted analogue of the Girsanov exponential associated to \(T\), where the Itô integral with respect to the Wiener process is replaced by the Skorokhod integral. He reviews sufficient conditions due to himself and M. Zakai [Probab. Theory Relat. Fields 73, 255-280 (1986; Zbl 0601.60053)], S. Kusuoka [J. Fac. Sci., Univ. Tokyo, Sect. I A 29, 567-597 (1982; Zbl 0525.60050)], and to R. Buckdahn [Probab. Theory Relat. Fields 89, No. 2, 211-238 (1991; Zbl 0722.60059)] in order that (a) the image of the measure \(P \circ T^{-1}\) is absolutely continuous with respect to \(P\) and (b) there exists a probability measure \(Q \ll P\) such that \(T\) is a Brownian motion under \(Q\).
The author applies these results to the solution of stochastic differential equations (SDE) with initial and final condition and to the Markov field property of their solution. For the first order SDE \(dX_ t+f (X_ t)=dW_ t\), the law of \(X_ t\) is identified by means of the law of the solution of \(dY_ t+\lambda Y_ t dt=dW_ t\) and the transformation \(T\) associated to \(u_ t=\overline f(Y_ t)\), where \(f(x)=\lambda x+\overline f(x)\). Similarly, the solution of the second order equation \(d^ 2X_ t/dt^ 2+g(X_ t)=dW_ t/dt\) with Neumann boundary conditions is identified.