SCALE TRANSFORMATIONS FOR PRESENT POSITION-INDEPENDENT CONDITIONAL EXPECTATIONS

Let C(0,t) denote a generalized Wiener space, the space of real-valued continuous functions on the interval (0,t) and define a random vector Zn : C(0,t) → R n by Zn(x) = ( R t 1 0 h(s)dx(s),..., R tn 0 h(s)dx(s)), where 0 < t1 < ··· < tn < t is a partition of (0,t) and h ∈ L2(0,t) with h 6 0 a.e. In this paper we will introduce a simple formula for a generalized conditional Wiener integral on C(0,t) with the conditioning function Zn and then evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function F(x) = f( R t 0 e(s)dx(s)) for x ∈ C(0,t), where f ∈ Lp(R)(1 ≤ p ≤ ∞) and e is a unit element in L2(0,t). Finally we express the generalized analytic conditional Feynman integral of F as two kinds of limits of non-conditional generalized Wiener integrals of polygonal functions and of cylinder functions using a change of scale transformation for which a normal density is the kernel. The choice of a complete orthonormal subset of L2(0,t) used in the transformation is independent of e and the conditioning function Zn does not contain the present positions of the generalized Wiener paths.