Analytic continuation of the change of variable formula for smooth measures

Abstract

We also show that, if the standard Feynman‐Kac formula is assumed to be known, then the classical Feynman‐Kac‐Ito theorem turns out to be an immediate corollary of the generalized Girsanov‐Maruyama formula. This result is an alternative interpretation of the remark on p. 172 of Simon’s book [4], which is incorrect in its original form. An important role in obtaining the results which we discuss is played by the notion of the logarithmic derivative of a function of a real argument taking values in the space of real (alternating) measures, which was introduced in [2]. We apply this notion to define the logarithmic derivative of such a measure along a vector field taking values in the complexification of the space of differentiability of this measure. In fact, we define the logarithmic derivative of a function of a real argument whose values are analytic continuations (to a suitable domain in the complex plane) of some functions of another real argument, taking values in the space of real measures. Such analytic extensions take values in the complexification of the space of real measures, i.e., in the space of complex-valued measures. In turn, the logarithmic derivative thus defined is an analytic continuation with respect to the same parameter of the logarithmic derivative of the just mentioned function taking values in the space of real measures. The logarithmic derivative is used to determine the corresponding trans