Radon-Nikodym Derivatives with Respect to Wiener Measure

Abstract

Let W(t;ω) denote the n-dimensional Wiener process, 0 ≤ t ≤ 1, inducing the Wiener measure on C= C(0, 1) as we have indicated. Let ϕ(ω) denote a (Borel) measurable function mapping into C into C. Then ϕ(·) induces a measure on the Borel sets of C given by:

$${p_\varphi }(B)\; = \;{p_W}({\varphi ^{ - 1}}(B))$$

where p_W denotes the Wiener measure. For example, often ϕ(·) may be defined by means of a stochastic integral:

$$x(t;\omega ) = L(t)\smallint _0^tM(s)dW(S;\omega )$$

where say the functions L(·), M(·) are continuous. Of particular importance is to determine when the induced measure is absolutely continuous with respect to Wiener measure, and then to evaluate the Radon-Nikodym derivative⁺. In the present volume we shall mostly be concerned with the case where ϕ(·) is a linear transformation (or affine transformation).