Conditional Wiener integral associated with Gaussian processes and applications

Abstract

Let $C_0[0,T]$ denote the one-parameter Wiener space and let $C_0'[0,T]$ be the Cameron--Martin space in $C_0[0,T]$. Given a function $k$ in $C_0'[0,T]$, define a stochastic process $\mathcal Z_k:C_0[0,T]\times[0,T]\to\mathbb R$ by $\mathcal Z_k(x,t)=\int_0^t Dk(s) d x(s)$, where $Dk\equiv \tfrac{d}{dt}k$. Let a random vector $X_{\mathcal G,k} : C_{0}[0,T]\to \mathbb R^n$ be given by
$$
X_{\mathcal G,k}(x)
=((g_1, \mathcal Z_k(x,\cdot))^{\sim},\ldots,(g_n,\mathcal Z_k(x,\cdot))^{\sim}),
$$
where $\mathcal G=\{g_1,\ldots, g_n\}$ is an orthonormal set with respect to the weighted inner product induced by the function $k$ on the space $C_0'[0,T]$, and $(g ,\mathcal Z_k(x,\cdot))^{\sim}$ denotes the Paley--Wiener--Zygmund stochastic integral. In this paper, using the reproducing kernel property of the Cameron--Martin space, we establish a very general evaluation formula for expressing conditional generalized Wiener integrals,
$E\big(F( \mathcal{Z}_k (x,\cdot)) \big\vert X_{\mathcal{G},k}(x)=\vec\eta \big)$, associated with the Gaussian processes $\mathcal Z_k$. As an application, we establish a translation theorem for the
conditional Wiener integral and then use it to obtain various conditional Wiener integration formulas on $C_0[0,T]$.