Conditional wiener integral associated with gaussian processes and applications

Jae Gil Choi

Let C0[0,T] denote the one-parameter Wiener space and let C′0[0,T] be the Cameron–Martin space in C0[0,T]. Given a function k in C′0[0,T], define a stochastic process Zk : C0[0,T] × [0,T] → R by Zk(x, t) = ∫ t 0 Dk(s)dx(s), where Dk ≡ ddt k. Let a random vector XG,k : C0[0,T] → Rn be given by XG,k(x) = ((11,Zk(x, ·))∼, . . . , (1n,Zk(x, ·))∼), where G = {11, . . . , 1n} is an orthonormal set with respect to the weighted inner product induced by the function k on the space C′0[0,T], and (1,Zk(x, ·))∼ 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 ( F(Zk(x, ·)) ∣∣∣XG,k(x) = η⃗ ) , associated with the Gaussian processes Zk. 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 C0[0,T].