Itô Formula for Generalized Functionals of Brownian Bridge

Abstract

Employing the calculus on the classical Wiener space (C′, C) we represent the Brownian motion {B(t)} by B(t, x) = (x, α t) for x ∈ C, where (·, ·) is the C−C * pairing and α t is a function in C* such that α t (s) = min{t, s} for t ∈ [0, 1] and for s < t. It follows that Brownian bridge is represented by X (t, x) = (x, βt) for x ∈ C, where βt = αt − tα 1. Using such a representation, we define and study the generalized functionals associated with the Brownian bridge. It is shown that Itˆo formula for Brownian bridge may be derived without using the classical stochastic integration theory. In order to compare the Itˆo formula of Brownian bridge with the formula under the scheme of semimartingale theory we also consider the semimartingale version of the Brownian bridge represented by \(\hat{X}(t,x) = (x,{{\hat{\beta }}_{t}})\) , for x ∈ C, where \({{\hat{\beta }}_{t}}(s) = - (1 - t)\ln (1 - s \wedge t)\) for t < 1 and \({{\hat{\beta }}_{1}} \equiv 0\) . Its is shown that the Ito formula depends only on the variance parameter t(1 − t) of the Brownian bridge.