Small deviations for integrators in terms of Itô--Wiener expansions

Abstract

In this paper, we investigate small deviations for a class of Gaussian processes, known as integrators. In general, such processes are non-Markovian and may even fail to be semimartingales. First, we investigate small deviations for the multi-dimensional Wiener process, focusing on the asymptotic behavior of the Fourier--Wiener transform of the associated hitting event. We analyze the Fourier--Wiener transform and the separate summands in its Itô--Wiener expansion. Then, we establish the asymptotic limits of the associated multiple integrals with heat kernels. Finally, we obtain the small deviation asymptotic result for the integrator $\eta$ associated with the continuous linear operator.