Tangential existence and comparison, with applications to single and multiple integration

TANGENTIAL EXISTENCE AND COMPARISON, WITH APPLICATIONS TO SINGLE AND MULTIPLE INTEGRATIONTwo semi-martingales with respect to a common filtration are said to be tangential if they have the same local characteristics. When the latter are non-random, the underlying semi-martingale is known to have independent increments. We show that every semi-martingale has a tangential process with conditionally independent increments. We also extend the Zinn–Hitchenko and related tangential comparison theorems to continuous time. Combining those results, we obtain some surprisingly general existence,convergence, and tightness criteria for broad classes of single and multiple stochastic integrals.

The Implementation of Milstein Scheme in Two-Dimensional SDEs Using the Fourier Method

Multiple stochastic integrals of higher multiplicity cannot always be expressed in terms of simpler stochastic integrals, especially when the Wiener process is multidimensional. In this paper we describe how the Fourier series expansion of Wiener process can be used to simulate a two-dimensional stochastic differential equation (SDE) using Matlab program. Our numerical experiments use Matlab to show how our truncation of Itô’-Taylor expansion at an appropriate point produces Milstein method for the SDE.

Minimax regression estimation for Poisson coprocess

For a Poisson point process X, Itô’s famous chaos expansion implies that every square integrable regression function r with covariate X can be decomposed as a sum of multiple stochastic integrals called chaos. In this paper, we consider the case where r can be decomposed as a sum of δ chaos. In the spirit of Cadre and Truquet [ESAIM: PS 19 (2015) 251–267], we introduce a semiparametric estimate of r based on i.i.d. copies of the data. We investigate the asymptotic minimax properties of our estimator when δ is known. We also propose an adaptive procedure when δ is unknown.

Transformations of Wiener measure and orthogonal expansions

In this paper we study the structure of square integrable functionals measurable with respect to coalescing stochastic flows. The case of the Wiener process stopped at the moment of hitting an irregular continuous level is considered. Relying on the change of measure technique, we present a new construction of multiple stochastic integrals with respect to stopped Wiener process. An intrinsic analogue of the Itô–Wiener expansion for the space of square integrable functionals measurable with respect to the stopped Wiener process is constructed.