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A generalization of the conservation integral

Volkmar Liebscher (1998)

Banach Center Publications

Starting from the scheme given by Hudson and Parthasarathy [7,11] we extend the conservation integral to the case where the underlying operator does not commute with the time observable. It turns out that there exist two extensions, a left and a right conservation integral. Moreover, Itô's formula demands for a third integral with two integrators. Only the left integral shows similar continuity properties to that derived in [11] used for extending the integral to more than simple integrands. In...

A Malliavin calculus method to study densities of additive functionals of SDE’s with irregular drifts

Arturo Kohatsu-Higa, Akihiro Tanaka (2012)

Annales de l'I.H.P. Probabilités et statistiques

We present a general method which allows to use Malliavin Calculus for additive functionals of stochastic equations with irregular drift. This method uses the Girsanov theorem combined with Itô–Taylor expansion in order to obtain regularity properties for this density. We apply the methodology to the case of the Lebesgue integral of a diffusion with bounded and measurable drift.

A Milstein-type scheme without Lévy area terms for SDEs driven by fractional brownian motion

A. Deya, A. Neuenkirch, S. Tindel (2012)

Annales de l'I.H.P. Probabilités et statistiques

In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second-order Taylor expansion, where the usual Lévy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds...

Analysis of the Rosenblatt process

Ciprian A. Tudor (2008)

ESAIM: Probability and Statistics

We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears as limit in the so-called Non Central Limit Theorem (Dobrushin and Majòr (1979), Taqqu (1979)). This process is non-Gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Itô multiple integral with respect to the Brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin...

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