Stochastic Integrals and Abelian Processes

Stochastic Integrals and Abelian Processes

We study triangulation schemes for the joint kernel of a diffusion process with uniformly continuous coefficients and an adapted, non-resonant Abelian process. The prototypical example of Abelian process to which our methods apply is given by stochastic integrals with uniformly continuous coefficients. The range of applicability includes also a broader class of processes of practical relevance,...

Nonlinear stochastic integrals for hyperfinite Lévy processes

We develop a notion of nonlinear stochastic integrals for hyperfinite Lévy processes, and use it to find exact formulas for expressions which are intuitively of the form Pt s=0 φ(ω, dls, s) and Qt s=0 ψ(ω, dls, s), where l is a Lévy process. These formulas are then applied to geometric Lévy processes, infinitesimal transformations of hyperfinite Lévy processes, and to minimal martingale measure...

Fast quantum algorithms for numerical integrals and stochastic processes

We discuss quantum algorithms that calculate numerical integrals and descriptive statistics of stochastic processes. With either of two distinct approaches, one obtains an exponential speed increase in comparison to the fastest known classical deterministic algorithms and a quadratic speed increase in comparison to classical Monte Carlo (probabilistic) methods. We derive a simpler and slightly ...

Stochastic Processes as Fourier Integrals and Dilation of Vector Measures

Abelian integrals in holomorphic foliations

The aim of this paper is to introduce the theory of Abelian integrals for holomorphic foliations in a complex manifold of dimension two. We will show the importance of Picard-Lefschetz theory and the classification of relatively exact 1-forms in this theory. As an application we identify some irreducible components of the space of holomorphic foliations of a fixed degree and with a center singu...