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Representation of Itô integrals by Lebesgue/Bochner integrals

Qi Lü, Jiongmin Yong, Xu Zhang (2012)

Journal of the European Mathematical Society

In [Yong 2004], it was proved that as long as the integrand has certain properties, the corresponding Itô integral can be written as a (parameterized) Lebesgue integral (or a Bochner integral). In this paper, we show that such a question can be answered in a more positive and refined way. To do this, we need to characterize the dual of the Banach space of some vector-valued stochastic processes having different integrability with respect to the time variable and the probability measure. The later...

Semimartingale decomposition of convex functions of continuous semimartingales by brownian perturbation

Nastasiya F. Grinberg (2013)

ESAIM: Probability and Statistics

In this note we prove that the local martingale part of a convex function f of a d-dimensional semimartingale X = M + A can be written in terms of an Itô stochastic integral ∫H(X)dM, where H(x) is some particular measurable choice of subgradient ∇ f ( x ) off at x, and M is the martingale part of X. This result was first proved by Bouleau in [N. Bouleau, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981) 87–90]. Here we present a new treatment of the problem. We first prove the result for X ˜ = X + ϵ B x10ff65;...

Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition

Marek T. Malinowski (2015)

Open Mathematics

We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors). The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect...

Set-valued stochastic integrals and stochastic inclusions in a plane

Władysław Sosulski (2001)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We present the concepts of set-valued stochastic integrals in a plane and prove the existence of a solution to stochastic integral inclusions of the form z s , t ∈ φ s , t + ∫ 0 s ∫ 0 t F u , v ( z u , v ) d u d v + ∫ 0 s ∫ 0 t G u , v ( z u , v ) d w u , v

Set-valued Stratonovich integral

Anna Góralczyk, Jerzy Motyl (2006)

Discussiones Mathematicae Probability and Statistics

The purpose of the paper is to introduce a set-valued Stratonovich integral driven by a one-dimensional Brownian motion. We discuss the existence of this integral and investigate its properties.

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