This paper introduces a novel approach to integrals with respect to capacities. Any random variable is decomposed as a combination of indicators. A prespecified set of collections of events indicates which decompositions are allowed and which are not. Each allowable decomposition has a value determined by the capacity. The decomposition-integral of a random variable is defined as the highest of...

This paper introduces a novel approach to integrals with respect to capacities. Any random variable is decomposed as a combination of indicators. A pre-speci ed set of collections of events indicates which decompositions are allowed and which are not. Each allowable decomposition has a value determined by the capacity. The decomposition-integral of a random variable is de ned as the highest of ...

Using time-reversal, we introduce the stochastic integration for zero-energy additive functionals of symmetric Markov processes, which extends an early work of S. Nakao. Various properties of such stochastic integrals are discussed and an Itô formula for Dirichlet processes is obtained. AMS 2000 Mathematics Subject Classification: Primary 31C25; Secondary 60J57, 60J55, 60H05.

We consider a real-valued path; it is possible to associate a tree to this path, and we explore the relations between the tree, the properties of p-variation of the path, and integration with respect to the path. In particular, the fractal dimension of the tree is estimated from the variations of the path, and Young integrals with respect to the path, as well as integrals from the rough paths t...

We consider nonlinear stochastic integrals of Itô-type w.r.t. a family of semimartingales which depend on a spatial parameter. These integrals were introduced by Carmona and Nualart [2], Kunita [8], and Le Jan [9]. The extension of the elementary nonlinear integral is based on the condition that the semimartingale kernel has nice continuity properties in the spatial parameter. We investigate th...

We consider a real-valued path; it is possible to associate a tree to this path, and we explore the relations between the tree, the properties of p-variation of the path, and integration with respect to the path. In particular, the fractal dimension of the tree is estimated from the variations of the path, and Young integrals with respect to the path, as well as integrals from the rough paths t...

Using key tools such as Itô’s formula for general semimartingales, Kunita’s moment estimates for Lévy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the stochastic differential equations (SDEs) driven by Lévy noise are stable in probability, almost surely and moment exponentially stable. Keywords; stochastic differential equ...

Stochastic linear programming (SLP) models involve multivariate integrals. Although in the discretely distributed case these integrals become sums they typically contain a large amount of terms. The purpose of this paper is twofold: On the one hand we discuss the usage of bounds concerning integrals for constructing SLP algorithms and secondly we point out the role of bounds–based algorithms fo...

By using Malliavin calculus and multiple Wiener-Itô integrals, we study the existence and the regularity of stochastic currents defined as Skorohod (divergence) integrals with respect to the Brownian motion and to the fractional Brownian motion. We consider also the multidimensional multiparameter case and we compare the regularity of the current as a distribution in negative Sobolev spaces wit...

In this contribution we define a new class of non-linear integrals based on decomposition integrals. These integrals are motivated by greediness of many real-life situations. Another view on this new class of integrals is that it is a generalization of both the Shilkret and PAN integrals. Moreover, it can be seen as an iterated Shilkret integral. Also, an example in time-series analysis is prov...