On extended stochastic integrals with respect to Lévy processes

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...

Set-Valued Stochastic Integrals with Respect to Finite Variation Processes

In a Euclidean space , the Lebesgue-Stieltjes integral of set-valued stochastic processes d R       , 0, t F F t T    with respect to real valued finite variation process       , 0, t A t T   t is defined directly by employing all integrably bounded selections instead of taking the decomposable closure appearing in some existed references. We shall show that this kind of integr...

Lebesgues-Stieltjes Integrals of Fuzzy Stochastic Processes with Respect to Finite Variation Processes

Let ( ) [ ] { } t G G t T = ∈ ω , 0, be a fuzzy stochastic process and ( ) [ ] { } t A t T ∈ ω , 0, be a real valued finite variation process. We define the Lebesgue-Stieltjes integral denoted by ( ) ( ) ∫ t s s G A ω ω 0 d for each t > 0 by using the selection method, which is direct, nature and different from the indirect definition appearing in some references. We shall show that this kind o...

A Probability Bound for Integrals with Respect to Stochastic Processes with Independent Increments1

Stochastic Integration with respect to Gaussian Processes

We construct a Stratonovitch-Skorohod-like stochastic integral for general Gaussian processes. We study its sample-paths regularity and one of its numerical approximating schemes. We also analyze the way it is transformed by an absolutely continuous change of probability and we give an Itô formula. c 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Intégrale stochasti...