This article proposes an optimal method for approximate answer of stochastic Ito-Voltrra integral equations, via rationalized Haar functions and their stochastic operational matrix of integration. Stochastic Ito-voltreea integral equation is reduced to a system of linear equations. This scheme is applied for some examples. The results show the efficiency and accuracy of the method.

In this paper, multi-dimensional Wiener-Liu process is proposed. Wiener-Liu process is a type of hybrid process, it corresponds to Brownian motion (Wiener process) in stochastic process and Liu process in fuzzy process. In classical analysis, the basic operations are differential and integral. Correspondingly, Ito-Liu formula plays the role of Ito formula in stochastic process and Liu formula i...

in this article,we present a wavelet method for solving stochastic volterra integral equations based on haar wavelets. first, we approximate all functions involved in the problem by haar wavelets then, by substituting the obtained approximations in the problem, using the it^{o} integral formula and collocation points then, the main problem changes into a system of linear or nonlinear equation w...

in this paper, a numerical efficient method based on two-dimensional block-pulse functions (bpfs) is proposed to approximate a solution of the two-dimensional linear stochastic volterra-fredholm integral equation. finally the accuracy of this method will be shown by an example.

1 Prliminaries 2 1.1 Basic Probability concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Stochastic Differential Equations (SDE’s) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Langevin equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Ito integral and basics of Ito calculus . . . . . . . . . . . . . . ....

The study of fluctuations in gene regulatory networks is extended to the case of Gaussian colored noise. First, the solution of the corresponding Langevin equation with colored noise is expressed in terms of an Ito integral. Then, two important lemmas concerning the variance of an Ito integral and the covariance of two Ito integrals are shown. Based on the lemmas, we give the general formulas f...

Stress-induced failure is a critical concern that influences the mechanical reliability of an indium tin oxide (ITO) film deposited on a transparently flexible polyethylene terephthalate (PET) substrate. In this study, a cycling bending mechanism was proposed and used to experimentally investigate the influences of compressive and tensile stresses on the mechanical stability of an ITO film depo...

2 Ito-Doeblin’s formula(s) 7 2.1 First formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Continuous semi-martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Integration by parts formula . . . . . . . . . . . ...