Given an integer m, a probability measure ν on [0, 1], a process X and a real function g, we define the m-order ν-integral having as integrator X and as integrand g(X). In the case of the fractional Brownian motion B , for any locally bounded function g, the corresponding integral vanishes for all odd indices m > 1 2H and any symmetric ν. One consequence is an Itô-Stratonovich type expansion fo...

We deduce a nonlinear and inhomogeneous Fokker-Planck equation within a generalized Stratonovich, or stochastic α, prescription (α=0, 1/2, and 1, respectively, correspond to the Itô, Stratonovich and anti-Itô prescriptions). We obtain its stationary state p(st)(x) for a class of constitutive relations between drift and diffusion and show that it has a q-exponential form, p(st)(x)=N(q)[1-(1-q)βV...

Spectral representations of iterated Itô and Stratonovich stochastic integrals arbitrary multiplicity, including from Taylor–Itô Taylor–Stratonovich expansions, are obtained by the spectral method. They required for implementation numerical methods solving differential equations with high orders mean-square strong convergence. The purpose such is modeling nonlinear dynamics in many fields. This...

Multiple Stratonovich integrals (MSI) with respect to the Wiener process and the Brownian bridge are defined for a class of kernels having k-th order t-traces which are, in general different from the traces investigated in earlier work. Asymptotic distributions of V-statistics are derived and the limiting distribution expressed in terms of appropriate MSI. Another application yields an alternat...

Introduction: Diffusion Weighted Magnetic Resonance Imaging (DWMRI) provides visual contrast, depends on Brownian motion of water molecules. The diffusive behavior of water in cells alters in many disease states. Dephasing is a factor of magnetic field inhomogeneity, heterogeneity of tissue and etc., which is associated with the signal amplitude. In a series of DWI acquisition...

We provide a detailed hands-on tutorial for the R Development Core Team [2014] add-on package Sim.DiffProc [Guidoum and Boukhetala, 2014], for symbolic and floating point computations in stochastic calculus and stochastic differential equations (SDEs). The package implement is introduced and it is explains how to use the snssde1d, snssde2d and snssde3d main functions in this package, for simula...