In this study, a new application of Taylor expansion is considered to estimate the solution of Volterra-Fredholm integral equations (VFIEs) and systems of Volterra-Fredholm integral equations (SVFIEs). Our proposed method is based upon utilizing the nth-order Taylor polynomial of unknown function at an arbitrary point and employing integration method to convert VFIEs into a system of linear equ...

In this paper, we apply the differential transform (DT) method for finding approximate solution of the system of linear and nonlinear Volterra integro-differential equations with variable coefficients, especially of higher order. We also obtain an error bound for the approximate solution. Since, in this method the coefficients of Taylor series expansion of solution is obtained by a recurrence r...

The relative entropy rate is a natural and useful measure of distance between two stochastic processes. In this paper we study the relative entropy rate between two Hidden Markov Processes (HMPs), which is of both theoretical and practical importance. We give new results showing analyticity, representation using Lyapunov exponents, and Taylor expansion for the relative entropy rate of two discr...

I survey developments over the last decade in harmonic analysis on Lie groups relative to a heat kernel measure. These include analogs of the Hermite expansion, the Segal-Bargmann transform, and the Taylor expansion. Some of the results can be understood from the standpoint of geometric quantization. Others are intimately related to stochastic analysis.

In this paper, by using a Taylor development type formula, we show how it is possible to associate differential operators with stochastic differential equations driven by a fractional Brownian motion. As an application, we deduce that invariant measures for such SDEs must satisfy an infinite dimensional system of partial differential equations.

We give a summary on the geometry of iterated Stratonovich integrals. For this exposition, we always have the connection to stochastic Taylor expansion in mind. In particular, we believe that “cubature on Wiener space” is best understood in the setting presented in this text. Besides cubature on Wiener space, we also give a second application regarding the heat kernel on nilpotent free Lie groups.

David J. Nott, Siew Li Tan, Mattias Villani and Robert Kohn, Regression density estimation with variational methods and stochastic approximation, 2012, Journal of Computational And Graphical Statistics, (21), 3, 797-820. Journal of Computational And Graphical Statistics is available online at informaworld TM : http://dx.doi.org/10.1080/10618600.2012.679897 Copyright: American Statistical Associ...

We consider the necessary conditions for backward doubly stochastic control system, via the second-order Taylor expansion we have obtained. All the results are got under no restriction on the convexity of control domain and the diffusion coefficient does not contain the control variable.