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...

As biomedical research advances there is an increasing need to model and simulate more complicated systems to better understand them. Since biochemical processes are inherently stochastic and often contain both continuous and discrete behavior, stochastic hybrid systems are an ideal modeling paradigm for capturing their dynamics. In this paper we present a framework for modeling biochemical sys...

in this paper, the reduced dierential transform method is investigated fora nonlinear partial dierential equation modeling nematic liquid crystals, itis called the hunter-saxton equation. the main advantage of this methodis that it can be applied directly to nonlinear dierential equations withoutrequiring linearization, discretization, or perturbation. it is a semi analytical-numerical metho...

In this study, a Taylor method is developed for numerically solving the high-order most general nonlinear Fredholm integro-differential-difference equations in terms of Taylor expansions. The method is based on transferring the equation and conditions into the matrix equations which leads to solve a system of nonlinear algebraic equations with the unknown Taylor coefficients. Also, we test the ...

The main idea of this paper is to demonstrate a stochastic computational technique consisting of the generalized stochastic perturbation method using the Taylor expansions of random variables and the classical Finite Difference Method based on regular grids. As it is documented by computational illustrations, it is possible to determine, using this approach, also higher probabilistic moments fo...

The construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied. An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order. In general the proposed symplectic schemes are fully implicit, and they become computationally expensive for mean square orders greater than two. However, for stochasti...

Optimal as well as recursive parameter estimation for semimartingales had been studied in Thavaneswaran and Thompson [1, 2]. Recently, there has been a growing interest in modeling volatility of the observed process by nonlinear stochastic processes (Taylor [3]). In this paper, we study the recursive estimates for various classes of discretely sampled continuous time stochastic volatility model...

In this article, we propose an analytical approximation for the pricing of European options for some lognormal stochastic volatility models. This approximation is a second-order Taylor series expansion of the Fourier transform with respect to the "volatility of volatility". We give, using these formulas, a new method of variance reduction for the Monte-Carlo simulation of the trajectories of th...