40,101 research outputs found
Development and Application of the Fourier Method to the Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals
The article is devoted to the mean-square approximation of iterated Ito and
Stratonovich stochastic integrals in the context of the numerical integration
of Ito stochastic differential equations. The expansion of iterated Ito
stochastic integrals of arbitrary multiplicity and
expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to
5 have been obtained. Considerable attention is paid to expansions based on
multiple Fourier-Legendre series. The exact and approximate expressions for the
mean-square error of approximation of iterated Ito stochastic integrals are
derived. The results of the article will be useful for numerical integration of
Ito stochastic differential equations with non-commutative noise.Comment: 56 pages. Minor changes along the text in the whol
Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 2. Combined Approach Based on Generalized Multiple and Iterated Fourier Series
The article is devoted to the expansion of iterated Stratonovich stochastic
integrals of multiplicity 2 on the base of the combined approach of generalized
multiple and iterated Fourier series. We consider two different parts of the
expansion of iterated Stratonovich stochastic integrals. The mean-square
convergence of the first part is proved on the base of generalized multiple
Fourier series converging in the mean-square sense in the space The mean-square convergence of the second part is proved on the base of
generalized iterated (double) Fourier series converging pointwise. At that, we
prove the iterated limit transition for the second part of the expansion on the
base of the classical theorems of mathematical analysis. The results of the
article can be applied to the numerical integration of Ito stochastic
differential equations.Comment: 18 pages. Sect. 3 was added. arXiv admin note: text overlap with
arXiv:1801.05654, arXiv:1801.00784, arXiv:1801.01564, arXiv:1712.09746,
arXiv:1801.03195, substantial text overlap with arXiv:1712.0951
The Hypotheses on Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity and Their Partial Proof
In this review article we collected more than ten theorems on expansions of
iterated Ito and Stratonovich stochastic integrals, which have been formulated
and proved by the author. These theorems open a new direction for study of
iterated Ito and Stratonovich stochastic integrals. The expansions based on
multiple and iterated Fourier-Legendre series as well as on multiple and
iterated trigonomectic Fourier series converging in the mean and pointwise are
presented in the article. Some of these theorems are connected with the
iterated stochastic integrals of multiplicities 1 to 5. Also we consider two
theorems on expansions of iterated Ito stochastic integrals of arbitrary
multiplicity based on generalized multiple Fourier
series converging in the sense of norm in Hilbert space as well
as two theorems on expansions of iterated Stratonovich stochastic integrals of
arbitrary multiplicity based on generalized iterated
Fourier series converging pointwise. On the base of the presented theorems we
formulate 3 hypotheses on expansions of iterated Stratonovich stochastic
integrals of arbitrary multiplicity based on generalized
multiple Fourier series converging in the sense of norm in Hilbert space
The mentioned iterated Stratonovich stochastic integrals are
part of the Taylor-Stratonovich expansion. Moreover, the considered expansions
from these 3 hypotheses contain only one operation of the limit transition and
substantially simpler than their analogues for iterated Ito stochastic
integrals. Therefore, the results of the article can be useful for the
numerical integration of Ito stochastic differential equations. Also, the
results of the article were reformulated in the form of theorems of the
Wong-Zakai type for iterated Stratonovich stochastic integrals.Comment: 35 pages. Section 12 was added. arXiv admin note: text overlap with
arXiv:1712.09516, arXiv:1712.08991, arXiv:1802.04844, arXiv:1801.00231,
arXiv:1712.09746, arXiv:1801.0078
Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 4 Based on Generalized Multiple Fourier Series
The article is devoted to the expansions of iterated Stratonovich stochastic
integrals of multiplicities 1 to 4 on the basis of the method of generalized
multiple Fourier series that are converge in the sense of norm in Hilbert space
Mean-square convergence of the expansions for the
case of multiple Fourier-Legendre series and for the case of multiple
trigonometric Fourier series is proved. The considered expansions contain only
one operation of the limit transition in contrast to its existing analogues.
This property is very important for the mean-square approximation of iterated
stochastic integrals. The results of the article can be applied to numerical
integration of Ito stochastic differential equations with multidimensional
non-commutative noises.Comment: 50 pages. Sect. 6 is added. Minor changes along the article in the
whole. arXiv admin note: text overlap with arXiv:1712.0899
Entropy computing via integration over fractal measures
We discuss the properties of invariant measures corresponding to iterated
function systems (IFSs) with place-dependent probabilities and compute their
Renyi entropies, generalized dimensions, and multifractal spectra. It is shown
that with certain dynamical systems one can associate the corresponding IFSs in
such a way that their generalized entropies are equal. This provides a new
method of computing entropy for some classical and quantum dynamical systems.
Numerical techniques are based on integration over the fractal measures.Comment: 14 pages in Latex, Revtex + 4 figures in .ps attached (revised
version, new title, several changes, to appear in CHAOS
Finite element computation of a viscous compressible free shear flow governed by the time dependent Navier-Stokes equations
A finite element algorithm for solution of fluid flow problems characterized by the two-dimensional compressible Navier-Stokes equations was developed. The program is intended for viscous compressible high speed flow; hence, primitive variables are utilized. The physical solution was approximated by trial functions which at a fixed time are piecewise cubic on triangular elements. The Galerkin technique was employed to determine the finite-element model equations. A leapfrog time integration is used for marching asymptotically from initial to steady state, with iterated integrals evaluated by numerical quadratures. The nonsymmetric linear systems of equations governing time transition from step-to-step are solved using a rather economical block iterative triangular decomposition scheme. The concept was applied to the numerical computation of a free shear flow. Numerical results of the finite-element method are in excellent agreement with those obtained from a finite difference solution of the same problem
Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 4. Combained Approach Based on Generalized Multiple and Iterated Fourier series
The article is devoted to the expansions of iterated Stratonovich stochastic
integrals of multiplicities 1 to 4 on the base of the combined approach of
generalized multiple and iterated Fourier series. We consider two different
parts of the expansion of iterated Stratonovich stochastic integrals. The
mean-square convergence of the first part is proved on the base of generalized
multiple Fourier series that are converge in the sense of norm in Hilbert space
The mean-square convergence of the second part is
proved on the base of generalized iterated Fourier series that are converge
pointwise. At that, we do not use the iterated Ito stochastic integrals as a
tool of the proof and directly consider the iterated Stratonovich stochastic
integrals. The cases of multiple Fourier-Legendre series and multiple
trigonometric Fourier series are considered in detail. The considered
expansions contain only one operation of the limit transition in contrast to
its existing analogues. This property is very important for the mean-square
approximation of iterated stochastic integrals. The results of the article can
be applied to the numerical integration of Ito stochastic differential
equations.Comment: 40 pages. Sect.7 was added. arXiv admin note: text overlap with
arXiv:1712.09746, arXiv:1801.00231, arXiv:1801.03195, arXiv:1712.09516,
arXiv:1801.06501, arXiv:1712.0899
Comparative Analysis of the Efficiency of Application of Legendre Polynomials and Trigonometric Functions to the Numerical Integration of Ito Stochastic Differential Equations
The article is devoted to comparative analysis of the efficiency of
application of Legendre polynomials and trigonometric functions to the
numerical integration of Ito stochastic differential equations in the framework
of the method of approximation of iterated Ito and Stratonovich stochastic
integrals based on generalized multiple Fourier series. On the example of
iterated Ito stochastic integrals of multiplicities 1 to 3, included in the
Taylor-Ito expansion, it is shown that expansions of stochastic integrals based
on Legendre polynomials are much easier and require significantly less
computational costs compared to their analogues obtained using the
trigonometric system of functions. The results of the article can be useful for
construction of strong numerical methods for Ito stochastic differential
equations.Comment: 25 pages. Some corrections in Sect. 5, 6. Minor changes along the
text in the whole. arXiv admin note: substantial text overlap with
arXiv:1801.08862, arXiv:1801.00231, arXiv:1807.02190, arXiv:1802.00643,
arXiv:1806.1070
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