14,944 research outputs found
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. 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
Analytic Evaluation of Four-Particle Integrals with Complex Parameters
The method for analytic evaluation of four-particle integrals, proposed by
Fromm and Hill, is generalized to include complex exponential parameters. An
original procedure of numerical branch tracking for multiple valued functions
is developed. It allows high precision variational solution of the Coulomb
four-body problem in a basis of exponential-trigonometric functions of
interparticle separations. Numerical results demonstrate high efficiency and
versatility of the new method.Comment: 13 pages, 4 figure
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
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
Properties of generalized univariate hypergeometric functions
Based on Spiridonov's analysis of elliptic generalizations of the Gauss
hypergeometric function, we develop a common framework for 7-parameter families
of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric
functions. In each case we derive the symmetries of the generalized
hypergeometric function under the Weyl group of type E_7 (elliptic, hyperbolic)
and of type E_6 (trigonometric) using the appropriate versions of the
Nassrallah-Rahman beta integral, and we derive contiguous relations using
fundamental addition formulas for theta and sine functions. The top level
degenerations of the hyperbolic and trigonometric hypergeometric functions are
identified with Ruijsenaars' relativistic hypergeometric function and the
Askey-Wilson function, respectively. We show that the degeneration process
yields various new and known identities for hyperbolic and trigonometric
special functions. We also describe an intimate connection between the
hyperbolic and trigonometric theory, which yields an expression of the
hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric
hypergeometric functions.Comment: 46 page
Limits of elliptic hypergeometric integrals
In math.QA/0309252, the author proved a number of multivariate elliptic
hypergeometric integrals. The purpose of the present note is to explore more
carefully the various limiting cases (hyperbolic, trigonometric, rational, and
classical) that exist. In particular, we show (using some new estimates of
generalized gamma functions) that the hyperbolic integrals (previously treated
as purely formal limits) are indeed limiting cases. We also obtain a number of
new trigonometric (q-hypergeometric) integral identities as limits from the
elliptic level.Comment: 41 pages LaTeX. Minor stylistic changes, statement of Theorem 4.7
fixe
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