8,826 research outputs found
Iterated integrals of Jacobi polynomials
Let P(α,β)n be the n-th monic Jacobi polynomial with α,β>−1. Given m numbers ω1,…,ωm∈C∖[−1,1], let Ωm=(ω1,…,ωm) and P(α,β)n,m,Ωm be the m-th iterated integral of (n+m)!n!P(α,β)n normalized by the conditions
dkP(α,β)n,m,Ωmdzk(ωm−k)=0, for k=0,1,…,m−1.
The main purpose of the paper is to study the algebraic and asymptotic properties of the sequence of monic polynomials {P(α,β)n,m,Ωm}n. In particular, we obtain the relative asymptotic for the ratio of the sequences {P(α,β)n,m,Ωm}n and {P(α,β)n}n. We prove that the zeros of these polynomials accumulate on a suitable ellipse.The research of H. Pijeira was supported by research Grant MTM2015-65888-C4-2-P Ministerio de Economía y Competitividad of Spain
Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Multiple Fourier Series Converging in the Mean
The article is devoted to the expansions of iterated Ito stochastic integrals
based on generalized multiple Fourier series converging in the sence of norm in
the space The method of generalized multiple
Fourier series for expansion and mean-square approximation of iterated Ito
stochastic integrals of arbitrary multiplicity () with
respect to components of the multidimensional Wiener process is proposed and
developed. The obtained expansions contain only one operation of the limit
transition in contrast to existing analogues. In the article it is also
obtained the generalization of the proposed method for discontinuous complete
orthonormalized systems of functions in the space
as well as for the complete orthonormalized systems of
functions with weight in the space
. The comparison of the considered method with the well-known
expansions of iterated Ito stochastic integrals based on Ito formula and
Hermite polynomials is given. The convergence in the mean of degree and with probability 1 of the proposed method is proved.Comment: 70 pages. Sect. 11 was adde
A Distributed Procedure for Computing Stochastic Expansions with Mathematica
The solution of a (stochastic) differential equation can be locally
approximated by a (stochastic) expansion. If the vector field of the
differential equation is a polynomial, the corresponding expansion is a linear
combination of iterated integrals of the drivers and can be calculated using
Picard Iterations. However, such expansions grow exponentially fast in their
number of terms, due to their specific algebra, rendering their practical use
limited.
We present a Mathematica procedure that addresses this issue by
re-parametrising the polynomials and distributing the load in as small as
possible parts that can be processed and manipulated independently, thus
alleviating large memory requirements and being perfectly suited for
parallelized computation. We also present an iterative implementation of the
shuffle product (as opposed to a recursive one, more usually implemented) as
well as a fast way for calculating the expectation of iterated Stratonovich
integrals for Brownian Motion.Comment: 15 pages, 2 figures. Submitte
A distributed procedure for computing stochastic expansions with Mathematica
The solution of a (stochastic) differential equation can be locally approximated by a (stochastic) expansion. If the vector field of the differential equation is a polynomial, the corresponding expansion is a linear combination of iterated integrals of the drivers and can be calculated using Picard Iterations. However, such expansions grow exponentially fast in their number of terms, due to their specific algebra, rendering their practical use limited.
We present a Mathematica procedure that addresses this issue by reparametrizing the polynomials and distributing the load in as small as possible parts that can be processed and manipulated independently, thus alleviating large memory requirements and being perfectly suited for parallelized computation. We also present an iterative implementation of the shuffle product (as opposed to a recursive one, more usually implemented) as well as a fast way for calculating the expectation of iterated Stratonovich integrals for Brownian motion
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
Decomposition of elliptic multiple zeta values and iterated Eisenstein integrals
We describe a decomposition algorithm for elliptic multiple zeta values,
which amounts to the construction of an injective map from the algebra
of elliptic multiple zeta values to a space of iterated Eisenstein integrals.
We give many examples of this decomposition, and conclude with a short
discussion about the image of . It turns out that the failure of
surjectivity of is in some sense governed by period polynomials of
modular forms.Comment: v2, minor change
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