scholarly journals The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence

2020 ◽  
Author(s):  
Horatio Boedihardjo ◽  
Joscha Diehl ◽  
Marc Mezzarobba ◽  
Hao Ni
Keyword(s):  
Brownian Motion ◽  
Radius Of Convergence ◽  
Finite Radius

A generalization of the Hurwitz composition theorem to irregular power series

1951 ◽  
Vol 47 (3) ◽  
pp. 477-482 ◽  
Author(s):  
H. G. Eggleston
Keyword(s):  
Power Series ◽  
Radius Of Convergence ◽  
Finite Radius ◽  
Image Position ◽  
Composition Theorem

When two functions are given, each with a finite radius of convergence, a theorem due independently to Hurwitz and Pincherle (1, 2) provides information about the position of the singularities of the functionin terms of the positions of the singularities of f(z) and g(z).

VI.—On Solvability of Some Two-Parameter Eigenvalue Problems in Hilbert Space

1968 ◽  
Vol 68 (1) ◽  
pp. 83-93
Author(s):  
Z. Bohte
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Radius Of Convergence ◽  
Finite Radius ◽  
Image Position ◽  
Adjoint Operators ◽  
Two Parameter

SynopsisThis paper studies two particular cases of the general 2-parameter eigenvalue problem namelywhere A, B, B1, B2, C, C1, C2 are self-adjoint operators in Hilbert space, all except A being bounded. The disposable parameters λ and μ have to be determined so that the equations have non-trivial solutions x, y.On the assumption that the solution is known for ∊ = o, solutions are constructed in the form of series for λ, μ, x, y as power series in ∊ with finite radius of convergence.

scholarly journals On a generalized three-parameter wright function of Le Roy type

2017 ◽  
Vol 20 (5) ◽  
Author(s):  
Roberto Garrappa ◽  
Sergei Rogosin ◽  
Francesco Mainardi
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Laplace Transform ◽  
Analytic Continuation ◽  
Special Function ◽  
Special Functions ◽  
Integral Representations ◽  
Radius Of Convergence ◽  
Wright Function ◽  
Finite Radius

AbstractRecently S. Gerhold and R. Garra – F. Polito independently introduced a new function related to the special functions of the Mittag-Leffler family. This function is a generalization of the function studied by É. Le Roy in the period 1895-1905 in connection with the problem of analytic continuation of power series with a finite radius of convergence. In our note we obtain two integral representations of this special function, calculate its Laplace transform, determine an asymptotic expansion of this function on the negative semi-axis (in the case of an integer third parameter

scholarly journals Convergent Power Series of sech⁡(x) and Solutions to Nonlinear Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
U. Al Khawaja ◽  
Qasem M. Al-Mdallal
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Series Expansion ◽  
Taylor Series ◽  
Nonlinear Differential Equations ◽  
Series Representation ◽  
Power Series Expansion ◽  
Detailed Comparison ◽  
Radius Of Convergence ◽  
Finite Radius

It is known that power series expansion of certain functions such as sech⁡(x) diverges beyond a finite radius of convergence. We present here an iterative power series expansion (IPS) to obtain a power series representation of sech⁡(x) that is convergent for all x. The convergent series is a sum of the Taylor series of sech⁡(x) and a complementary series that cancels the divergence of the Taylor series for x≥π/2. The method is general and can be applied to other functions known to have finite radius of convergence, such as 1/(1+x2). A straightforward application of this method is to solve analytically nonlinear differential equations, which we also illustrate here. The method provides also a robust and very efficient numerical algorithm for solving nonlinear differential equations numerically. A detailed comparison with the fourth-order Runge-Kutta method and extensive analysis of the behavior of the error and CPU time are performed.

Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

2007 ◽  
Vol 44 (02) ◽  
pp. 393-408 ◽  
Author(s):  
Allan Sly
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
White Noise ◽  
Gaussian Process ◽  
Discrete Data ◽  
Hölder Exponent ◽  
Scaling Properties ◽  
Multifractional Brownian Motion ◽  
Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

Brownian motion with quadratic killing and some implications

1986 ◽  
Vol 23 (04) ◽  
pp. 893-903 ◽  
Author(s):  
Michael L. Wenocur
Keyword(s):  
Brownian Motion ◽  
Weibull Distribution ◽  
Special Function ◽  
Function Theory ◽  
Killing Rate

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

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