On nonoscillating integrals  for computing  inhomogeneous Airy functions

Gil, Amparo; Segura, Javier; Temme, Nico

doi:10.1090/s0025-5718-00-01268-0

Cited by 35 publications

(24 citation statements)

References 15 publications

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“…In some cases, the integral representation for Gi(y > 0) given in (Gil, Segura and Temme, 2001) may be useful for evaluating Gi ′ (y > 0).…”

Section: A Some Properties Of Airy and Scorer Functionsmentioning

confidence: 99%

Coping With Negative Short Rates

Kakushadze¹

2016

Wilmott

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We discuss a simple extension of the Ho and Lee model with generic timedependent drift in which: 1) we compute bond prices analytically; 2) the yield curve is sensible and the asymptotic yield is positive; and 3) our analytical solution provides a clean and simple way of separating volatility from the drift in the short-rate process. Our extension amounts to introducing one or two reflecting barriers for the underlying Brownian motion (as opposed to the short-rate), which allows to have more realistic time-dependent drift (as opposed to constant drift). In our model the spectrum -or, roughly, the set of short-rate values contributing to bond and other claim prices -is discrete and positive. We discuss how to calibrate our model using empirical yield data by fitting three parameters and then read off the time-dependent drift.

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“…In some cases, the integral representation for Gi(y > 0) given in (Gil, Segura and Temme, 2001) may be useful for evaluating Gi ′ (y > 0).…”

Section: A Some Properties Of Airy and Scorer Functionsmentioning

confidence: 99%

Coping With Negative Short Rates

Kakushadze¹

2016

Wilmott

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“…We use a representation of the integral in (11) similar to the one for Gi(x) in (3.18) of [15]. To do so, notice that the exponential function in the integrand in (11) has a saddle point at ξ = i √ x.…”

Section: Proofmentioning

confidence: 99%

Asymptotic expansions for Riesz potentials of Airy functions and their products

Temme

Варламов²

2009

Phys. Scr.

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Abstract. Riesz potentials of a function are defined as fractional powers of the Laplacian. Asymptotic expansions for x → ±∞ are derived for the Riesz potentials of the Airy function Ai(x) and the Scorer function Gi(x). Reduction formulas are provided that allow to compute Riesz potentials of the products of Airy functions Ai 2 (x) and Ai(x)Bi(x), where Bi(x) is the Airy function of the second type, via the Riesz potentials of Ai(x) and Gi(x). Integral representations are given for the function A 2 (a, b; x) = Ai (x − a) Ai (x − b) with a, b ∈ R, and its Hilbert transform. Combined with the above asymptotic expansions they can be used for obtaining asymptotics of the Hankel transform of Riesz potentials of A 2 (a, b; x). The study of the above Riesz fractional derivatives can be used for establishing new properties of Korteweg-de Vries-type equations.

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“…It is not difficult to verify that the representations in (3.20), (3.21), (3.28) and (3.29) can be obtained, with G j , H j replaced with G j , H j (j = 1, 2), where (1) r (p) g (1) j (p) dp (2) 1 (u) = (u + 1 − t 2 )g (2) 2 (u), h (2) 2 (u) = −(u + 1 − t 2 )g (2) 1 (u).…”

Section: The Case T ∼mentioning

confidence: 99%

“…In particular, we can use the new representations for large parameter cases. In earlier papers [8] and [2] we have used these methods for obtaining stable integral representations for modified Bessel functions with pure imaginary order and for inhomogeneous Airy functions (Scorer functions).…”

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confidence: 99%

Integral representations for computing real parabolic cylinder functions

2004

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C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationIntegral representations for computing real parabolic cylinder functions A. Gil, J. Segura, N.M. Temme Integral representations for computing real parabolic cylinder functions ABSTRACT Integral representations are derived for the parabolic cylinder functions U(a,x), V(a,x) and W(a,x) and their derivatives. The new integrals will be used in numerical algorithms based on quadrature. They follow from contour integrals in the complex plane, by using methods from asymptotic analysis (saddle point and steepest descent methods), and are stable starting points for evaluating the functions U(a,x), V(a,x) and W(a,x) and their derivatives by quadrature rules. In particular, the new representations can be used for large parameter cases. Relations of the integral representations with uniform asymptotic expansions are also given. The algorithms will be given in a future paper. REPORT MAS-E0508 JUNE 2005

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On nonoscillating integrals for computing inhomogeneous Airy functions

Cited by 35 publications

References 15 publications

Coping With Negative Short Rates

Coping With Negative Short Rates

Asymptotic expansions for Riesz potentials of Airy functions and their products

Integral representations for computing real parabolic cylinder functions

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