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27 pages, 431 KB

Open AccessArticle

On the Generalized Inverse Gaussian Volatility in the Continuous Ho–Lee Model

by Roman V. Ivanov

Computation 2025, 13(4), 100; https://doi.org/10.3390/computation13040100 - 19 Apr 2025

Cited by 1 | Viewed by 1177

This paper presents a new model of the term structure of interest rates that is based on the continuous Ho–Lee one. In this model, we suggest that the drift and volatility coefficients depend additionally on a generalized inverse Gaussian (GIG) distribution. Analytical expressions for the bond price and its moments are found in the new GIG continuous Ho–Lee model. Also, we compute in this model the prices of European call and put options written on bond. The obtained formulas are determined by the values of the Humbert confluent hypergeometric function of two variables. A numerical experiment shows that the third and fourth moments of the bond prices differentiate substantially in the continuous Ho–Lee and GIG continuous Ho–Lee models. Full article

(This article belongs to the Section Computational Social Science)

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17 pages, 354 KB

Open AccessArticle

On Voigt-Type Functions Extended by Neumann Function in Kernels and Their Bounding Inequalities

by Rakesh K. Parmar, Tibor K. Pogány and Uthara Sabu

Axioms 2024, 13(8), 534; https://doi.org/10.3390/axioms13080534 - 7 Aug 2024

Cited by 1 | Viewed by 1296

Abstract

The principal aim of this paper is to introduce the extended Voigt-type function

V_{μ, ν} (x, y)

and its counterpart extension

W_{μ, ν} (x, y)

, involving the Neumann function

Y_{ν}

in [...] Read more.

The principal aim of this paper is to introduce the extended Voigt-type function

V_{μ, ν} (x, y)

and its counterpart extension

W_{μ, ν} (x, y)

, involving the Neumann function

Y_{ν}

in the kernel of the representing integral. The newly defined integral reduces to the classical Voigt functions

K (x, y)

and

L (x, y)

, and to their generalizations by Srivastava and Miller, by the unification of Klusch. Following an approach by Srivastava and Pogány, we also present the multiparameter and multivariable versions

V_{μ, ν}^{(r)} (x, y), W_{μ, ν}^{(r)} (x, y)

and the r positive integer of the initial extensions

V_{μ, ν} (x, y), W_{μ, ν} (x, y)

. Several computable series expansions are obtained for the discussed Voigt-type functions in terms of Humbert confluent hypergeometric functions

Ψ_{2}^{(r)}

. Furthermore, by transforming the input extended Voigt-type functions by the Grünwald–Letnikov fractional derivative, we establish representation formulae in terms of the associated Legendre functions of the second kind

Q_{η}^{- ν}

in the two-parameter and two-variable cases. Finally, functional bounding inequalities are given for

V_{μ, ν} (x, y)

and

W_{μ, ν} (x, y)

. Particularly interesting results are presented for the Neumann function

Y_{ν}

and for the Struve

H_{ν}

function in the form of several functional bounds. The article ends with a thorough discussion and closing remarks. Full article

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

14 pages, 277 KB

Open AccessArticle

Another Method for Proving Certain Reduction Formulas for the Humbert Function ψ₂ Due to Brychkov et al. with an Application

by Asmaa O. Mohammed, Adem Kilicman, Mohamed M. Awad, Arjun K. Rathie and Medhat A. Rakha

Symmetry 2022, 14(5), 868; https://doi.org/10.3390/sym14050868 - 23 Apr 2022

Cited by 1 | Viewed by 2755

Abstract

Recently, Brychkov et al. established several new and interesting reduction formulas for the Humbert functions (the confluent hypergeometric functions of two variables). The primary objective of this study was to provide an alternative and simple approach for proving four reduction formulas for the [...] Read more.

ψ_{2}

. We construct intriguing series comprising the product of two confluent hypergeometric functions as an application. Numerous intriguing new and previously known outcomes are also achieved as specific instances of our primary discoveries. It is well-known that the hypergeometric functions in one and two variables and their confluent forms occur naturally in a wide variety of problems in applied mathematics, statistics, operations research, physics (theoretical and mathematical) and engineering mathematics, so the results established in this paper may be potentially useful in the above fields. Symmetry arises spontaneously in the abovementioned functions. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

11 pages, 817 KB

Open AccessArticle

Note on the Hurwitz–Lerch Zeta Function of Two Variables

by Junesang Choi, Recep Şahin, Oğuz Yağcı and Dojin Kim

Symmetry 2020, 12(9), 1431; https://doi.org/10.3390/sym12091431 - 28 Aug 2020

Cited by 4 | Viewed by 3086

Abstract

A number of generalized Hurwitz–Lerch zeta functions have been presented and investigated. In this study, by choosing a known extended Hurwitz–Lerch zeta function of two variables, which has been very recently presented, in a systematic way, we propose to establish certain formulas and representations for this extended Hurwitz–Lerch zeta function such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to yield corresponding results for several less generalized Hurwitz–Lerch zeta functions than the extended Hurwitz–Lerch zeta function considered here. For further investigation, among possibly various more generalized Hurwitz–Lerch zeta functions than the one considered here, two more generalized settings are provided. Full article

(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅲ)

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