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Analytically Computing the Moments of a Conic Combination of Independent Noncentral Chi-Square Random Variables and Its Application for the Extended Cox–Ingersoll–Ross Process with Time-Varying Dimension

Author

Listed:
  • Sanae Rujivan

    (Center of Excellence in Data Science for Health Study, Division of Mathematics and Statistics, School of Science, Walailak University, Nakhon Si Thammarat 80161, Thailand)

  • Athinan Sutchada

    (Center of Excellence in Data Science for Health Study, Division of Mathematics and Statistics, School of Science, Walailak University, Nakhon Si Thammarat 80161, Thailand)

  • Kittisak Chumpong

    (Division of Computational Science, Faculty of Science, Prince of Songkla University, Songkhla 90110, Thailand
    Statistics and Applications Research Unit, Faculty of Science, Prince of Songkla University, Songkhla 90110, Thailand)

  • Napat Rujeerapaiboon

    (Department of Industrial Systems Engineering and Management, National University of Singapore, Singapore 117576, Singapore)

Abstract

This paper focuses mainly on the problem of computing the γ th , γ > 0 , moment of a random variable Y n : = ∑ i = 1 n α i X i in which the α i ’s are positive real numbers and the X i ’s are independent and distributed according to noncentral chi-square distributions. Finding an analytical approach for solving such a problem has remained a challenge due to the lack of understanding of the probability distribution of Y n , especially when not all α i ’s are equal. We analytically solve this problem by showing that the γ th moment of Y n can be expressed in terms of generalized hypergeometric functions. Additionally, we extend our result to computing the γ th moment of Y n when X i is a combination of statistically independent Z i 2 and G i in which the Z i ’s are distributed according to normal or Maxwell–Boltzmann distributions and the G i ’s are distributed according to gamma, Erlang, or exponential distributions. Our paper has an immediate application in interest rate modeling, where we can explicitly provide the exact transition probability density function of the extended Cox–Ingersoll–Ross (ECIR) process with time-varying dimension as well as the corresponding γ th conditional moment. Finally, we conduct Monte Carlo simulations to demonstrate the accuracy and efficiency of our explicit formulas through several numerical tests.

Suggested Citation

  • Sanae Rujivan & Athinan Sutchada & Kittisak Chumpong & Napat Rujeerapaiboon, 2023. "Analytically Computing the Moments of a Conic Combination of Independent Noncentral Chi-Square Random Variables and Its Application for the Extended Cox–Ingersoll–Ross Process with Time-Varying Dimens," Mathematics, MDPI, vol. 11(5), pages 1-29, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:5:p:1276-:d:1089395
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    References listed on IDEAS

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    2. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
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