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

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

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This paper focuses mainly on the problem of computing the ${\gamma }^{\mathrm{th}}$, , moment of a random variable $Y_n:={\sum }_{i=1}^n{\alpha }_i{X}_i$ in which the ’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 , superdry baseball top especially when not all ${\alpha }_i$’s are equal.

We analytically solve this problem by showing that the ${\gamma }^{\mathrm{th}}$ moment of can be expressed in terms of generalized hypergeometric functions.Additionally, we extend our result to computing the ${\gamma }^{\mathrm{th}}$ moment of when $X_i$ is a combination of statistically independent weboost splitter and $G_i$ in which the ’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 ${\gamma }^{\mathrm{th}}$ conditional moment.Finally, we conduct Monte Carlo simulations to demonstrate the accuracy and efficiency of our explicit formulas through several numerical tests.