Finite-Dimensional Distributions of a Square-Root Diffusion

2014 ◽  
Vol 51 (04) ◽  
pp. 930-942
Author(s):  
Michael B. Gordy
Keyword(s):  
Gamma Distribution ◽  
Joint Distribution ◽  
Sample Path ◽  
Moment Generating Function ◽  
Square Root ◽  
Discrete Sample ◽  
Multivariate Gamma Distribution ◽  
Chi Squared ◽  
Cir Process ◽  
The Moment

We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t 1,…,t n , we find the conditional joint distribution of (X(t 1),…,X(t n )) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.

scholarly journals Finite-Dimensional Distributions of a Square-Root Diffusion

2014 ◽  
Vol 51 (4) ◽  
pp. 930-942
Author(s):  
Michael B. Gordy
Keyword(s):  
Gamma Distribution ◽  
Joint Distribution ◽  
Sample Path ◽  
Moment Generating Function ◽  
Square Root ◽  
Discrete Sample ◽  
Multivariate Gamma Distribution ◽  
Chi Squared ◽  
Cir Process ◽  
The Moment

We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t1,…,tn, we find the conditional joint distribution of (X(t1),…,X(tn)) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.

scholarly journals A Study of Properties and Applications of Gamma Distribution

2021 ◽  
Vol 4 (2) ◽  
pp. 52-65
Author(s):  
Eric U. ◽  
Oti M.O.O. ◽  
Francis C.E.
Keyword(s):  
Probability Distribution ◽  
Gamma Distribution ◽  
Generating Function ◽  
Real Life ◽  
Moment Generating Function ◽  
Reliability Theory ◽  
Burn Out ◽  
Electrical Devices ◽  
The Moment ◽  
Special Case

The gamma distribution is one of the continuous distributions; the distributions are very versatile and give useful presentations of many physical situations. They are perhaps the most applied statistical distribution in the area of reliability. In this paper, we present the study of properties and applications of gamma distribution to real life situations such as fitting the gamma distribution into data, burn-out time of electrical devices and reliability theory. The study employs the moment generating function approach and the special case of gamma distribution to show that the gamma distribution is a legitimate continuous probability distribution showing its characteristics.

scholarly journals SIFAT-SIFAT DAN KEJADIAN KHUSUS DISTRIBUSI GAMMA

2021 ◽  
Vol 15 (1) ◽  
pp. 047-058
Author(s):  
Royke Yohanes Warella ◽  
Henry Junus Wattimanela ◽  
Venn Yan Ishak Ilwaru
Keyword(s):  
Gamma Distribution ◽  
Shape Parameter ◽  
Random Variable ◽  
Moment Generating Function ◽  
Continuous Random Variable ◽  
Positive Real ◽  
Chi Square ◽  
Special Cases ◽  
Distribution Parameters ◽  
The Moment

The gamma distribution is one of special continuous random variable distribution with scale parameter  and shape parameter  where  is positive real numbers. On some conditions the gamma distribution astablishes other continuous distributions which are then called special cases of the gamma distribution. Therefore, this study was conducted to determine the properties of gamma distribution and the characteristics of the special cases of gamma distribution by analyzed the theories from literatures. The properties of gamma distribution include expectation value, variance, moment generating function, characteristic function, and estimation of gamma distribution parameters with the moment method to earn the special cases of the gamma distribution are Erlang, exponential, chi-square, and beta distributions.

Analytical pricing of discrete arithmetic Asian options under generalized CIR process with time change

2018 ◽  
Vol 05 (01) ◽  
pp. 1850002 ◽  
Author(s):  
Zhigang Tong ◽  
Allen Liu
Keyword(s):  
Recursive Formula ◽  
Moment Generating Function ◽  
Time Change ◽  
Jump Processes ◽  
Asian Options ◽  
Option Prices ◽  
New Model ◽  
Expansion Technique ◽  
Cir Process ◽  
The Moment

Starting from CIR process, we build a new model for pricing discrete arithmetic Asian options with nonlinear transformation and stochastic time change. The new model introduces the nonlinearity in both drift and diffusion components of the underlying process and allows for flexible jump processes. We are able to derive the recursive formula for the moment generating function of average price by employing the eigenfunction expansion technique. The Asian option prices can then be implemented through a Fourier transform. We also investigate the sensitivities of option prices with respect to the parameters of the new model.

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