Recursions and Fast Fourier Transforms for Certain Bivariate Compound Distributions

Tao Jin and Jiandong Ren

Journal of Operational Risk

Abstract: ABSTRACT We study methods for computing the distribution functions of bivariate compound random variables. In particular, we consider three classes of bivariate counting distributions and the corresponding compound distributions introduced in a 1996 paper by Hesselager.We implement the recursive methods for computing the joint probability functions derived by Hesselager and then compare the results with those obtained from fast Fourier transform (FFT) methods. In applying the FFT methods, we extend the concept of exponential tilting for univariate FFT proposed by Grüubel and Hermesmeier to the bivariate case. Our numerical results show that although the recursive methods yield the exact compound distributions if the floating-point representation error is ignored, they generally consume more computation time than the FFT methods.With appropriate tilting, the error associated with the FFT method is ignorable and the method is therefore a viable alternative to the recursive method for computing joint probabilities.

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