Abstract
In this paper, we study multiple shot noise process and its integral. We analyse these two processes systematically for their theoretical distributions, based on the piecewise deterministic Markov process theory developed by Davis [1] and the martingale methodology used by Dassios and Jang [2]. The analytic expressions of the Laplace transforms of these two processes are presented. We also obtain the multivariate probability generating function for the number of jumps, for which we use a multivariate Cox process. To derive these, we assume that the Cox processes jumps, intensity jumps and primary event jumps are independent of each other. Using the Laplace transform of the integral of multiple shot noise process, we obtain the tail of multivariate distributions of the first jump times of the Cox processes, i.e. the multivariate survival functions. Their numerical calculations and other relevant joint distributions’ numerical values are also presented.
Highlights
Due to global warming and climate changes, there have been increases in the frequency and intensity of floods in one area and draught in the other
We introduced multiple shot noise process, where each process acts as a jump intensity for the one, and its integral
Based on the piecewise deterministic Markov process theory developed by Davis [1] and the martingale methodology used by Dassios and Jang [2], we derived the Laplace transforms of these two processes
Summary
Due to global warming and climate changes, there have been increases in the frequency and intensity of floods in one area and draught in the other. A Cox process with shot noise intensity has been suggested to use to predict claims arising from catastrophic events by Dassios and Jang [2]. The prevalence of above financial contagion has led to further bankruptcies and default of mortgage lenders in US announcing their significant losses in 2008 This subprime mortgage meltdown has led to new ownership for Bears Stern and Merrill Lynch and the bankruptcy of Lehman Brothers. JANG governments requiring their interventions to ameliorate these contagious effects to the larger economy [6] To these effects, in this paper we introduce multiple shot noise process [7]. In this paper we introduce multiple shot noise process [7] It consists of d component processes λt(d ) , λt(d −1) , λt(d −2) , , λt(1) , where each process acts as a jump intensity for the one.
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