Abstract:In this paper, we study the family of renewal shot-noise processes. The Feynmann-Kac formula is obtained based on the piecewise deterministic Markov process theory and the martingale methodology. We then derive the Laplace transforms of the conditional moments and asymptotic moments of the processes. In general, by inverting the Laplace transforms, the asymptotic moments and the first conditional moments can be derived explicitly, however, other conditional moments may need to be estimated numerically. As an e… Show more
“…This work can be extended by incorporating interarrival jump times with renewal processes, where the moments of renewal shot-noise processes have been shown in Jang et al (2018a) and Dassios et al (2015) studied a risk model with renewal shot-noise Cox process.…”
Section: Discussionmentioning
confidence: 99%
“…Shot-noise Poisson process is another extension of the Poisson process, where it is capable of displaying the frequency, magnitude and time period needed to determine the effect of points. The applications of shot-noise Poisson process in insurance and credit risk context can be noticed in Klüppelberg & Mikosch (1995), Jang (2004), Jang & Krvavych (2004), Herbertsson et al (2011) and Jang et al (2018a).…”
The Poisson process is an essential building block to move up to complicated counting processes, such as the Cox (“doubly stochastic Poisson”) process, the Hawkes (“self-exciting”) process, exponentially decaying shot-noise Poisson (simply “shot-noise Poisson”) process and the dynamic contagion process. The Cox process provides flexibility by letting the intensity not only depending on time but also allowing it to be a stochastic process. The Hawkes process has self-exciting property and clustering effects. Shot-noise Poisson process is an extension of the Poisson process, where it is capable of displaying the frequency, magnitude and time period needed to determine the effect of points. The dynamic contagion process is a point process, where its intensity generalises the Hawkes process and Cox process with exponentially decaying shot-noise intensity. To facilitate the usage of these processes in practice, we revisit the distributional properties of the Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes. We provide simulation algorithms for these processes, which would be useful to statistical analysis, further business applications and research. As an application of the compound processes, numerical comparisons of value-at-risk and tail conditional expectation are made.
“…This work can be extended by incorporating interarrival jump times with renewal processes, where the moments of renewal shot-noise processes have been shown in Jang et al (2018a) and Dassios et al (2015) studied a risk model with renewal shot-noise Cox process.…”
Section: Discussionmentioning
confidence: 99%
“…Shot-noise Poisson process is another extension of the Poisson process, where it is capable of displaying the frequency, magnitude and time period needed to determine the effect of points. The applications of shot-noise Poisson process in insurance and credit risk context can be noticed in Klüppelberg & Mikosch (1995), Jang (2004), Jang & Krvavych (2004), Herbertsson et al (2011) and Jang et al (2018a).…”
The Poisson process is an essential building block to move up to complicated counting processes, such as the Cox (“doubly stochastic Poisson”) process, the Hawkes (“self-exciting”) process, exponentially decaying shot-noise Poisson (simply “shot-noise Poisson”) process and the dynamic contagion process. The Cox process provides flexibility by letting the intensity not only depending on time but also allowing it to be a stochastic process. The Hawkes process has self-exciting property and clustering effects. Shot-noise Poisson process is an extension of the Poisson process, where it is capable of displaying the frequency, magnitude and time period needed to determine the effect of points. The dynamic contagion process is a point process, where its intensity generalises the Hawkes process and Cox process with exponentially decaying shot-noise intensity. To facilitate the usage of these processes in practice, we revisit the distributional properties of the Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes. We provide simulation algorithms for these processes, which would be useful to statistical analysis, further business applications and research. As an application of the compound processes, numerical comparisons of value-at-risk and tail conditional expectation are made.
“…More generally, our model may be applicable to modeling many events of other types which are gradually disappearing. Recently, alternative intensity functions for Cox processes have been specified, see [1,2,14,25,37] and just to name a few. Figure 2.Piecewise-constant decreasing intensity process .…”
Innovations in medicine provide us longer and healthier life, leading lower mortality. Sooner rather than later, much greater longevity would be possible for us due to artificial intelligence advances in health care. Similarly, Advanced Driver Assistance Systems (ADAS) in highly automated vehicles may reduce or even eventually eliminate accidents by perceiving dangerous situations, which would minimize the number of accidents and lead to fewer loss claims for insurance companies. To model the survivor function capturing greater longevity as well as the number of claims reflecting less accidents in the long run, in this paper, we study a Cox process whose intensity process is piecewise-constant and decreasing. We derive its ultimate distributional properties, such as the Laplace transform of intensity integral process, the probability generating function of point process, their associated moments and cumulants, and the probability of no more claims for a given time point. In general, this simple model may be applicable in many other areas for modeling the evolution of gradually disappearing events, such as corporate defaults, dividend payments, trade arrivals, employment of a certain job type (e.g., typists) in the labor market, and release of particles. In particular, we discuss some potential applications to insurance.
“…Among these studies we find Albrecher and Boxma (2004), Albrecher and Teugels (2006) and Boudreault et al (2006) which analyze ruin-related problems; , Adékambi (2011, 2012), investigate the risk aggregation and the distribution of the discounted aggregate amount of claims; Garrido (2001a, 2001b) use the renewal theory to derive a closed expressions for the first two moments of the discounted aggregated claims; and study the aggregate discount payment and expenses process for medical malpractice insurance. Most recently, Jang et al (2018) study the family of renewal shot-noise processes. Based on the piecewise deterministic Markov process theory and the martingale methodology, they obtained the Feynmann-Kac formula and then derived the Laplace transforms of the conditional moments and asymptotic moments of the processes.…”
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