scholarly journals Kalman-Bucy Filtering for Linear Systems Driven by the Cox Process with Shot Noise Intensity and Its Application to the Pricing of Reinsurance Contracts

2005 ◽  
Vol 42 (1) ◽  
pp. 93-107 ◽  
Author(s):  
Angelos Dassios ◽  
Ji-Wook Jang
Keyword(s):  
Gaussian Process ◽  
Shot Noise ◽  
Three Dimensional ◽  
Arrival Rate ◽  
Cox Process ◽  
Noise Process ◽  
Shot Noise Process ◽  
Convergence Results ◽  
Insurance Portfolio ◽  
Aggregate Loss

In practical situations, we observe the number of claims to an insurance portfolio but not the claim intensity. It is therefore of interest to try to solve the ‘filtering problem’; that is, to obtain the best estimate of the claim intensity on the basis of reported claims. In order to use the Kalman-Bucy filter, based on the Cox process incorporating a shot noise process as claim intensity, we need to approximate it by a Gaussian process. We demonstrate that, if the primary-event arrival rate of the shot noise process is reasonably large, we can then approximate the intensity, claim arrival, and aggregate loss processes by a three-dimensional Gaussian process. We establish weak-convergence results. We then use the Kalman-Bucy filter and we obtain the price of reinsurance contracts involving high-frequency events.

scholarly journals Kalman-Bucy Filtering for Linear Systems Driven by the Cox Process with Shot Noise Intensity and Its Application to the Pricing of Reinsurance Contracts

2005 ◽  
Vol 42 (01) ◽  
pp. 93-107 ◽  
Author(s):  
Angelos Dassios ◽  
Ji-Wook Jang
Keyword(s):  
Gaussian Process ◽  
Shot Noise ◽  
Three Dimensional ◽  
Arrival Rate ◽  
Cox Process ◽  
Noise Process ◽  
Shot Noise Process ◽  
Convergence Results ◽  
Insurance Portfolio ◽  
Aggregate Loss

In practical situations, we observe the number of claims to an insurance portfolio but not the claim intensity. It is therefore of interest to try to solve the ‘filtering problem’; that is, to obtain the best estimate of the claim intensity on the basis of reported claims. In order to use the Kalman-Bucy filter, based on the Cox process incorporating a shot noise process as claim intensity, we need to approximate it by a Gaussian process. We demonstrate that, if the primary-event arrival rate of the shot noise process is reasonably large, we can then approximate the intensity, claim arrival, and aggregate loss processes by a three-dimensional Gaussian process. We establish weak-convergence results. We then use the Kalman-Bucy filter and we obtain the price of reinsurance contracts involving high-frequency events.

scholarly journals The Distribution of the Interval between Events of a Cox Process with Shot Noise Intensity

2008 ◽  
Vol 2008 ◽  
pp. 1-14 ◽  
Author(s):  
Angelos Dassios ◽  
Jiwook Jang
Keyword(s):  
Laplace Transform ◽  
Generating Function ◽  
Shot Noise ◽  
Noise Intensity ◽  
Probability Generating Function ◽  
Cox Process ◽  
Noise Process ◽  
Shot Noise Process ◽  
Piecewise Deterministic Markov Processes ◽  
The Laplace Transform

Applying piecewise deterministic Markov processes theory, the probability generating function of a Cox process, incorporating with shot noise process as the claim intensity, is obtained. We also derive the Laplace transform of the distribution of the shot noise process at claim jump times, using stationary assumption of the shot noise process at any times. Based on this Laplace transform and from the probability generating function of a Cox process with shot noise intensity, we obtain the distribution of the interval of a Cox process with shot noise intensity for insurance claims and its moments, that is, mean and variance.

scholarly journals Estimation of probability of exceeding a curve by a strictly φ-sub-Gaussian quasi shot noise process

2020 ◽  
pp. 49-56
Author(s):  
O. I. Vasylyk ◽  
R. E. Yamnenko ◽  
T. O. Ianevych
Keyword(s):  
Continuous Function ◽  
Gaussian Process ◽  
Shot Noise ◽  
Gaussian Processes ◽  
Random Processes ◽  
Telecommunication Networks ◽  
Financial Mathematics ◽  
Class V ◽  
Noise Process ◽  
Shot Noise Process

In this paper, we continue to study the properties of a separable strictly φ-sub-Gaussian quasi shot noise process $X(t) = \int_{-\infty}^{+\infty} g(t,u) d\xi(u), t\in\R$, generated by the response function g and the strictly φ-sub-Gaussian process ξ = (ξ(t), t ∈ R) with uncorrelated increments, such that E(ξ(t)−ξ(s))^2 = t−s, t>s ∈ R. We consider the problem of estimating the probability of exceeding some level by such a process on the interval [a;b], a,b ∈ R. The level is given by a continuous function f = {f(t), t ∈ [a;b]}, which satisfies some given conditions. In order to solve this problem, we apply the theorems obtained for random processes from a class V (φ, ψ), which generalizes the class of φ-sub-Gaussian processes. As a result, several estimates for probability of exceeding the curve f by sample pathes of a separable strictly φ-sub-Gaussian quasi shot noise process are obtained. Such estimates can be used in the study of shot noise processes that arise in the problems of financial mathematics, telecommunication networks theory, and other applications.

scholarly journals On the intensity of crossings by a shot noise process

1987 ◽  
Vol 19 (3) ◽  
pp. 743-745 ◽  
Author(s):  
Tailen Hsing
Keyword(s):  
Shot Noise ◽  
Marginal Probability ◽  
Noise Process ◽  
Shot Noise Process

The crossing intensity of a level by a shot noise process with a monotone response is studied, and it is shown that the intensity can be naturally expressed in terms of a marginal probability.

Inequalities for the M/G/∞ queue and related shot noise processes

1987 ◽  
Vol 24 (04) ◽  
pp. 978-989 ◽  
Author(s):  
Fred W. Huffer
Keyword(s):  
Poisson Process ◽  
Shot Noise ◽  
Convex Functions ◽  
Sample Path ◽  
Noise Process ◽  
Shot Noise Process

Suppose that pulses arrive according to a Poisson process of rate λ with the duration of each pulse independently chosen from a distribution F having finite mean. Let X(t) be the shot noise process formed by the superposition of these pulses. We consider functionals H(X) of the sample path of X(t). H is said to be L-superadditive if for all functions f and g. For any distribution F for the pulse durations, we define H(F) = EH(X). We prove that if H is L-superadditive and for all convex functions ϕ, then . Various consequences of this result are explored.

scholarly journals A Renewal Shot Noise Process with Subexponential Shot Marks

Risks ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 63
Author(s):  
Yiqing Chen
Keyword(s):  
Asymptotic Formula ◽  
Shot Noise ◽  
Crucial Role ◽  
Random Variables ◽  
Tail Probability ◽  
Noise Process ◽  
Shot Noise Process ◽  
Precise Asymptotic

We investigate a shot noise process with subexponential shot marks occurring at renewal epochs. Our main result is a precise asymptotic formula for its tail probability. In doing so, some recent results regarding sums of randomly weighted subexponential random variables play a crucial role.

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