Risk Processes with Non-stationary Hawkes Claims Arrivals

Stabile, Gabriele; Torrisi, Giovanni Luca

doi:10.1007/s11009-008-9110-6

Cited by 73 publications

(54 citation statements)

References 20 publications

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“…Bacry et al [1] and Bacry et al [2] and the large deviations result has been applied to study the ruin probabilities in insurance, see e.g. Stabile and Torrisi [16] and Zhu [22].…”

Section: Hawkes Processmentioning

confidence: 99%

Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

Zhu

2014

J. Appl. Probab.

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Abstract. In this paper, we propose a stochastic process, which is a CoxIngersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. Laplace transforms and limit theorems have been obtained, including law of large numbers, central limit theorems and large deviations.

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“…Bacry et al [1] and Bacry et al [2] and the large deviations result has been applied to study the ruin probabilities in insurance, see e.g. Stabile and Torrisi [16] and Zhu [22].…”

Section: Hawkes Processmentioning

confidence: 99%

Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

Zhu

2014

J. Appl. Probab.

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“…The Hawkes process (Hawkes, 1971) is a simple point process that has a self-exciting property, clustering effect, and long memory. It has been widely applied in seismology, neuroscience, DNA modeling, and many other fields, including finance (Embrechts et al, 2011) and insurance (Stabile and Torrisi, 2010).…”

Section: Introductionmentioning

confidence: 99%

Risk Model Based on General Compound Hawkes Process

Swishchuk

2017

SSRN Journal

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In this paper, we introduce a new model for the risk process based on the general compound Hawkes process (GCHP) for the arrival of claims. We call it the risk model based on the general compound Hawkes process (RMGCHP). The law of large numbers (LLN) and the functional central limit theorem (FCLT) are proved. We also study the main properties of this new risk model: net profit condition, premium principle, and ruin time (including ultimate ruin time) applying the LLN and FCLT for the RMGCHP. We also present, as applications of our results, similar results for the risk model based on the compound Hawkes process (RMCHP) and apply them to the classical risk model based on the compound Poisson process (RMCPP). It follows that all previous results for RMCHP and classical results for RMCPP follow from our results.Definition 1 (counting process). . A counting process is a stochastic process N(t), t ≥ 0, taking positive integer values and satisfying: N(0) = 0. It is almost surely finite, and is a right-continuous step function with increments of size +1. (See, e.g., Daley and Vere-Jones, 1988).Denote by  N (t), t ≥ 0, the history of the arrivals up to time t, that is, { N (t), t ≥ 0} is a filtration (an increasing sequence of -algebras). A counting process N(t) can be interpreted as a cumulative count of the number of arrivals into a system up to the current time t.The counting process can also be characterized by the sequence of random arrival times (T 1 , T 2 , ...) at which the counting process N(t) has jumped. The process defined by these arrival times is called a point process.Definition 2 (point process). If a sequence of random variables (T 1 , T 2 , ...), taking values in [0, +∞), has P(0 ≤ T 1 ≤ T 2 ≤ ...) = 1, and the number of points in a 50 WILMOTT magazine

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“…It is worth mentioning that asymptotic ruin probability results with respect to the initial capital, under some non-stationary claim arrival processes (e.g. Hawkes and Cox processes with shot-noise intensity) have recently been obtained by Stabile and Torrisi (2010) and Zhu (2013).…”

Section: Introductionmentioning

confidence: 99%