Risk Model Based on General Compound Hawkes Process

Abstract

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

General Compound Hawkes Processes in Limit Order Books

“…We also present some numerical examples. The general compound Hawkes process was first introduced in Swishchuk (2017) to model the risk process in insurance and studied in detail there. In the paper Swishchuk et al (2019), we obtained functional CLTs and LLNs for general compound Hawkes processes with dependent orders and regime-switching compound Hawkes processes.…”

General Compound Hawkes Processes in Limit Order Books

“…Lemma 5 (LLN for GCHPnSDO). The process S nt in (74) satisfies the following weak convergence in the Skorokhod topology (see [40]):…”

“…Following the martingale method from [43], we have the following weak convergence in the Skorokhod topology (see [40]):…”

“…We also present some numerical examples. The general compound Hawkes process was first introduced in [40] to model the risk process in insurance. In the paper [41] we obtained functional CLTs and LLNs for so-called compound Hawkes process with dependent orders and regime-switching compound Hawkes process.…”

Merton Investment Problems in Finance and Insurance for the Hawkes-based Models

“…Merton optimal investment and consumption stochastic problem is one of the most studied classical problem in finance ( [22,23,24,6,19]). In this paper, we will show how to solve the Merton optimal investment stochastic control problem for Hawkes-based models in finance and insurance, i.e., for a wealth portfolio X(t) consisting of a bond and a stock price described by general compound Hawkes process (GCHP) ( [33,34,28]), and for a capital R(t) of an insurance company with the amount of claims described by risk model based on GCHP ( [30,35]).…”