Studies on a Double Poisson-Geometric Insurance Risk Model with Interference

This paper mainly studies a generalized double Poisson-Geometric insurance risk model. By martingale and stopping time approach, we obtain adjustment coefficient equation, the Lundberg inequality, and the formula for the ruin probability. Also the Laplace transformation of the time when the surplus reaches a given level for the first time is discussed, and the expectation and its variance are obtained. Finally, we give the numerical examples.


Introduction
In insurance mathematics, the classical risk model has been the center of focus for decades [1][2][3].The surplus () in the classical model at time  can be expressed as where  = (0) > 0 is the initial capital,  > 0 is the constant rate of premium, and { 1 (),  ≥ 0} is a Poisson process, with Poisson rate  1 > 0 denoting the number of claims up to time .The individual claim sizes  1 ,  2 , . .., independent of { 1 (),  ≥ 0}, are independent and identically distributed nonnegative random variables with common distribution function () with mean   , variance  2  , and moment generating function   () = [  ].
But in the Poisson process, the expectation and variance are equal.This is obviously not consistent with actual situation.So recently there is a huge amount of literature devoted to the generalization of the classical model in different ways.Lu and Li [4] consider a Markov-modulated risk model in which the claim interarrivals, claim sizes, and premiums are influenced by an external Markovian environment process.Tan and Yang [5] discuss the compound binomial risk model with an interest on the surplus under a constant dividend barrier and periodically paying dividends.Vellaisamy and Upadhye [6] study the convolution of compound negative binomial distributions with arbitrary parameters.The exact expression and also a random parameter representation are obtained.Cossette et al. [7] present a compound Markov binomial model, which is an extension of the compound binomial model.The compound Markov binomial model is based on the Markov Bernoulli process which introduces dependency between claim occurrences.Recursive formulas are provided for the computation of the ruin probabilities over finite-and infinite-time horizons.A Lundberg exponential bound is derived for the ruin probability, and numerical examples are also provided.Yang and Zhang [8] investigate a Sparre Andersen risk model in which the inter-claim times are generalized Erlang(n) distributed.Czarna and Palmowski [9] focus on a general spectrally negative Levy insurance risk process.For this class of processes, they analyze the socalled Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time  > 0.
In this paper, we will consider a double Poisson-Geometric risk model with diffusion in which the arrival of policies is a Poisson-Geometric process and the claims process follows the compound Poisson-Geometric process.For more details and new developments on the Poisson-Geometric risk model, the interested readers can refer to [10][11][12][13].
The rest of the paper is organized as follows.In Section 2, the risk model is introduced.In Section 3, we obtain the adjustment coefficient equation and the formula of ruin probability.Then we present the effect of the related parameters on the adjustment coefficient.In Section 4, using the martingale method, the time when the surplus reaches a level firstly is considered, and the expectation and its variance are obtained.Numerical illustrations are also given.

The Risk Model
Definition 1 (see [10]).A distribution is said to be Poisson-Geometric distributed, denoted by (, ), if its generating function is where  > 0, 0 ≤  < 1.Note that if  = 0, then the Poisson-Geometric distribution degenerates into Poisson distribution.
Definition 2 (see [10]).Let  > 0 and 0 ≤  < 1, then {(),  ≥ 0} is said to be a Poisson-Geometric process with parameters ,  if it satisfies (1) (0) = 0; (2) {(),  ≥ 0} has stationary and independent increments; (3) for all  > 0, () is a Poisson-Geometric distributed with parameters , , and The corresponding moment generating function of () Then the double Poisson-Geometric risk model with interference is defined as where  2 () is the number of premium up to time  and follows a Poisson-Geometric distribution with parameters  2 and  2 ;  3 () is the number of claims up to time  and follows a Poisson-Geometric distribution with parameters  3 and  3 .() is the standard Brownian motion and  is a constant, representing the diffusion volatility parameters.Throughout this paper, we assume that  2 (),  3 (), (), and {  } are mutually independent.In order to ensure the insurance company's stable operation, we assume which implies Then  > 0 is the relative security loading factor.
For the risk model (3), the time to ruin, denoted by , is defined as And define the ruin probability with an initial surplus  > 0 by (), namely,

𝐸 [𝑒
Lemma 9.The ruin time  is the stopping time of    .
Theorem 10.For for all , the ultimate ruin probability satisfies where  Proof.For a fixed time  0 ,  0 ∧  is a bounded stopping time; using the theorem of martingale and stopping time, we have     Proof. is a ruin time and for a fixed time  0 ,   ∧  0 is a bounded stopping time.Using the theorem of martingale and stopping time, we have

Figure 1 :
Figure 1: The impact of  on .

Theorem 5 .
Equation  () = 0 (16) has a unique positive solution  =  > 0, and (16) is said to be an adjustment coefficient equation of the risk model (3) and  > 0 is said to be an adjustment coefficient.

Figure 3 :
Figure 3: The impact of  3 on .

Figure 7 :
Figure 7: The impact of  on .

Figure 8 :
Figure 8: The impact of  on ruin probability ().