Comparison of Ruin Probabilities in Compound Poisson Risk Model

Compound Poisson risk model has been simulated. It has started with exponential claim sizes. The simulations have checked for infinite ruin probabilities. An appropriate time window has been chosen to estimate and compare ruin probabilities. The infinite ruin probabilities of two-compound Poisson risk process have estimated and compared them with standard theoretical results.


Introduction
The Compound Poisson risk model can be modeled by the quotation, ( ) ( ) where ( ) N t and { } i X are independent.
where 0 θ > , loading coefficient and λ is the rate of compound Poisson Process.
In this paper, the theoretical assumptions are: 1) In a Compound Poisson model, the ruin probability with initial reserve satisfies, ( ) and 2) In a compound Poisson model with exponential claim, ( ) ( ) It has assumed that the probability of ruin at time window of T = 1000 after 100,000 independent runs which has set just to make algorithm convenience.
For computational convenience, it has chosen the rate 0.5 λ = , loading coefficient 0.1 θ = , initial surplus 0 u = and claim size which are exponentially dis- tributed with mean 1.After simulation, an approximate 95% confidence interval based on 1,000,000 independent runs, and the confidence interval of ( ) It helped to claim that the simulation which has done so far has very high accuracy since it is approximately equal to theoretical value of 0.909 from above given condition.
It has repeated to construct more confidence interval for each where For simulation purpose set and

Compound Poisson Risk Model
Suppose that at time 0 t = , the insurer has an amount of money set aside for the portfolio.This amount of money is called the initial surplus and is denoted by u.
We can further assume that throughout this work that 0 u ≥ .The insurer needs this initial surplus because the future premium income on its own may not be sufficient to cover the future claims.We are also ignoring the expenses.The insurer's surplus at any time 0 t > is a random variable since its value depends on the claims experience up to time t.
We have an additional assumption that . Then c can be expressed as where 0 θ > is a loading coefficient and λ is the rate of compound Poisson , the ruin occurs with probability 1 which is generally not interested by insurer [1].Intuitively, that means, averagely the premium should exceed the claim rate on one unit time period.We have from Wald's Identity, which can be proved by conditioning ( )

Theoretical Results
Theorem 3.1.In a compound Poisson model defined by (1), the ruin proba-bility with no initial reserve satisfies ( ) ( ) Theorem 3.2.In a compound Poisson model defined by (1) with

Simulation
The purpose of simulation is to estimate infinite ruin probabilities at ( ) to its theoretical value defined by (15).The steps are repeated for different u value to develop ( ) First, it has focused on estimating ( ) , u τ Ψ using simulation.In each process is simulated up to time τ and the result of the i-th simulation run i Z is set to 1 if ruin occurs at or before time τ and to 0 if this is not the case.After  runs, the Monte Carlo estimator is given by, The following algorithm has used to obtain of a single simulation run.
Step 1: Initialize the simulation: set Step 2: Draw an exponential inter-arrival time I with parameter λ and draw a claim size X F .
For computational convenience, it has assumed that the probability of ruin time at time window of T = 1000 after 1,000,000 independent runs which are assigned just to make algorithm convenience.The parameters which are defined in (1) are chosen: 0.5 λ = , 0.1 θ = with initial surplus 0 u = and claim size are exponentially distributed with mean 1.An approximate 95% confidence interval of ( ) based on 1,000,000 independent runs, is given by [0.9083541, 0.9094819].
We expect to see that the theoretical curve lies in between the confidence intervals.The plot from above procedure has expected to see a good estimate.However, it is not the case always since the time window chosen may not be the sufficient to provide an accurate estimate.The simulation has repeated to see different confidence interval for each  .Then four different ruin probabilities plots are estimated from simulation (Figure 1).
One main issue is that the Monte Carlo simulation used doesn't work for small ruin probabilities.It could be the reason the graph doesn't display the probabilities of ruin smaller than 0.15.We need to test for more surplus with sufficient time window to address this issue.For this, important sampling or other method is required.But the scope of this paper is to extend this estimation to compare ruin probabilities

Ruin Probabilities Comparison of Two Compound Poisson Model with Different Claims
Definition: Let X and Y be two random variable where X is smaller than Y in convex order denoted by for all convex function : φ →  , provided the expectation exists.

Theoretical Results
Theorem 6.1.For two random variable X and Y defined from (18 provided that the expectation exist [3].Theorem 6.2.For two random variable X and Y defined from (18 provided that expectation exist [3].
The above result implies that cx X Y ≤ means Y has more dispersion.The above result is important because this guarantees that the premium rate, which makes easier to compare the ruin probabilities.A typical way to construct X and Y following order cx ≤ has seen from Ohlin's Lemma.

Ohlin's Lemma
Lemma 7.1.Let Z be a positive random variable and f and g be increasing functions.If g crosses f only once from below, i.e., there exist o x , such that ( ) ( ) ( ) ( ) The above setting is motivated from reinsurance problem.We can think Z as the original risk and f, g are different insurance strategies.In this context, besides increasing, f and g are assumed to be continuous and satisfy ( ) (5) and (6).We are interested in two quantities: the ruin time and infinite ruin probability of two different models defined by ( 5) and ( 6) respectively.

Compound Poisson Risk Models with Different Claims
. u: initial surplus at time 0 t = .c: rate of premium income per unit time.An additional assumption: simulation.The compound Poisson risk model has been estimated.
defined by theorem 8.1 which has demonstrated on Figure 2. Open Journal of Statistics
defined in(4) and (5) respectively with generic claim size X and Y.If to see the above theoretical results by using simulation.The theoretical result has observed in simulation for different values.For simulation purpose we can set, of loss insurance form.For example, customer's loss in a car accident with an auto insurance where only the part in excess of certain level is covered by insurance company.The maximum loss for the customer has capped at d. Again the setting applicable on quota share (proportional insurance).Graph of has estimated as Figure2.
The time window and independent runs are changed as required.It has also The insurer's surplus at time t is i X are independent.Again u is initial surplus at time 0 t = .The ruin probability of compound Poisson surplus process is denoted by