Simulation for Ruin Probabilities in Insurance with Sequence Markov Dependence Random Variables

The aim of this paper is to calculate ruin probabilities using Monte Carlo method for two models: i) classical risk model with claim amounts are homogeneous Markov chains; ii) generalized risk models with premiums amounts, claim amounts are homogeneous Markov chains. The sequence of random variables in the article is considered as a series of Markov dependent random variables. The main results of this paper are Lemma 3.1, Lemma 3.2 and Lemma 3.3, which have built mathematical formulas for the simulation of the probability of insurance models considered in this paper. From those lemmas, we build algorithms to simulate ruin probability for insurance models considered in this paper. From these algorithms, we build numerical results illustrating the problems posed in the paper. These results all show that when the initial capital increases, the ruin probability will decrease, and when the time increases, the ruin probability will increase. This result is consistent with the theory of the risk problem in insurance.


Introduction
In risk theory, the premiums amount U(t) at time t: t N i i1 U(t) u rt X      , where u > 0 is the initial capital of that company, and r is the premium rate per a unit of time.
The number of claim amounts to time t, N t is the pure Poisson process with intensity  and claim amount series {X i } is a series of independent random variables having the same distribution as the probability distribution function F, which have finite mean  . The ruin probability with finite time t, denoted (u, t)  , is defined by: Cramer -Lundberg and R is called exponential constant Lundberg. (see H. U. Gerber [3] and Grandell [4]). For these dependency structure models, it would often be very hard to calculate the approximation of exponential constant R (see Phung Duy Quang [5], [6]). In [1], authors considered the numerical solution to one type of integro-differential equation by a probability method based on the fundamental martingale of mixed Gaussian processes. As an application, we try to simulate the estimation of ruin probability with an unknown parameter driven not by the classical Lé vy process, but by the mixed fractional Brownian motion. In [2], authors studied based on a discrete version of the Pollaczeck-Khinchine formula, a general method of calculating the ultimate ruin probability in the Gerber-Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Analytical results and numerical results are often unknown. Simulation Monte Carlo methods can provide tools for calculating approximately probabilities (u, t)  . The aim of this paper is to approximately calculate ruin probability (u, t)  in two cases using Monte Carlo simulation method: i) the claim amounts is an homogeneous Markov chain in classical model; ii) premiums amounts, claim amounts are homogeneous Markov chains in the general model does not have effects of interests.
In the second section of this paper, authors will introduce the classical model, the general model that has no effect of interest rates with a series of Markov dependent random variables. In the third section of this paper, authors will introduce simulation algorithms to calculate ruin probability in the models introduced in the second section of this paper. In the fourth section of this paper, authors will introduce simulation results with different homogeneous Markov chain dependent models. Finally, the fifth section concludes the paper.

Insurance Model with Homogeneous
Markov Chain Dependent Random Variables

Classical Risk Model
In the classical risk model, we assume that the capital of the insurance company at time t is: Where u is the initial capital, r is the cost of credit, X t is the claim amount at time t; N t is the number of claims up to time t (N t is the pure Poisson process with intensity  , the interval between two claims, which is independent and co-distributed, following an exponential distribution with parameter  , expectation 1  ); X t is a homogeneous Markov chain independent of N t ; the total claim amounts up to time t is Ruin probability to time t is determined by:

The General Risk Model where There Is No Interest Rate Effect
In the general risk model where there is no interest rate effect, we assume that the capital of the insurance company at time t is: Where u is the initial capital; the series of premium amounts X 1 , X 2 , …, X n depends on homogeneous Markov chain; series of claim amounts Y 1 , Y 2 , …, Y n depends on homogeneous Markov chain (X t is independent on Y t ); 1 t N is the number of premium amounts up to time t with 1 t N is the pure Poisson process with intensity 1  > 0 (the time interval between two premium is independent and co-distributed, following an exponential distribution with parameter 1  , the expectation is 1 1  ), X t is independent on 1 t N ; 2 t N is the number of claim amounts to time t with 2 t N is the pure Poisson process with intensity 2  > 0 (the time interval between two claims is independent and co-distributed, following an exponential distribution with parameter 2  , the expectation is The ruin probability to time t is determined by:

The Monte Carlo Simulation Method
Approximates the Ruin Probabilities in the Insurance Model

The Algorithm to Simulate A Homogeneous Markov Chain
We assume that X n are defined on the probability space X  is an homogeneous Markov chain, such that for any n the values of X n are taken from a set of Let us assume we use inversion to generate such a Y i .
 . In the following algorithm, whenever we say "generate a Y i ", we mean doing so using this inverse transform method using uniform distribution. We will build a homogeneous Markov chain simulation algorithm. , generate Y i , set n = n + 1 and set X n = Y i ; otherwise stop.
Step 4. Go back to Step3.

The Algorithm to Simulate Ruin Probability for the Model (2.1)
We see model ( In which, random numbers v i ( i1  ) is independent. We, now, consider event A(t) (up to time t) of the problem (2.1): The basis for simulating event A(t) is the following proposition:

Prove:
Without losing generality, we assume t N1  , we set To point out that: On the other hand: From Lemma 3.1, the ruin probability at (2.4) is estimated as: Where M is the number of occurrences of event A(t) in N simulations and M is determined by the following algorithm. Notice that: the loop will stop when i = N t (xem (3.1)) and finish the n th simulation of event A(t).
Step B. After simulating N times event A(t) (repeat N times step A, approximately calculate the probability of risk: M (u, t) . N 

Algorithm to Simulate Ruin Probability for the Model (2.2)
To describe the method, we consider the model (2.2) with the assumption that: series of amounts   Then we have: Where 12 j 12 j1 2) In case For each 2 j 2 N (t)  , we rely on equations (3.6) and (3.12) to represent (3.11) in the form: That mean, we have (3.6) for all Since u > 0 and i X ( i1  ) are non-negative random variables, from the above formula, we directly deduce (3.8).
Now we consider the risky event A(t) (up to time t) of problem (2.2): The basis for simulating event A(t) is the following proposition:  2. Event A(t) also does not occur, if: v~ U(0, 1)) (3.16)

Prove:
In the case of 2 N (t) 1  , we assign 2 2 1 Firstly, we consider the case j= 1 meaning that (see That means, we obtained (3.19) with j = 1. Next, we consider case On this basis (3.21) and (3.22) we get: So, (3.14) is proven. When letting: We rely on (3.14) and the De Morgan duality rule to infer: Therefore, in condition (3.15) event A(t) will not occur and conclusion number 1. is completely proved.
To prove the rest, we rely on (3.4) and (3.5) to deduce the equivalence of the following events: 2 1 v~ U(0, 1).
When the above event has occurred, from (3.8) and (3.13) we find that event A(t) will not happen and we get the conclusion number 2.
Since random variables     Then we finish the n th time simulation of event A(t) (see (3.15)). In case N 2 (t) = 0 (see (3.16), the n th time simulation of event A(t) will end immediately at step A1 with j = 1.
We have compiled calculation software in Python environment to demonstrate algorithm 3.2, when running this program, we obtain simulation results of ruin probability for model (2.1) with hypothesis (4.1) given in table 4.1 below:

Simulation Ruin Probability of the Model (2.3)
With input data: u = 2; u = 3; u = 4; u = 5; u = 6; u = 7; time t gets values: t = 4, t = 6, t = 10; number of simulations N = 1000; parameter 1  We have compiled calculation software in Python environment to demonstrate algorithm 3.3, when running this program, we obtain simulation results of ruin probability for model (2.3) with hypothesis (4.2) and (4.3) given in table 4.3 below:

Conclusions
This paper study two models: i)the claim amounts is an homogeneous Markov chain in classical model; ii) premiums amounts, claim amounts are homogeneous Markov chains in the general model does not have effects of interests.
The sequence of random variables in the article is considered as a series of Markov dependent random variables. The main results of this paper are Lemma 3.1, Lemma 3.2 and Lemma 3.3, which have built mathematical formulas for the simulation of the probability of insurance models considered in this article. This paper has built the theoretical basis of simulation for ( When increasing the initial capital u of insurance companies, the ruin probability will decrease. For each level of capital u, as time t increases, the ruin probability will increase.
This study is a result of a research with the title 'Mathematical Models in Economics and Application in