3. ARTÍCULOS DE APLICACIÓN

Transcription

1 3. ARTÍCULOS DE APLICACIÓN SOME PROBABILITY APPLICATIONS TO THE RISK ANALYSIS IN INSURANCE THEORY Jinzhi Li College of Science Central University for Nationalities, China Shixia Ma School of Sciences Hebei University of Technology, China Abstract This work concerns with some probability applications in insurance theory. The problem to determine the ruin probability of an insurance company is considered. We show that by using some stochastic models considering several probabilistic procedures such a probability can be approached. As illustration, an application to car insurance is provided. Keywords: Risk analysis, Stochastic modelling, Insurance theory applications. 1. Introduction Risk analysis in insurance theory is an important area for probability statistical applications. In particular, the probabilistic modelling of the surplus evolution in an insurance company has received some attention in the specialized literature. In fact, denoting by Ut) the surplus of an insurance company at time t, the classical model establishes that: Ut) = u + ct St), t, 1.1) where u > is the initial surplus, c is the constant rate at which the premiums are received per unit time St) is the aggregate amount concerning the claims reported in the time interval [, t]. It is assumed that {St), t } is a compound Poisson process, St) = Nt) X i, t {Nt), t } being a Poisson process {X n, n = 1,,...} a sequence of independent identically distributed rom variables, both assumed to be independent. The variable Nt) represents the number of reported claims in [, t] X i is the amount corresponding to the ith claim. In general, from model.1), some results in risk Corresponding Author. mashixia1@163.com theory have been derived but such a model it is not exible enough in order to describe the probabilistic evolution of Ut) in more complex real situations. In an attempt to contribute some solution to this problem, several classes of stochastic models have been introduced some theory applications about them developed. We will quote, for example: a) Stochastic models considering claim arrivals governed by non-poissonian processes. See e.g. [4] where it is assumed that the number of claims is described through a Cox process. b) Stochastic models allowing a rate c ) which change through a Markov process. See e.g. [1] or [6] where it is considered that the rate changes modulated by an underlying irreducible Markov chain or a premium rate in a Markovian environment, respectively. c) Stochastic models including a diusion component which represents uncertainties in both the premium income the costs. Firstly studied by Gerber, such models have been applied in several risk problems, see e.g. [], [3] or [8]. In this work, we will focus our interest in a class of risk models which allows premium rate

2 diusion component depending on an underlying continuous-time Markov chain. Moreover, the occurrence of claims is assumed to be well-described by a Cox process. It is usually referred as the class of risk models with Markov modulated speed. We will center our attention in the determination of the ruin probability for an insurance company. This problem is an important objective in many research developed in actuarial risk theory. We will show by using some classical probabilistic techniques that it is possible to obtain the ruin probability. The paper is organized as follow: In Section, we provide the probabilistic descriptions of such a class of risk models. In Section 3, we develop a probabilistic procedure to determine the ruin probability. Finally, Section 4 is devoted to considering an application in car insurance.. Stochastic modelling Let us consider the following stochastic modelling for Ut), t : Ut) = u + Nt) cis))ds X i + σis))dw s.1) where: a) {Nt), t }, N) =, is a Cox point process. Nt) represents the number of claims received for the insurance company during the interval [, t]. b) {X n, n =, 1,...} is a sequence of independent identically distributed rom variables which represents the amount corresponding to the successive claims received by the company in [, t]. Let us write F x) = P X 1 x), x >, F ) =, µ = E[X 1 ]. c) {W t), t } is a stard Wiener process with diusion coecient σ ). This process represents the disturbances originated from several tiny stochastic factors. d) {It), t } is an underlying continuoustime homogeneous Markov chain with state space S = {1,..., n} assumed to be irreducible. Notice that in.1) the premium rate c ) the diusion coecient σ ) depend on the current state of the Markov chain {It), t }. In fact, if at time t it is veried that It) = i then, by simplicity, cit)) σit)) will be denoted, respectively, as c i σ i, assumed to be positive. On the other h, we are considering that the number of reported claims is governed by a Cox process, hence if Is) = i, s [, t] then the number of claims received in [, t] has a Poisson distribution with mean λ i >. According to [5], we shall denote by q i the rate at which the Markov chain {It), t } leaves the state i, by q ij p ij, respectively, the transition intensity the transition probability that it leaves the state i for the rst time enter into the state j immediately. Assuming that p ii =, i S, one deduces that q ij = q i p ij for i j q ii = q i. Since all the states communicate, π i q i = π j q j p ji..) where π 1, π,..., π n denotes a stationary distribution corresponding to {It), t }. Also, we will assume that the named safety loading is positive, namely c λµ > where: c = π i c i λ = π i λ i. 3. Ruin probability In this section we are interested in the determination of the ruin probability dened by: ψu) = π i ψ i u) 3.1) where ψ i u) = P Ut) U) = u, I) = i) for some t. First, by considering the evolution of Ut) in a short interval [, h), h >, we shall determine a system of equations for R i u) = 1 ψ i u), i S. In fact, assuming that I) = i, we can consider the following possibilities during the interval [, h): a) There is not reported claims Is) = i for s [, h). b) One claim is produced but the amount to be paid for such a claim does not cause ruin Is) = i for s [, h). c) There is not reported claims, from the state i a change to other state it is produced. 1

3 d) At least one claim is reported in at least one change of state, from i, is produced. Consequently, see for more details [], on has for t [, h), R i u) = 1 λ i h q i h + oh))e[r i ϕ i u, h)] +λ i h + oh))1 q i h + oh)) E[ ϕi u,h) R i ϕ i u, h) x)df x)] +1 λ i h + oh))q i h + oh)) p ij E[R j ϕ i u, h))] + oh). 3.) where ϕ i u, h) = u + c i h + σ i W h. By considering the Taylor expression in u of E[R i ϕ i u, h))], dividing by h, taking limit as h, it is matter of some straightforward calculation to deduce, σ i R i u) + c i R iu) = λ i + q i )R i u) u λ i R i u x)df x) n q i p ij R j u). 3.3) By integration of 3.3) on [, t] using the fact that R i ) =, σi it) + c R i R i t) = σ i i) R + λ i + q i ) λ i q i n, taking into account that u u R i u)du R i u x)df x)du p ij R j u)du. R i u x)df x)du = R i u)du + R i t x)f x)dx. where F x) = 1 F x), considering the fact that ψ i t) = 1 R i t), i = 1,..., n, σ i ψ it) + c i ψ i t) = c i + σ i ψ i) + λ i λ i ψ i t x)f x)dx +q i ψ i u)du q i F x)dx n p ij ψ j u)du. 3.4) Finally, taking limit as t, we deduce the following system of equations for ψ i u), i = 1,..., n: ψ i) = σ i c i λ i µ q i ψ i u)du n +q i p ij ψ j u)du ). 3.5) Using the numerical solutions of 3.5), from 3.1) we may determine the corresponding ruin probability. Note that when c i = c σ i = σ, i = 1,..., n, by 3.4) 3.5), taking into account.) we deduce: σ ψ t) + cψt) = c + σ ψ ) + λ π i λ i ψ i t x)f x)dx, ψ ) = c + λµ) σ F x)dx respectively. 4. Application to car insurance It is well-known the inuence that certain factors, for example the inclemency of the weather, the conditions of the roads, so on, have in the occurrence of trac accidents. Next we shall consider an application of the model.1) in car insurance. In a rst approximation, we will assume that

4 the underlying Markov chain {It), t } has a two-states space, namely S = {1, }, where: The state 1 represents the risk under normal conditions. The state represents the risk under bad conditions for e.g. slippery roads, foggy days or high trac volume. We refer the reader to [5] [7] for more details. Also, we will consider that the variable X i has exponential distribution with mean µ that p 1 = p 1 = 1, p 11 = p =. Then, q 11 = q 1, q = q, q 1 = q 1, q 1 = q π 1 = q q 1 + q ) 1, π = q 1 q 1 + q ) 1. Let us denote by φ i s) = φ s) = Taking into account that e st ψ i t)dt e st F t)dt. e st ψ it)dt = sφ i s) 1, e st ψ i u)dudt = 1 s φ is). by using the Laplace transformation in 3.4), ones deduces, for i = 1,, c i + σ i s q ) i s + λ iφ s) φ i s)+ q i s hence, p ij φ j s) = σ i + 1 s c i + σ i ψ i)) + λ i s φ s) c 1 + σ 1 s q 1 s + λ ) 1µ φ 1 s) + q 1 sµ + 1 s φ s) = σ1 + 1 ) c 1 + σ 1 s ψ 1) + λ 1µ ssµ + 1) c + σ s q s + λ ) µ φ s) + q sµ + 1 s φ 1s) = σ + 1 ) c + σ s ψ ) + λ µ ssµ + 1). Conclusion: The surplus evolution corresponding to an insurance company could be suitably described in terms of the general model given in.1). From a practical point of view, by solving the system given in 3.5) taking into account expression 3.1), it is possible to determine the ruin probability. This parameter plays a crucial role in research about risk analysis in insurance theory. Acknowledgements: The authors would like to thank Professor Molina from the Department of Mathematics at Extremadura University for his constructive suggestions which have improved this paper. This research has been supported by the Funded Projects of the Education Department of Hunnan Province, grant number 7C745. References [1] Asmussen, S. Kella, O. 1996). Rate modulation in dams ruin problems, Journal of Applied Probability 33, [] Dufresne, F. Gerber, H. U. 1991). Risk theory for the compound poisson process that is perturbed by diusion. Insurance: Mathematics Economics. 1, [3] Gerber, H. U. Lry, B. 1998). On the discounted penalty at ruin a jump-diusion the perpetual put option. Insurance: Mathematics Economics., [4] Gerber, H. U. Shiu, E. S. W. 1998). On the time value of ruin. North American Actuarial Journal. 1), [5] Grell, J. 1991). Aspects of Risk Theory. Springer, Berlin. [6] Jasiulewicz, H. 1). Probability of ruin with variable premium rate in a Markovian environ- 3

5 ment. Insurance: Mathematics Economics. 9, [7] Reinhard, J.M. 1984). On a class of semi- Markov risk models obtained as classical risk models in Markovian environment. ASTIN Bulletin XIV, [8] Wang, G. 1). A decomposition of the ruin probability for the risk process perturbed by diffusion. Insurance: Mathematics Economics. 8,