Deriving the Equation for the Non-Ruin Probability of the Insurance Company in ( B , S )-market. Stochastic Claims and S
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DERIVING THE EQUATION FOR THE NON-RUIN PROBABILITY OF THE INSURANCE COMPANY IN ( B, S ) -MARKET. STOCHASTIC CLAIMS AND STOCHASTIC PREMIUMS B. V. Bondareva† and V. O. Boldyrevaa‡
UDC 519.21
Abstract. The integro-differential equations for the non-ruin probability, on finite and infinite time intervals, for the insurance company operating in the ( B , S )-market are derived for the Cramer–Lundberg model with stochastic premiums. To derive the equations, no smooth distribution densities of premiums and claims are required. The examples are considered where the problem is reduced to the solution of differential or integral equations. Keywords: Samuelson model, non-ruin probability, stochastic premiums and claims, transition probability density, Ito equation. INTRODUCTION Finding the non-ruin probability of an insurance company is one of the major problems of actuarial and financial mathematics [1, 2] since insurance activity requires the influence of various random factors to be taken into account as fully as possible. To obtain the model of a company that is most close to reality, stochastic premiums and claims are used that take into account the random nature of time and of the amount of insurance claims and premiums arriving at the company. It is also necessary to consider the finances obtained by the insurance company due to investments of spare cash in bank accounts and in shares. The problem of finding the non-ruin probability of an insurance company was solved for some simple models with special cases of distributions. Integro-differential and integral equations for finite and infinite time intervals were obtained in the general case. The absence of a diffusion component in the main Samuelson process required smooth density functions of the amounts of premiums, claims, and share prices, which imposed additional constraints [3]. The purpose of the present study is to obtain a new mode to derive the equations for the non-ruin probability of insurance company. METHOD OF DERIVING THE INTEGRO-DIFFERENTIAL EQUATIONS Let us consider the Cramer–Lundberg model for an insurance company that operates on ( B , S )-market, i.e., where the available assets are allocated on a bank account (risk-free assets B) and in shares (risk assets S ). Denote by x x ( t ) the company capital at time t provided that at the initial time the company capital is x x ( 0) = x. Assume that the insurance claims and the premiums are stochastic. The number of claims obeys the Poisson distribution law Z ( t ) , and the number of arriving premiums obeys the Poisson distribution law Z1 ( t ). The total claims
Z (t )
0
k =1
k =1
å h k (assume that å h k
= 0), where h k are the values
of claims, P ( h k < x ) = F ( x ) , make a complex Poisson process with the parameter l, which is representable as stochastic a
Donetsk National University, Donetsk, Ukraine, a†[email protected]; a‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2014, pp. 113–121. Original article submitted November 21,
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