A generalized model of a classical risk process describing the evolution of the capital of an insurance company in a random environment is considered. A system of integral equations for the bankruptcy probability if a function of initial state. The possibility of applying the method of successive approximation to solve the system is analyzed. The method generates approximations that converge from above and below to the solution.Keywords: actuarial mathematics, risk process, Markovian chain, bankruptcy probability, system of integral equations, method of successive approximations.
RISK PROCESSES IN A MARKOVIAN ENVIRONMENTA classical risk process describes the stochastic evolution of the capital of an insurance company and is formally specified by the equationwhere t is time; u ³ 0 is the initial capital of the insurance company; c is the premium income rate; S t is the claim payment process, S Y t k N k t = = å 1 , Y k are independent equally distributed random variables (claims) with distribution function F z ( ) and mean m = -+ ¥ ò 0 1 ( ( )) F z dz; N t is the number of premium payments by a time t (a Poison process independent of S t ). One of the most important stability parameters of an insurance company is (non-)bankruptcy probability. It is generally known [1] that the non-bankruptcy probability j x ( ) u P t t = ³ ³ {inf } 0 0 on an infinite time interval for the classical risk process (1) satisfies the integral equation j am a j ( ) ( )( ( )) u c c u z F z dz u = -+ --ò 1 1 0 .(2)The classical model assumes that the parameters c, , , a m and F of the process (1) do not vary with time. A more general risk process in a random (Markovian) environment is considered in [2][3][4][5][6][7][8][9][10][11]. The parameters c i i i , , , a m and F i of the process may depend on a state i of the environment (for example, weather conditions, traffic intensity in case of motor insurance, parameters of economic environment, etc.). For a fixed state of the environment, the random moments of claim arrival and claimed premium states are assumed independent. Let us consider a risk process x t in a Markovian environment 917
519.21Necessary and sufficient conditions of existence and uniqueness of solutions of integral equations are established for ruin probability as a function of the initial capital of an insurance company. Several sufficient conditions and correctness conditions of the problem of finding ruin probability are also established. A general method of successive approximations for solution of this problem is substantiated.
Keywords: actuarial mathematics, insurance mathematics, risk process, ruin probability, integral equation, existence and uniqueness of a solution, necessary and sufficient conditions, method of successive approximations, correctness.We call a risk process a random process x t ³ 0 describing the stochastic evolution of the capital of an insurance company and denote by x 0 = u the initial capital of the company [1]. We denote by Y( ) u the ruin probability for the insurance company as a function of u ³ 0 and by j x ( ) ( ) u u P t t
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