Method of Successive Approximations for Solving Integral Equations of the Theory of Risk Processes

2004

Cybern Syst Anal

Self Cite

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

“…The author derived the non-bankruptcy probability j( ) u of a classical risk process in [14,15] from the integral equation (2) …”

Section: Methods Of Successive Approximationsmentioning

confidence: 99%

The method of successive approximations for calculating the probability of bankruptcy of a risk process in a Markovian environment

2004

Cybern Syst Anal

Self Cite

“…It was shown in [2,3] that the function of nonruin probability ϕ ( u ) satisfies the following integral equation [the basic equation of actuarial mathematics (see also Eq. (3.74) in [4] with U ( u, t ) = u + at and F ( 0 ) = 0 )]:…”

mentioning

confidence: 99%

On the solution of the basic integral equation of actuarial mathematics by the method of successive approximations

2007

Ukr Math J

Self Cite

We study the basic integral equation of actuarial mathematics for the probability of (non)ruin of an insurance company regarded as a function of the initial capital. We establish necessary and sufficient conditions for the existence of a solution of this equation, general sufficient conditions for its existence and uniqueness, and conditions for the uniform convergence of the method of successive approximations for finding the solution.Consider a random risk process (with random claims and deterministic and random premiums) ξ t that describes the evolution of the capital of an insurance company in time t and satisfies the stochastic equation [1]where u ≥ 0 is the initial capital, c( ) ⋅ is a nonnegative piecewise-continuous function that describes the intensity of arrival of deterministic premiums as a function of the current capital, S t = z k k N t = ∑ 1 are aggregated random insurance claims and premiums, z k are independent random variables (claims in the case z k ≥ 0 or random premiums in the case z k ≤ 0 ) with common distribution function F ( z ), and N t is the number of random claims and premiums arrived by time t (an ordinary renewal process with time distribution function between successive events K ( t )). We consider the ruin probability of the insurance company ψ ( u ) = P{ } :and the corresponding nonruin probability ϕ ( u ) = 1 -ψ ( u ) as functions of the initial capital u ≥ 0.We define the function of growth of capital in the absence of insurance claims U ( u, t ) as the solution of the following Cauchy problem for an ordinary differential equation:For example, if c( ) ⋅ ≡ a, then U ( u, t ) = u + at. If the capital of the insurance company is deposited with continuous interest rate δ and c t ( ) ξ ≡ a t + δξ , δ > 0, then

“…As is shown in [2][3][4][5][6], these assumptions are true for each concrete integral operator of insurance mathematics. Assumption A.…”

mentioning

confidence: 99%

Necessary and sufficient conditions of existence and uniqueness of solutions to integral equations of actuarial mathematics

2006

Cybern Syst Anal

Self Cite

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