Research of the amount of demands for insurance paymentin a company with arbitrary distribution of duration of the insurance contract

The model of insurance company with an unlimited insurance field in the form of a queuingsystem with unlimited number of servers instruments is considered. The risks coming to the company form the elementary flow of events with intercity ƒ. Every risk that has been in the companyduring the period of validity of insurance police regardless of other risk generates a demand forinsurance payment with ƒ intensity. And these demands also form the elementary flow of events.The distribution B(x) of duration of the insurance contract is assumed to be arbitrary. The totalamount of demands for insurance payment by all the insured risks in the company is studied inthis model.Maximum decomposition method is used to find the formula for generating function of theamount of demands for insurance payment:( ) { ( )} ( )( )( )( )( ) ( )11 10exp exp 1 1 11x t tG x,t F x,t m ex t e t e x τ t h x, dx− ƒ− ƒ −ƒ − − = = ⎧⎪⎨⎪⎩ƒ ⎡⎣ − ⎤⎦− ƒ⎛⎝⎜⎜ ƒ− − + ⎞⎠⎟⎟− ⎡⎣ − ⎤⎦ ƒ ƒ⎫⎪⎬⎪⎭ ,where the function h(x,t) is( ) (1) ( ( )) (1) ( )01 ,th x,t ex− ƒt B t e−x− ƒƒ b t d ⎛ ⎞= ⎜⎜ƒ − +ƒ ƒ −ƒ ƒ⎟⎟⎝ ⎠and ( ( ))0m 1Bxdx=  − −is a mean value of insurance contract duration.Also, formulas for expectation and variance of S(t) - the total sum of insurance payments -were found{ ( )} 2( )2 22 3 2 2 ( ) ( )1 2 1 10 01 23tD S t m at ma t a t a t sb s dsdƒ= ƒƒ + ƒ ƒ − ƒƒ + ƒƒ  − ƒ  ƒ − ƒ ,where a1 - mean value, a2 - the second initial moment of value of payment for a singleoccurrence insured.

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