Ruin probability of an insurance company under double stochastic insurance premium and insurance payment currents

The aim of this paper is to generalize the classical model of an insurance company for thesituation when the intensity of the insurance premiums flow ƒ(t) is the homogeneous Markovchain with continuous time, the states ƒ(t) =ƒi (i= 1,m) and the matrix of infinitesimal characteristicsƒ =[ƒi j], which has the rank m −1 . Insurance premiums are independent identicallydistributed with the distribution density ϕ(x) , the mean M{x} =a and the second moment2M{x}=a2 . Analogously the insurance payments flow ƒ(t) is the homogeneous Markovchain with continuous time, the states ƒ(t) =ƒi (i= 1,n) and the matrix of infinitesimal characteristicsƒ = [ƒi j ] , which has the range n −1 . Insurance payments are independent identicallydistributed with the distribution density ƒ(x), the mean M{x} =b and the second moment2M{x}=b2 . Let ƒ be the loading of insurance premium, ƒ0 and ƒ0 be the average intensitiesof premiums' and payments' flows correspondingly. Then ƒ0a= (1+ƒ)ƒ0b.Let Gi j(s) be the ruin probability of an insurance company if at the beginning moment itscapital is s and the intensities quantities are ƒi and ƒ j correspondingly. It is shown, that for ƒ << 1ƒ ⎨− ƒ ⎬= ⎩ ⎭ + ƒƒ + ƒ − ƒ  ϕwhere( ) ( ) ( ) ( ) 1 1 1 10 2 0 22 21 0 0 0 01 1 1 1,2m m n nk k kj j k k kj jk j k jA a b a R b Q− − − −= = = =ƒ +ƒ= − ƒƒ ƒ − ƒ ƒ ƒ −ƒ − ƒƒ ƒ −ƒ ƒ ƒ −ƒ A2= ƒ0b ,ƒi are the final distribution of the premiums' intensity quantity, ƒi are the final distribution ofthe payments' intensity quantity, the matrixes1( ) 1( ) R ij i,j 1,m 1,Q ij i,j 1,n 1 − − = ⎡⎣ƒ ⎤⎦ = − = ⎡⎣ƒ ⎤⎦ = − .

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