Simple approximation of an insurance company ruin probability for cramer-lundberg model with stochastic premiums

In this paper the calculated formulas which allow to construct the simple approximation for the ruin probability of an insurance company for Cramer-Lundberg model with stochastic premiums are obtained. The approximation is based on the replacement of the true risk process with arbitrary distributions of the insurance premiums and the insurance payments m(t) n (t) s (t) = S (0) + £ x -x yj, i=1 j=1 where S(0) is the initial capital, m(t) is the number of received insurance premiums to the time t, n(t) is the number of insurance payments, x_t is the value of i -th insurance premiums, y. is the value of j-th insurance payment, a_k = M {x_i^k } , b_t = M {yf } , the flow of insurance premiums is Poisson with intensity X, the flow of insurance payments is Poisson with intensity on the risk process with the insurance premiums and the insurance payments having the exponential distributions. At that the premiums flow intensives X₀, the payments flow intensives ц₀, the average values of the premiums a and the payments b in the approximating process are chosen so that the first four moments of the true and approximating risk processes coincided. This requires the fulfillment of the conditions dj = Xa_x -цЪ_г = X₀a -ц₀Ъ, d₂ = 2(Xa₂ + цЪ₂) = X₀a² + ц₀Ъ², d₃ = ¹(Xa₃ - цЬ₃) = X₀a³ - ц₀Ъ³, d₄ = -(Xa₄ + цЬ₄) = X₀a⁴ + ц₀Ь ⁴. 6 24 The parameters a, b, X₀, ц₀ are determined by the ratios d₁b + d₂ d 2 - d₁a Xq =-, Ц0 =-, a(a + b) b(a + b) v Wv² + 4m -v Wv² + 4m where d d4 d d do d 2d 4 v = -, M =--=-2- . d1 do d 2 d1 do d 2 For the rating of the ruin probability is taken the ruin probability of the approximating process ₍^X0^a-^0^b_S P(S) = M-0^(a + ^b)_e (Xq+Ц0)ab . (XQ +h))a

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