Insurance: Mathematics and Economics 40 (2007) 485–497 www.elsevier.com/locate/ime Moments of claims in a Markovian environment Bara Kim a , Hwa-Sung Kim b,∗ a Department of Mathematics, Korea University, Anam-dong, Sungbuk-ku, Seoul, Republic of Korea b Department of Business Administration, Kwangwoon University, Wolgye-dong, Nowon-ku, Seoul, Republic of Korea Received February 2005; received in revised form January 2006; accepted 1 July 2006 Abstract This paper considers discounted aggregate claims when the claim rates and sizes fluctuate according to the state of the risk business. We provide a system of differential equations for the Laplace–Stieltjes transform of the distribution of discounted aggregate claims under this assumption. Using the differential equations, we present the first two moments of discounted aggregate claims in a Markovian environment. We also derive simple expressions for the moments of discounted aggregate claims when the Markovian environment has two states. Numerical examples are illustrated when the claim sizes are specified. c  2006 Elsevier B.V. All rights reserved. MSC: IM11 Keywords: Discounted aggregate claims; Laplace–Stieltjes transform; Moments; Circumstance process 1. Introduction In the risk model the discounted value of aggregate claims is represented as follows: Let s n be the ith epoch of claim occurrence, (n = 1, 2, 3,...). Set s 0 = 0 for convenience. Let X n be the size of the claim that occurs at s n (n = 1, 2, 3,...). Let δ be the instantaneous rate of net interest. Then the discounted value of aggregate claims up to time t is given by L (t ) = N (t )  n=1 X n e −δs n , (1) where N (t ) is the number of claim occurrences during (0, t ], that is, N (t ) = sup{n ≥ 0 : s n ≤ t }. (2) The classical risk model assumes zero rate of net interest, that is, δ = 0 in Eq. (1). The discount rate of aggregate claims is the difference of the claim inflation and the interest earned on investments. As stated in Jang (2004), in real phenomena these two components of the net interest rate do not have the same level. In contrast with the classical ∗ Corresponding author. Tel.: +82 2 940 5595. E-mail address: fstar@kw.ac.kr (H.-S. Kim). 0167-6687/$ - see front matter c  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2006.07.004