# The ruin problem for a class of correlated aggregated claims model

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```					   Scientia Magna
Vol. 5 (2009), No. 4, 96-102

The ruin problem for a class of correlated
aggregated claims model
Wei Liu

School of Sciences, Linyi Normal University, Linyi, Shandong 276005, P.R.China
E-mail: liuwei@lytu.edu.cn liuweide3544@sina.com

Abstract In this paper, the author studies a risk model. Under this model the two claim pro-
cesses are correlated. Claim occurrence relate to Poisson and Erlang processes. The formulae
is derived for the distribution of the surplus immediately before ruin, for the distribution of
the surplus after ruin and the joint distribution of the surplus immediately before and after
ruin. The asymptotic property of these ruin functions is also discussed.
Keywords Double risk model, Poisson process, Erlang(2) process, correlated aggregate cla-
im, ruin function.

§1. Introduction
Resently, many authors studied various correlated aggregate claims models. In this model,
the two claim number processes are correlated. Ambagaspitiya(1998) considered a general
method of constructing a vector of p dependent claim numbers from a vector of independent
random variables, derived formulae to get the correlated claims distribution.
On the other hand, Erlang(2) distribution is also one of the mostly commonly used distribu-
tions in risk theory, for example, Diskson and Hipp(1998)considered the inﬁnite time survival
probability as a compound geometric random variable under the Erlang(2) risk model. Sun
and Yang(2004) derived the integro-diﬀerential equation and Laplace transform of the joint
distributions of the surplus immediately before and after ruin for Erlang(2) risk processes.
In this paper we consider a correlated risk model. Under the assumed risk model the
claim number processed involve Poisson and Erlang(2) process. We derived the formulae for
the distribution of the surplus immediately before ruin, for the distribution after ruin and the
joint distribution of the surplus immediately before and after ruin. The asymptotic property of
these ruin functions is also studied.

§2. Model set-up and model transformation
We deﬁne the surplus process

N1 (t)          N2 (t)
U (t) = u + ct −            Xi −            Yi , t ≥ 0,                      (1)
i=1             i=1
Vol. 5              The ruin problem for a class of correlated aggregated claims model        97

where u is the initial surplus, c > 0 is the premium rate. {Xi , i ≥ 1} and {Yi , i ≥ 1}are
independent random variables with distributions FX (x), FY (y), density functions fX (x), fY (y).

 N (t) = M (t) + M (t),
1        1        2
(2)
 N2 (t) = M (t) + M2 (t),

here {M1 (t), t ≥ 0}, {M2 (t), t ≥ 0} are Poisson processes with parameters λ1 and λ2 respec-
tively. {M (t), t ≥ 0} is a Erlang(2) process with parameter β. We also assume {M1 (t), t ≥
0}, {M2 (t), t ≥ 0} and {M (t), t ≥ 0} are three independent renewal processes.
Let T denote the time of ruin, so that

T = inf{t ≥ 0 : U (t) < 0}.

Then the probability of ultimate ruin with initial surplus is deﬁned as

ψ(u) = P (T < ∞|U (0) = u), u ≥ 0.

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Description: In this paper, the author studies a risk model. Under this model the two claim processes are correlated. Claim occurrence relate to Poisson and Erlang processes. The formulae is derived for the distribution of the surplus immediately before ruin, for the distribution of the surplus after ruin and the joint distribution of the surplus immediately before and after ruin. The asymptotic property of these ruin functions is also discussed. [PUBLICATION ABSTRACT]
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