Ruin Theory In Some Renewal Risk Models

In actuarial mathematics, ruin theory is the core in risk theory. Many literatures on ruin theory are concerned with the classical risk model, in which claims occur as a Poisson process. Sparre Andersen (1957) considered the situation in which claims occur as a general renewal process, and an explicit result for the ultimate ruin probability was derived for a particular case. The explicit expression for Laplace transform of the ruin time with exponentially claim amount distribution is obtained by Malinovskii (1998);and Wang and Liu (2002) generalized the result to the case when claim amount has the mixed distribution of two exponentials.During the recent years, in the study of insurance risk analysis, many people have been interested in the Erlang risk model in which the interclaim times have Erlang distribution, see e.g. Dickson and Hipp (1998), Dickson and Hipp (2001), Cheng and Tang (2003a,b), Gerber and Shiu (2005) Li and Garrido (2005), among others. Gerber and Shiu (2003b) got the perfect result for generalized Erlang process. Li and Carrido (2005) obtained Integro-differential equation for generalized Erlang process with perturbed by diffusion.In Chapter 1, we study the ruin probability and the Laplace transform of ruin time for renewal risk model in which the claim amount is Erlang and mixed Erlang random variables. In particular, we consider some solutions for the moments of the time to ruin by first solving for Φ.In Chapter 2, we consider a Sparre Anderson risk processwhere u > 0 is initial surplus;c is the positive constant premium income rate;{X_i,i ≥ 1}, {Yj,j ≥ 1} are i.i.d random variables with p.d.f fx{x),fy{x),respectively;K\{t) is Poisson process with parameter /x;K2(t) = max{k > 1 : Wx + ■ ■ ■ + Wk < t},where the i.i.d. claim waiting times W, have a common generalized Erlang(n) distribution;{B(t) : t > 0} is a standard Brownian motion that is independent of {Xiti > 1}, {Yj,j > 1}, Ki(t) and K2{t), o > 0. We get an Integro-differential equation of Gerber-Shiu function. In particular, when n = 2, we obtain the explicit expression for Gerber-Shiu function.