Generalizing earlier works of Delbaen & Haezendonck [5] as well as of [18] and [16] for given compound mixed renewal process S under a probability measure P, we characterize all those probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound mixed renewal process under Q with improved properties. As a consequence, we prove that any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures. Applications related to the ruin problem and to the computation of premium calculation principles in an insurance market without arbitrage opportunities are discussed in [26] and [27], respectively.
Large deviations principle is obtained for terminating multidimensional compound renewal processes. We also obtained the asymptotic of large deviations for the case when a Gibbs change of the original probability measure takes place. The random processes mentioned in the paper are widely used in polymer pinning models.
Dealing with compound renewal process with generally distributed jump sizes and inter-renewal intervals, we focus on the approximation for the fixed-probability level, which is the core of inverse level crossing problem. We are developing an analytical technique presented in [15]-[17] and based on Kendall's identity; this yields (see [18]) inverse Gaussian approximation in the direct level crossing problem. These issues are of great importance in risk theory.
In the present paper the change of measures technique for compound mixed renewal processes, developed in Tzaninis & Macheras [24], is applied to the ruin problem in order to compute the ruin probability and to find upper and lower bounds for it.
Local theorems for arithmetic compound renewal processes, when Cramer's condition holds
我们继续研究复合更新过程（c. r. p.），其中Cramer条件的时刻成立（参见[1]-[10]，c. r. p.的研究开始的地方）。本文研究了算术c. r. p. Z（n）。在这样的过程中，随机向量=（τ，τ）具有算术分布，其中τ > 0定义跳跃之间的距离，τ定义跳跃的值。对于这一过程，在x $\in$ Z的Cramer偏差区域内，得到了概率P（Z（n）= x）的局部极限定理的精细渐近性.在[6]-[10]中，对于非格点c. r. p.，已经解决了类似的问题，当向量τ =（τ，τ）具有非格点分布时。
In this manuscript we consider the dual risk model with financial application, where the random gains occur under a renewal process. We particularly work the Erlang(n) case for common distribution of the inter-arrival times, from there it is easy to understand that our method or procedures can be generalised to other cases under the matrix-exponential family case. We work several and different problems involving future dividends and ruin. We also show that our results are valid even if the usual income condition is not satisfied. In most known works under the dual model, the main target under …
