论文标题
随机利率的相型表示与人寿保险的申请
Phase-type representations of stochastic interest rates with applications to life insurance
论文作者
论文摘要
本文的目的是将随机利率纳入多国家人寿保险的矩阵诉讼中，在该保险中，可以根据产品积分或矩阵指数的指数来获得储量的公式，未来付款的时刻和等价保费。为此，我们考虑了马尔可夫兴趣模型，其中速率是马尔可夫跳跃过程不同状态中的分段确定性（甚至是恒定），并且显示出来自然集成到矩阵框架中。然后，折现因子成为零息债券的价格，该债券可能与生物识别保险过程相关，也可能不会相关。马尔可夫兴趣模型的另一个不错的功能是，债券的价格与相型分布式随机变量的生存函数一致。特别是，这允许使用观察到的数据（价格）或诸如例如VASICEK型号。由于相型分布的密度，我们可以通过选择足够大的可能利率值的数量来近似任何零息债券的价格行为，并从下面限制。对于几乎没有数据点的观察到的数据模型，较低的维度通常就足够了，而对于理论模型，维度只是一个计算问题。
The purpose of the present paper is to incorporate stochastic interest rates into a matrix-approach to multi-state life insurance, where formulas for reserves, moments of future payments and equivalence premiums can be obtained as explicit formulas in terms of product integrals or matrix exponentials. To this end we consider the Markovian interest model, where the rates are piecewise deterministic (or even constant) in the different states of a Markov jump process, and which is shown to integrate naturally into the matrix framework. The discounting factor then becomes the price of a zero-coupon bond which may or may not be correlated with the biometric insurance process. Another nice feature about the Markovian interest model is that the price of the bond coincides with the survival function of a phase-type distributed random variable. This, in particular, allows for calibrating the Markovian interest rate models using a maximum likelihood approach to observed data (prices) or to theoretical models like e.g. a Vasicek model. Due to the denseness of phase-type distributions, we can approximate the price behaviour of any zero-coupon bond with interest rates bounded from below by choosing the number of possible interest rate values sufficiently large. For observed data models with few data points, lower dimensions will usually suffice, while for theoretical models the dimensionality is only a computational issue.
