马尔可夫随机利率模型下的年金精算

Actuarial Calculation of Annuities under Markov Stochastic Interest Rate Model

年金的精算与利率模型密切相关.标准型年金中,每期的利率是一个固定的常数值.在实际中,每期的利率可以是变化的,甚至是一个随机变量.这些随机变量组成一个利率随机过程.许多情形下,利率随机过程是马尔可夫过程.本文在马尔可夫随机利率模型下研究年金的精算.证明了利率过程是时齐马尔可夫链时其折现过程也是时齐马尔可夫链,且与利率马尔可夫链过程具有\相同的"初始分布和一步转移概率矩阵.借助利率折现过程,计算了马尔可夫随机利率模型下年金现值的期望和方差.引进了年金多项式和一些算子以及年金的算子多项式,使得年金现值的期望和方差表达式非常简洁,且易于编程计算.

The actuarial calculation of annuities is closely related to the interest rate model.In standard annuities,the interest rate for each period is a xed constant.In practice,the interest rate for each period can be a variable or even a random variable.These random variables constitute a stochastic process of interest rate.In many cases,the stochastic process of interest rate is a Markov process.This article studies the actuarial calculation of annuities under the Markov stochastic interest rate model.It is proved that if the interest rate process is a time-homogeneous Markov chain,then the discounting process is also a time-homogeneous Markov chain,and they have the`same'initial distribution and the`same'one-step transition probability matrix.With the help of the interest rate discounting process,the expectation and variance of the present value of annuities under the Markov stochastic interest rate model are calculated.This article introduced annuity polynomials,operators,and annuity operator polynomials.It makes the expressions of expectation and variance for annuities very concise,and easy to program and calculate.