Annuities Under Random Interest Rate

Annuity is one of the most important concepts in economic,finance and insurance, the study of annuities with random interests has been received a lot of attention by many authors in applied probability and statistic fields in resent years,and a series of profound research results are provided.Because of the complexity and importance of the random interests,the research about this problem is still be in the ascendant.In this paper,we try to investigate the numerical characteristics of present value of annuities with random interests.Based on the idea and technical in the references[16],[17]and[26]as well as the theories of martingale,Brownian motion and stochastical orders,the expectation of present value of annuities with controlled random interests is obtained,and the upper bounds of present value function of discrete stochastic annuities with random interests in sense of convex order are derived,and the numerical characteristics of upper bounds are also obtained,furthermore,the expectation of present value functions of discrete stochastic annuities under random interests is provided.The main results of this paper can be summarized as following:1.Under the assumption that the random interests vary as reflected Brownian motion, a mistake of Perry and Stadje is corrected,and the expectation of present value of continuous annuities is obtained,meanwhile,the expectation of present value of continuous annuities with controlled random interests is derived under the condition that the active life of annuity is random variable and the ruling rate of interest is immediately adjusted when the interests reach the lowest interests level a,the middle interests level b or the highest interests level c.2.For m independent random vectors(?)=(X_1,1,X_1,2,…,X_1,n),(?)= (X_2,1,X_2,2,…,X_2,n),…(?)=(X_m,1,X_m,2,…,X_m,n),the upper and lower bounds for the random variable S^(m)=∑_i=1ⁿ X_1,iX_2,i…X_m,iin sense of convex order are obtained,and the distribution function and stop-loss premium of the upper and lower bounds are derived for the case m=3.At the last,the discrete stochastic annuities with random interests is investigated and the upper bounds in sense of convex order are pro-vided for the present value function of discrete stochastic annuities with random interests, and the distribution function and stop-loss premium for the upper bounds are derived, furthermore,the expectation of present value functions of discrete stochastic annuities is obtained.