Notes on the variance of a widow's pension

1974 ◽  
Vol 101 (1) ◽  
pp. 103-107 ◽  
Author(s):  
P. P. Boyle
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Stochastic Approach ◽  
Higher Order ◽  
Expected Value ◽  
Confidence Limits ◽  
Higher Order Moments ◽  
Variable Time ◽  
Second Moments ◽  
Constant Rate

Pollard and Pollard have considered a stochastic approach to certain actuarial problems. In their notation ãx denotes the random variable whose expected value is ax. The random variable ãx depends on the random variable, time until death and the (assumed constant) rate of interest. The authors show how to calculate the second and higher-order moments of some common actuarial random variables such as ãx and Ãx. The second moments, in particular, are useful in estimating confidence limits for the total liability in respect of a group of contracts.

scholarly journals Uniform renewal theory with applications to expansions of random geometric sums

2007 ◽  
Vol 39 (4) ◽  
pp. 1070-1097 ◽  
Author(s):  
J. Blanchet ◽  
P. Glynn
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Renewal Theory ◽  
Random Variable ◽  
Higher Order ◽  
Order Correction ◽  
Diffusion Approximations ◽  
Geometric Sums ◽  
Correction Terms

Consider a sequence X = (Xn: n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable SM = X1 + ∙ ∙ ∙ + XM is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of SM as p ↘ 0. If EX1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pSM > x) ≈ exp(-x/EX1). Conversely, if EX1 = 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

scholarly journals Uniform renewal theory with applications to expansions of random geometric sums

2007 ◽  
Vol 39 (04) ◽  
pp. 1070-1097 ◽  
Author(s):  
J. Blanchet ◽  
P. Glynn
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Renewal Theory ◽  
Random Variable ◽  
Higher Order ◽  
Order Correction ◽  
Diffusion Approximations ◽  
Geometric Sums ◽  
Correction Terms

Consider a sequenceX= (Xn:n≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variableMwith parameterp. The random variableSM=X1+ ∙ ∙ ∙ +XMis called a geometric sum. In this paper we obtain asymptotic expansions for the distribution ofSMasp↘ 0. If EX1> 0, the asymptotic expansion is developed in powers ofpand it provides higher-order correction terms to Renyi's theorem, which states that P(pSM>x) ≈ exp(-x/EX1). Conversely, if EX1= 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

MAKING SIMULATIONS MORE EFFICIENT WHEN ANALYZING POISSON ARRIVAL SYSTEMS AND MEANS OF MONOTONE FUNCTIONS

2006 ◽  
Vol 20 (2) ◽  
pp. 251-256
Author(s):  
Sheldon M. Ross
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Random Variable ◽  
Expected Value ◽  
Stratified Sampling ◽  
Arrival Process ◽  
Monotone Functions ◽  
New Approach ◽  
Poisson Arrival ◽  
Increasing Function

For a system in which arrivals occur according to a Poisson process, we give a new approach for using simulation to estimate the expected value of a random variable that is independent of the arrival process after some specified time t. We also give a new approach for using simulation to estimate the expected value of an increasing function of independent uniform random variables. Stratified sampling is a key technique in both cases.

MIXING AND MOMENT PROPERTIES OF VARIOUS GARCH AND STOCHASTIC VOLATILITY MODELS

2002 ◽  
Vol 18 (1) ◽  
pp. 17-39 ◽  
Author(s):  
Marine Carrasco ◽  
Xiaohong Chen
Keyword(s):  
Stochastic Volatility ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Decay Rates ◽  
Stochastic Volatility Models ◽  
Higher Order Moments ◽  
Second Moments ◽  
Volatility Models ◽  
Strict Stationarity ◽  
Autoregressive Conditional Duration

This paper first provides some useful results on a generalized random coefficient autoregressive model and a generalized hidden Markov model. These results simultaneously imply strict stationarity, existence of higher order moments, geometric ergodicity, and β-mixing with exponential decay rates, which are important properties for statistical inference. As applications, we then provide easy-to-verify sufficient conditions to ensure β-mixing and finite higher order moments for various linear and nonlinear GARCH(1,1), linear and power GARCH(p,q), stochastic volatility, and autoregressive conditional duration models. For many of these models, our sufficient conditions for existence of second moments and exponential β-mixing are also necessary. For several GARCH(1,1) models, our sufficient conditions for existence of higher order moments again coincide with the necessary ones in He and Terasvirta (1999, Journal of Econometrics 92, 173–192).

An Application of Mean Square Calculus to Sliding Wear

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
Cláudio R. Ávila da Silva ◽  
Giuseppe Pintaude ◽  
Hazim Ali Al-Qureshi ◽  
Marcelo Alves Krajnc
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Sliding Wear ◽  
Random Variables ◽  
Random Variable ◽  
Expected Value ◽  
Mean Square ◽  
Cauchy Problems ◽  
Numerical Examples ◽  
Correlation Models

In this paper the Archard model and classical results of mean square calculus are used to derive two Cauchy problems in terms of the expected value and covariance of the worn height stochastic process. The uncertainty is present in the wear and roughness coefficients. In order to model the uncertainty, random variables or stochastic processes are used. In the latter case, the expected value and covariance of the worn height stochastic process are obtained for three combinations of correlation models for the wear and roughness coefficients. Numerical examples for both models are solved. For the model based on a random variable, a larger dispersion in terms of worn height stochastic process was observed.

scholarly journals Distributions in Life Insurance

1990 ◽  
Vol 20 (1) ◽  
pp. 81-92 ◽  
Author(s):  
Jan Dhaene
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Life Insurance ◽  
Wide Class ◽  
Stochastic Theory ◽  
Stochastic Approach ◽  
Higher Order ◽  
Loss Functions ◽  
Higher Order Moments

AbstractIn most textbooks and papers that deal with the stochastic theory of life contingencies, the stochastic approach is restricted to the computation of expectations and higher order moments. For a wide class of insurances on a single life, we derive the distribution and the probability density function of the benefit and the loss functions. Both the continuous and the discrete case are considered.

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