Article Type: Research Article Article Citation: Rajani P. Agadi, and
A. S. Talawar. (2020). STOCHASTIC DIFFERENTIAL
EQUATION: AN APPLICATION TO MORTALITY DATA. International Journal of Research
-GRANTHAALAYAH, 8(6), 229 – 235. https://doi.org/10.29121/granthaalayah.v8.i6.2020.538 Received Date: 08 May 2020 Accepted Date: 29 June 2020 Keywords: Stochastic Differential
Equation Mortality Estimation Confidence Interval and
Prediction In the present paper we consider an application of stochastic differential equation to model age-specific mortalities. We use New Zealand mortality data for the period 1948–2015 to fit the model. The point predictions of mortality rates at ages 40, 60 and 80 are quite good, almost undistinguishable from the true mortality rates observed.
1. INTRODUCTIONHuman mortality rates are very important for prediction purposes in social security systems, life insurance, public and private pension plans, etc. The mortality rates of all age and sex groups have a tendency to decline over time, but they clearly show random fluctuations. Many authors have contributed towards the modelling of mortality data through stochastic differential equations (Braumann, 1993; Braumann, 1999a; Braumann, 1999b; Kessler et al., 2012 and Aït-Sahalia, (2002, 2008)). These authors also have contributed towards estimation and other statistical inferential issues on stochastic differential equations. Braumann (1993) and Braumann (1999a) use Black–Scholes model for estimation, testing and prediction, including comparison tests among average return/growth rates of different stocks or populations. Ordinary differential equation (ODE) models have been used to study the growth of individual living beings, like farm animals, trees or fish. When the growth occurs in environments with random variations, stochastic differential equation (SDE) models have been proposed. Braumann (1999b) shows that qualitative results very similar to those obtained for specific models also hold for the general stochastic population growth model. Lagarto and Braumann (2014) used SDE model (stochastic Malthusian) for mortality data of Portugal and considered precisely the joint evolution of the crude death rates of two age-sex groups. Lagarto (2014) has considered a study of alternative structures and has determined, among those, which ones have a good performance. These can then be used in predictions and in applications. 2.
STOCHASTIC MORTALITY MODEL
The random fluctuations are mostly explained by environmental stochasticity and
use a stochastic differential equation (SDE) model (Black–Scholes or stochastic
Malthusian model) to represent Consider a Stochastic differential equation (SDE)
Where By
Or in differential form
It is the Various models describe the pattern of human mortality, the one such model that we consider is a stochastic Gompertz model also called Malthusian growth model.
Where The solution of equation (3) is obtained by
Where We conclude that 1) If
2) If
3) If
To fit model (4), we made it age-specific by writing
Where We apply the methodology used in (Lagarto
and Braumann, 2014; Braumann,
2019; Talawar and Agadi,
2020) for parameters estimation. The value of
Therefore the 95% confidence intervals of the parameters 3. APPLICATIONS OF THE MODEL For the application of model we considered the mortality data of New Zealand from
1948-2015. In Figure 1 it shows that age-specific death rates of 60-year-old
females and males of the New Zealand population for each year of the period
1948–2015 (Source: https://www.mortality.org/).
For convenience, we start counting time in 1948, so initial time, Figures 2-4, show
the results of using model (6) for the age-specific mortality rates of 60 year-old New Zealand males and females. Notice the
decline from Therefore the estimates of the parameters are
The 95% confidence intervals of the parameters are The parameter value and an approximate 95% confidence prediction interval for From which extremes, by taking exponentials, one gets an
approximate confidence prediction interval for Figure 1: Forecasted mortality rate of New Zealand male and female Figure 2: Forecasted mortality rate of New Zealand female Figure 3: Forecasted mortality rate of New Zealand male Figure 4: Forecasted mortality rate of New Zealand male-female difference Figure 5: Forecasted mortality rate of New Zealand Infant Figure 6: Forecasted mortality rate of New Zealand at age 40 Figure 7: Forecasted mortality rate of New Zealand at age 80 Table 1: The parameter
values for males and females of New Zealand mortality data.
4. CONCLUSIONSThis stochastic mortality model gives good prediction
intervals and the point prediction for each age. The point predictions of
mortality rates at ages 40, 60 and 80 are quite good, almost undistinguishable
from the true mortality rates observed. The similar procedure can be used to
estimate parameters at each age SOURCES OF FUNDINGNone. CONFLICT OF INTERESTNone. ACKNOWLEDGMENTNone. REFERENCES[1] Aït-Sahalia, Y. (2002). Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes: Comment. Journal of Business and Economic Statistics. 20(3):317-21 DOI: 10.1198/073500102288618405 [2] Aït-Sahalia, Y. (2008). Closed-form likelihood expansions for multivariate diffusions. Ann. Statist. 36 (2008), no. 2, 906--937. doi:10.1214/009053607000000622. https://projecteuclid.org/euclid.aos/ [3] 1205420523 [4] Braumann, C.A. (1993) Model fitting and prediction in stochastic population growth models in random environments. Bulletin of the International Statistical Institute, LV (CP1), 163–164. Braumann, C.A. (1999a) Comparison of geometric Brownian motions and applications to population growth and finance. Bulletin of the International Statistical Institute, LVIII (CP1), 125–126. [5] Braumann, C. A., Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance, John Wiley & Sons Ltd, 2019. [6] Itô, K. (1951) On Stochastic Differential Equations, American Mathematical Society Memoirs, No. 4. [7] Kessler,M., Lindner A. and, Sorensen, M. Statistical Methods for Stochastic Differential Equations, Chapman and Hall/CRC, 2012. [8] Lagarto, S. and Braumann, C.A. Modeling human population death rates: A bi-dimensional stochastic Gompertz model with correlated Wiener processes, in New Advances in Statistical Modeling and Applications (eds, A. Pacheo, R. Santos, M.R. Oliveira, and C.D. Paulino), Springer, Heidelberg, pp. 95–103, doi:10.1007/978-3-319-05323-3_9, 2014. [9] Øksendal, B. Stochastic Differential Equations. An Introduction with Applications, 6th edition, Springer, New York, 2003. [10] Talawar, A. S. and Agadi, R.P. (2020). Novel corona virus pandemic disease (covid-19): Some applications to south asian countries. International Journal of Research and Analytical Reviews, 7(2), 905-912.
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