American Institute of Mathematical Sciences

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September  2016, 21(7): 2073-2089. doi: 10.3934/dcdsb.2016037

Environmental variability and mean-reverting processes

Edward Allen 1,

1. 

Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409-1042, United States

Received  February 2015 Revised  February 2016 Published  August 2016

Environmental variability is often incorporated in a mathematical model by modifying the parameters in the model. In the present investigation, two common methods to incorporate the effects of environmental variability in stochastic differential equation models are studied. The first approach hypothesizes that the parameter satisfies a mean-reverting stochastic process. The second approach hypothesizes that the parameter is a linear function of Gaussian white noise. The two approaches are discussed and compared analytically and computationally. Properties of several mean-reverting processes are compared with respect to nonnegativity and their asymptotic stationary behavior. The effects of different environmental variability assumptions on population size and persistence time for simple population models are studied and compared. Furthermore, environmental data are examined for a gold mining stock. It is concluded that mean-reverting processes possess several advantages over linear functions of white noise in modifying parameters for environmental variability.
Keywords: Environmental variability, population dynamics, mean-reverting process, white noise, stochastic differential equation, mathematical biology..
Mathematics Subject Classification: Primary: 60H10, 92D25; Secondary: 34F05, 92B0.
Citation: Edward Allen. Environmental variability and mean-reverting processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2073-2089. doi: 10.3934/dcdsb.2016037
References:
[1]

A. Alfonsi, On the discretization schemes for the CIR (and Bessel squared) processes, Monte Carlo Methods and Applications, 11 (2005), 355-384. doi: 10.1515/156939605777438569.  Google Scholar

[2]

A. Alfonsi, Strong convergence of some drift implicit Euler scheme, application to the CIR process, arXiv:1206.3855, (2012). Google Scholar

[3]

E. J. Allen, L. J. S. Allen and H. Schurz, A comparison of persistence-time estimation for discrete and continuous stochastic population models that include demographic and environmental variability, Mathematical Biosciences, 196 (2005), 14-38. doi: 10.1016/j.mbs.2005.03.010.  Google Scholar

[4]

E. J. Allen, Modeling With Itô Stochastic Differential Equations, Springer, Dordrecht, The Netherlands, 2007.  Google Scholar

[5]

L. J. S. Allen and E. J. Allen, A comparison of three different stochastic population models with regard to persistence time, Theoretical Population Biology, 64 (2003), 439-449. doi: 10.1016/S0040-5809(03)00104-7.  Google Scholar

[6]

L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Second edition. CRC Press, Boca Raton, FL, 2011.  Google Scholar

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F. Black and P. Karasinski, Bond and option pricing when short rates are lognormal, Financial Analysts Journal, 47 (1991), 52-59. doi: 10.2469/faj.v47.n4.52.  Google Scholar

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Y. Cai, X. Wang, W. Wang and M. Zhao, Stochastic dynamics of an SIRS epidemic model with ratio-dependent incidence rate, Abstract and Applied Analysis, 2013 (2013), Article ID 172631, 11 pages.  Google Scholar

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T. C. Gard, Introduction to Stochastic Differential Equations, Marcel Decker, New York, 1988.  Google Scholar

[10]

A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902. doi: 10.1137/10081856X.  Google Scholar

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P. Hänggi and P. Jung, Colored noise in dynamical systems, Advances in Chemical Physics, 89 (1995), 239-326. Google Scholar

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A. S. Hurn, K. A. Lindsay and V. L. Martin, On the efficacy of simulated maximum likelihood for estimating the parameters of stochastic differential equations, Journal of Time Series Analysis, 24 (2003), 45-63. doi: 10.1111/1467-9892.00292.  Google Scholar

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K. Kladívko, Maximum likelihood estimation of the Cox-Ingersoll-Ross process: The MATLAB implementation, Technical Computing Prague, 2007. Google Scholar

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P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, New York, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[15]

P. E. Kloeden, E. Platen and H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer, Berlin, 1994. doi: 10.1007/978-3-642-57913-4.  Google Scholar

[16]

C. Kou and S. G. Kou, Modeling growth stocks via birth-death processes, Advances in Applied Probability, 35 (2003), 641-664. Google Scholar

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A. G. Ladde and G. S. Ladde, An Introduction to Differential Equations: Stochastic Modeling, Methods and Analysis, Volume 2, World Scientific Publishing Company, Singapore, 2013. doi: 10.1142/8384.  Google Scholar

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Y. Lin and D. Jiang, Long-time behavior of a perturbed SIR model by white noise, Discrete and Continuous Dynamical Systems Series B, 18 (2013), 1873-1887. doi: 10.3934/dcdsb.2013.18.1873.  Google Scholar

[19]

Q. Lv, M. K. Schneider and J. W. Pitchford, Individualism in plant populations: Using stochastic differential equations to model individual neighbourhood-dependent plant growth, Theoretical Population Biology, 74 (2008), 74-83. doi: 10.1016/j.tpb.2008.05.003.  Google Scholar

[20]

X. Mao, C. Yuan and J. Zou, Stochastic differential delay equations of population dynamics, Journal of Mathematical Analysis and Applications, 304 (2005), 296-320. doi: 10.1016/j.jmaa.2004.09.027.  Google Scholar

[21]

G. Marion and E Renshaw, Stochastic modelling of environmental variation for biological populations, Theoretical Population Biology, 57 (2000), 197-217. doi: 10.1006/tpbi.2000.1450.  Google Scholar

[22]

M. Montero, Predator-Prey Model for Stock Market Fluctuations, arXiv:0810.4844, (2008). doi: 10.2139/ssrn.1290728.  Google Scholar

[23]

F. Rao, Dynamical analysis of a stochastic predator-prey model with an alee effect, Abstract and Applied Analysis, 2013 (2013), Article ID 340980, 10 pages.  Google Scholar

[24]

S. Solomon, Generalized lotka-volterra (GLV) models of stock markets, Advances in Complex Systems, 3 (2000), 301-322. doi: 10.1142/S0219525900000224.  Google Scholar

[25]

T. V. Ton and A. Yagi, Dynamics of a stochastic predator-prey model with the Beddington-DeAngelis functional response, Communications on Stochastic Analysis, 5 (2011), 371-386.  Google Scholar

show all references

References:
[1]

A. Alfonsi, On the discretization schemes for the CIR (and Bessel squared) processes, Monte Carlo Methods and Applications, 11 (2005), 355-384. doi: 10.1515/156939605777438569.  Google Scholar

[2]

A. Alfonsi, Strong convergence of some drift implicit Euler scheme, application to the CIR process, arXiv:1206.3855, (2012). Google Scholar

[3]

E. J. Allen, L. J. S. Allen and H. Schurz, A comparison of persistence-time estimation for discrete and continuous stochastic population models that include demographic and environmental variability, Mathematical Biosciences, 196 (2005), 14-38. doi: 10.1016/j.mbs.2005.03.010.  Google Scholar

[4]

E. J. Allen, Modeling With Itô Stochastic Differential Equations, Springer, Dordrecht, The Netherlands, 2007.  Google Scholar

[5]

L. J. S. Allen and E. J. Allen, A comparison of three different stochastic population models with regard to persistence time, Theoretical Population Biology, 64 (2003), 439-449. doi: 10.1016/S0040-5809(03)00104-7.  Google Scholar

[6]

L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Second edition. CRC Press, Boca Raton, FL, 2011.  Google Scholar

[7]

F. Black and P. Karasinski, Bond and option pricing when short rates are lognormal, Financial Analysts Journal, 47 (1991), 52-59. doi: 10.2469/faj.v47.n4.52.  Google Scholar

[8]

Y. Cai, X. Wang, W. Wang and M. Zhao, Stochastic dynamics of an SIRS epidemic model with ratio-dependent incidence rate, Abstract and Applied Analysis, 2013 (2013), Article ID 172631, 11 pages.  Google Scholar

[9]

T. C. Gard, Introduction to Stochastic Differential Equations, Marcel Decker, New York, 1988.  Google Scholar

[10]

A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902. doi: 10.1137/10081856X.  Google Scholar

[11]

P. Hänggi and P. Jung, Colored noise in dynamical systems, Advances in Chemical Physics, 89 (1995), 239-326. Google Scholar

[12]

A. S. Hurn, K. A. Lindsay and V. L. Martin, On the efficacy of simulated maximum likelihood for estimating the parameters of stochastic differential equations, Journal of Time Series Analysis, 24 (2003), 45-63. doi: 10.1111/1467-9892.00292.  Google Scholar

[13]

K. Kladívko, Maximum likelihood estimation of the Cox-Ingersoll-Ross process: The MATLAB implementation, Technical Computing Prague, 2007. Google Scholar

[14]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, New York, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[15]

P. E. Kloeden, E. Platen and H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer, Berlin, 1994. doi: 10.1007/978-3-642-57913-4.  Google Scholar

[16]

C. Kou and S. G. Kou, Modeling growth stocks via birth-death processes, Advances in Applied Probability, 35 (2003), 641-664. Google Scholar

[17]

A. G. Ladde and G. S. Ladde, An Introduction to Differential Equations: Stochastic Modeling, Methods and Analysis, Volume 2, World Scientific Publishing Company, Singapore, 2013. doi: 10.1142/8384.  Google Scholar

[18]

Y. Lin and D. Jiang, Long-time behavior of a perturbed SIR model by white noise, Discrete and Continuous Dynamical Systems Series B, 18 (2013), 1873-1887. doi: 10.3934/dcdsb.2013.18.1873.  Google Scholar

[19]

Q. Lv, M. K. Schneider and J. W. Pitchford, Individualism in plant populations: Using stochastic differential equations to model individual neighbourhood-dependent plant growth, Theoretical Population Biology, 74 (2008), 74-83. doi: 10.1016/j.tpb.2008.05.003.  Google Scholar

[20]

X. Mao, C. Yuan and J. Zou, Stochastic differential delay equations of population dynamics, Journal of Mathematical Analysis and Applications, 304 (2005), 296-320. doi: 10.1016/j.jmaa.2004.09.027.  Google Scholar

[21]

G. Marion and E Renshaw, Stochastic modelling of environmental variation for biological populations, Theoretical Population Biology, 57 (2000), 197-217. doi: 10.1006/tpbi.2000.1450.  Google Scholar

[22]

M. Montero, Predator-Prey Model for Stock Market Fluctuations, arXiv:0810.4844, (2008). doi: 10.2139/ssrn.1290728.  Google Scholar

[23]

F. Rao, Dynamical analysis of a stochastic predator-prey model with an alee effect, Abstract and Applied Analysis, 2013 (2013), Article ID 340980, 10 pages.  Google Scholar

[24]

S. Solomon, Generalized lotka-volterra (GLV) models of stock markets, Advances in Complex Systems, 3 (2000), 301-322. doi: 10.1142/S0219525900000224.  Google Scholar

[25]

T. V. Ton and A. Yagi, Dynamics of a stochastic predator-prey model with the Beddington-DeAngelis functional response, Communications on Stochastic Analysis, 5 (2011), 371-386.  Google Scholar

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