American Institute of Mathematical Sciences

Advanced
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.  doi: 10.1515/156939605777438569.  Google Scholar

[2]

A. Alfonsi, Strong convergence of some drift implicit Euler scheme,, application to the CIR process, (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.  doi: 10.1016/j.mbs.2005.03.010.  Google Scholar

[4]

E. J. Allen, Modeling With Itô Stochastic Differential Equations,, Springer, (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.  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, (2011).   Google Scholar

[7]

F. Black and P. Karasinski, Bond and option pricing when short rates are lognormal,, Financial Analysts Journal, 47 (1991), 52.  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).   Google Scholar

[9]

T. C. Gard, Introduction to Stochastic Differential Equations,, Marcel Decker, (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.  doi: 10.1137/10081856X.  Google Scholar

[11]

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

[17]

A. G. Ladde and G. S. Ladde, An Introduction to Differential Equations: Stochastic Modeling, Methods and Analysis,, Volume 2, (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.  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.  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.  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.  doi: 10.1006/tpbi.2000.1450.  Google Scholar

[22]

M. Montero, Predator-Prey Model for Stock Market Fluctuations,, , (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).   Google Scholar

[24]

S. Solomon, Generalized lotka-volterra (GLV) models of stock markets,, Advances in Complex Systems, 3 (2000), 301.  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.   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.  doi: 10.1515/156939605777438569.  Google Scholar

[2]

A. Alfonsi, Strong convergence of some drift implicit Euler scheme,, application to the CIR process, (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.  doi: 10.1016/j.mbs.2005.03.010.  Google Scholar

[4]

E. J. Allen, Modeling With Itô Stochastic Differential Equations,, Springer, (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.  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, (2011).   Google Scholar

[7]

F. Black and P. Karasinski, Bond and option pricing when short rates are lognormal,, Financial Analysts Journal, 47 (1991), 52.  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).   Google Scholar

[9]

T. C. Gard, Introduction to Stochastic Differential Equations,, Marcel Decker, (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.  doi: 10.1137/10081856X.  Google Scholar

[11]

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

[17]

A. G. Ladde and G. S. Ladde, An Introduction to Differential Equations: Stochastic Modeling, Methods and Analysis,, Volume 2, (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.  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.  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.  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.  doi: 10.1006/tpbi.2000.1450.  Google Scholar

[22]

M. Montero, Predator-Prey Model for Stock Market Fluctuations,, , (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).   Google Scholar

[24]

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

[1]

Hoi Tin Kong, Qing Zhang. An optimal trading rule of a mean-reverting asset. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1403-1417. doi: 10.3934/dcdsb.2010.14.1403
[2]

Yan Wang, Lei Wang, Yanxiang Zhao, Aimin Song, Yanping Ma. A stochastic model for microbial fermentation process under Gaussian white noise environment. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 381-392. doi: 10.3934/naco.2015.5.381
[3]

Yanqing Hu, Zaiming Liu, Jinbiao Wu. Optimal impulse control of a mean-reverting inventory with quadratic costs. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1685-1700. doi: 10.3934/jimo.2018027
[4]

Qihong Chen. Recovery of local volatility for financial assets with mean-reverting price processes. Mathematical Control & Related Fields, 2018, 8 (3&4) : 625-635. doi: 10.3934/mcrf.2018026
[5]

Weiwei Wang, Ping Chen. A mean-reverting currency model with floating interest rates in uncertain environment. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1921-1936. doi: 10.3934/jimo.2018129
[6]

Xinfu Chen, Carey Caginalp, Jianghao Hao, Yajing Zhang. Effects of white noise in multistable dynamics. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1805-1825. doi: 10.3934/dcdsb.2013.18.1805
[7]

Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103
[8]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887
[9]

Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain, Glenn W.A. Rowe. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology. Mathematical Biosciences & Engineering, 2006, 3 (4) : 571-582. doi: 10.3934/mbe.2006.3.571
[10]

Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210
[11]

Boris P. Belinskiy, Peter Caithamer. Stochastic stability of some mechanical systems with a multiplicative white noise. Conference Publications, 2003, 2003 (Special) : 91-99. doi: 10.3934/proc.2003.2003.91
[12]

Luis J. Roman, Marcus Sarkis. Stochastic Galerkin method for elliptic spdes: A white noise approach. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 941-955. doi: 10.3934/dcdsb.2006.6.941
[13]

Qi Wang, Lifang Huang, Kunwen Wen, Jianshe Yu. The mean and noise of stochastic gene transcription with cell division. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1255-1270. doi: 10.3934/mbe.2018058
[14]

Deena Schmidt, Janet Best, Mark S. Blumberg. Random graph and stochastic process contributions to network dynamics. Conference Publications, 2011, 2011 (Special) : 1279-1288. doi: 10.3934/proc.2011.2011.1279
[15]

Eugene Kashdan, Dominique Duncan, Andrew Parnell, Heinz Schättler. Mathematical methods in systems biology. Mathematical Biosciences & Engineering, 2016, 13 (6) : i-ii. doi: 10.3934/mbe.201606i
[16]

Avner Friedman. Conservation laws in mathematical biology. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081
[17]

Yanzhao Cao, Li Yin. Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Communications on Pure & Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607
[18]

Tianlong Shen, Jianhua Huang, Caibin Zeng. Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1523-1533. doi: 10.3934/dcdsb.2018056
[19]

Ying Hu, Shanjian Tang. Nonlinear backward stochastic evolutionary equations driven by a space-time white noise. Mathematical Control & Related Fields, 2018, 8 (3&4) : 739-751. doi: 10.3934/mcrf.2018032
[20]

Shujuan Lü, Hong Lu, Zhaosheng Feng. Stochastic dynamics of 2D fractional Ginzburg-Landau equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 575-590. doi: 10.3934/dcdsb.2016.21.575

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences