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

Environmental variability and mean-reverting processes

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.
Citation: Edward Allen. Environmental variability and mean-reverting processes. Discrete and 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.

[2]

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

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[24]

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

[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.

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.

[2]

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

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[24]

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

[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.

[1]

Yin Li, Xuerong Mao, Yazhi Song, Jian Tao. Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial and Management Optimization, 2022, 18 (1) : 75-93. doi: 10.3934/jimo.2020143

[2]

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

[3]

Wan-Hua He, Chufang Wu, Jia-Wen Gu, Wai-Ki Ching, Chi-Wing Wong. Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2077-2094. doi: 10.3934/jimo.2021057

[4]

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 and Optimization, 2015, 5 (4) : 381-392. doi: 10.3934/naco.2015.5.381

[5]

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

[6]

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

[7]

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

[8]

Pengxu Xie, Lihua Bai, Huayue Zhang. Optimal pairs trading of mean-reverting processes over multiple assets. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022014

[9]

Guanggan Chen, Qin Li, Yunyun Wei. Approximate dynamics of a class of stochastic wave equations with white noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 73-101. doi: 10.3934/dcdsb.2021033

[10]

Shang Wu, Pengfei Xu, Jianhua Huang, Wei Yan. Ergodicity of stochastic damped Ostrovsky equation driven by white noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1615-1626. doi: 10.3934/dcdsb.2020175

[11]

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

[12]

Miao Tian, Xiangfeng Yang, Yi Zhang. Lookback option pricing problem of mean-reverting stock model in uncertain environment. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2703-2714. doi: 10.3934/jimo.2020090

[13]

Boris Anicet Guimfack, Conrad Bertrand Tabi, Alidou Mohamadou, Timoléon Crépin Kofané. Stochastic dynamics of the FitzHugh-Nagumo neuron model through a modified Van der Pol equation with fractional-order term and Gaussian white noise excitation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2229-2243. doi: 10.3934/dcdss.2020397

[14]

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

[15]

Leonid Shaikhet. Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3651-3657. doi: 10.3934/dcdsb.2020077

[16]

Xiang Lv. Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2313-2323. doi: 10.3934/dcdsb.2021133

[17]

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

[18]

Xingni Tan, Fuqi Yin, Guihong Fan. Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3153-3170. doi: 10.3934/dcdsb.2020055

[19]

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

[20]

Ling Xu, Jianhua Huang, Qiaozhen Ma. Random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021318

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (468)
  • HTML views (0)
  • Cited by (20)

Other articles
by authors

[Back to Top]