April  2022, 9(2): 203-217. doi: 10.3934/jdg.2022004

Robust policy selection and harvest risk quantification for natural resources management under model uncertainty

1. 

Department of Naval Sciences, Section of Mathematics, Mathematical Modeling and Applications Laboratory, Hellenic Naval Academy, Piraeus Greece

2. 

Stochastic Modeling and Applications Laboratory, Athens University of Economics and Business, Athens Greece

*Corresponding author: Georgios I. Papayiannis

Received  January 2022 Published  April 2022 Early access  February 2022

In this work the problem of optimal harvesting policy selection for natural resources management under model uncertainty is investigated. Under the framework of the neoclassical growth model dynamics, the associated optimal control problem is investigated by introducing the concept of model uncertainty on the initial conditions of the operational procedure. At this stage, the notion of convex risk measures, and in particular the class of Fréchet risk measures, is employed in order to quantify the total operational and marginal risk, whereas simultaneously obtaining robust to model uncertainty harvesting strategies.

Citation: Georgios I. Papayiannis. Robust policy selection and harvest risk quantification for natural resources management under model uncertainty. Journal of Dynamics and Games, 2022, 9 (2) : 203-217. doi: 10.3934/jdg.2022004
References:
[1] D. Acemoglu, Introduction to Modern Economic Growth, Princeton University Press, NY, 2009. 
[2]

P. C. Álvarez-EstebanE. Del BarrioJ. A. Cuesta-Albertos and C. Matrán, A fixed-point approach to barycenters in Wasserstein space, Journal of Mathematical Analysis and Applications, 441 (2016), 744-762.  doi: 10.1016/j.jmaa.2016.04.045.

[3]

I. BaltasA. Xepapadeas and A. N. Yannacopoulos, Robust portfolio decisions for financial institutions, Journal of Dynamics & Games, 5 (2018), 61-94.  doi: 10.3934/jdg.2018006.

[4]

I. BaltasA. Xepapadeas and A. N. Yannacopoulos, Robust control of parabolic stochastic partial differential equations under model uncertainty, European Journal of Control, 46 (2019), 1-13.  doi: 10.1016/j.ejcon.2018.04.004.

[5]

I. Baltas, L. Dopierala, K. Kolodziejczyk, M. Szczepański, G.-W. Weber and A. N. Yannacopoulos, Optimal management of defined contribution pension funds under the effect of inflation, mortality and uncertainty, European Journal of Operational Research, 298 (2022), 1162-1174. doi: 10.1016/j.ejor.2021.08.038.

[6]

T. R. BieleckiT. ChenI. CialencoA. Cousin and M. Jeanblanc, Adaptive robust control under model uncertainty, SIAM Journal on Control and Optimization, 57 (2019), 925-946.  doi: 10.1137/17M1137917.

[7]

R. BoucekkineC. Camacho and G. Fabbri, Spatial dynamics and convergence: The spatial AK model, Journal of Economic Theory, 148 (2013), 2719-2736.  doi: 10.1016/j.jet.2013.09.013.

[8]

W. A. BrockA. Xepapadeas and A. N. Yannacopoulos, Optimal control in space and time and the management of environmental resources, Annu. Rev. Resour. Econ., 6 (2014), 33-68.  doi: 10.1146/annurev-resource-100913-012411.

[9]

W. A. BrockA. Xepapadeas and A. N. Yannacopoulos, Robust control and hot spots in spatiotemporal economic systems, Dynamic Games and Applications, 4 (2014), 257-289.  doi: 10.1007/s13235-014-0109-z.

[10]

W. A. BrockA. Xepapadeas and A. N. Yannacopoulos, Optimal agglomerations in dynamic economics, Journal of Mathematical Economics, 53 (2014), 1-15.  doi: 10.1016/j.jmateco.2014.04.005.

[11]

C. W. Clark, Mathematical Bioeconomics: The Mathematics of Conservation, John Wiley & Sons, NJ, 2010.

[12]

G. Fabbri, S. Faggian and G. Freni, On competition for spatially distributed resources on networks, University Ca' Foscari of Venice, Dept. of Economics Research Paper Series No 7, (2020). doi: 10.2139/ssrn.3604337.

[13]

H. Föllmer and A. Schied, Convex measures of risk and trading constraints, Finance and Stochastics, 6 (2002), 429-447.  doi: 10.1007/s007800200072.

[14]

M. Fréchet, Les éléments aléatoires de nature quelconque dans un espace distancié, Annales de l'Institut de Henri Poincaré, 10 (1948), 215-310. 

[15]

M. Frittelli and E. R. Giannin, Putting order in risk measures, Journal of Banking & Finance, 26 (2002), 1473-1486.  doi: 10.1016/S0378-4266(02)00270-4.

[16]

L. Hansen and T. J. Sargent, Robust control and model uncertainty, American Economic Review, 91 (2001), 60-66.  doi: 10.1257/aer.91.2.60.

[17]

Z. LiZ. DuanL. Xie and X. Liu, Distributed robust control of linear multi-agent systems with parameter uncertainties, International Journal of Control, 85 (2012), 1039-1050.  doi: 10.1080/00207179.2012.674644.

[18]

F. MaccheroniM. Marinacci and A. Rustichini, Ambiguity aversion, robustness, and the variational representation of preferences, Econometrica, 74 (2006), 1447-1498.  doi: 10.1111/j.1468-0262.2006.00716.x.

[19]

G. I. Papayiannis and A. N. Yannacopoulos, A learning algorithm for source aggregation, Mathematical Methods in the Applied Sciences, 41 (2018), 1033-1039.  doi: 10.1002/mma.4086.

[20]

G. I. Papayiannis and A. N. Yannacopoulos, Convex risk measures for the aggregation of multiple information sources and applications in insurance, Scandinavian Actuarial Journal, 2018 (2018), 792-822.  doi: 10.1080/03461238.2018.1461129.

[21]

E. V. Petracou, A. Xepapadeas and A. N. Yannacopoulos, Decision Making Under Model Uncertainty: Fréchet–Wasserstein Mean Preferences, Management Science, (2021). doi: 10.1287/mnsc.2021.3961.

[22]

F. Santambrogio, Optimal Transport for Applied Mathematicians, Birkäuser, Cham, 2015. doi: 10.1007/978-3-319-20828-2.

[23]

R. M. Solow, Neoclassical growth theory, Handbook of macroeconomics, Vol. 1,637–667, Elsevier Science B.V., 1999. doi: 10.1016/S1574-0048(99)01012-5.

[24]

D. Tasche, Capital allocation to business units and sub-portfolios: The Euler principle, arXiv preprint, arXiv: 0708.2542, (2007).

[25]

C. Villani, Topics in Optimal Transportation, American Mathematical Soc., 2021. doi: 10.1090/gsm/058.

[26]

A. Xepapadeas and A. N. Yannacopoulos, Spatial growth with exogenous saving rates, Journal of Mathematical Economics, 67 (2016), 125-137.  doi: 10.1016/j.jmateco.2016.09.010.

show all references

References:
[1] D. Acemoglu, Introduction to Modern Economic Growth, Princeton University Press, NY, 2009. 
[2]

P. C. Álvarez-EstebanE. Del BarrioJ. A. Cuesta-Albertos and C. Matrán, A fixed-point approach to barycenters in Wasserstein space, Journal of Mathematical Analysis and Applications, 441 (2016), 744-762.  doi: 10.1016/j.jmaa.2016.04.045.

[3]

I. BaltasA. Xepapadeas and A. N. Yannacopoulos, Robust portfolio decisions for financial institutions, Journal of Dynamics & Games, 5 (2018), 61-94.  doi: 10.3934/jdg.2018006.

[4]

I. BaltasA. Xepapadeas and A. N. Yannacopoulos, Robust control of parabolic stochastic partial differential equations under model uncertainty, European Journal of Control, 46 (2019), 1-13.  doi: 10.1016/j.ejcon.2018.04.004.

[5]

I. Baltas, L. Dopierala, K. Kolodziejczyk, M. Szczepański, G.-W. Weber and A. N. Yannacopoulos, Optimal management of defined contribution pension funds under the effect of inflation, mortality and uncertainty, European Journal of Operational Research, 298 (2022), 1162-1174. doi: 10.1016/j.ejor.2021.08.038.

[6]

T. R. BieleckiT. ChenI. CialencoA. Cousin and M. Jeanblanc, Adaptive robust control under model uncertainty, SIAM Journal on Control and Optimization, 57 (2019), 925-946.  doi: 10.1137/17M1137917.

[7]

R. BoucekkineC. Camacho and G. Fabbri, Spatial dynamics and convergence: The spatial AK model, Journal of Economic Theory, 148 (2013), 2719-2736.  doi: 10.1016/j.jet.2013.09.013.

[8]

W. A. BrockA. Xepapadeas and A. N. Yannacopoulos, Optimal control in space and time and the management of environmental resources, Annu. Rev. Resour. Econ., 6 (2014), 33-68.  doi: 10.1146/annurev-resource-100913-012411.

[9]

W. A. BrockA. Xepapadeas and A. N. Yannacopoulos, Robust control and hot spots in spatiotemporal economic systems, Dynamic Games and Applications, 4 (2014), 257-289.  doi: 10.1007/s13235-014-0109-z.

[10]

W. A. BrockA. Xepapadeas and A. N. Yannacopoulos, Optimal agglomerations in dynamic economics, Journal of Mathematical Economics, 53 (2014), 1-15.  doi: 10.1016/j.jmateco.2014.04.005.

[11]

C. W. Clark, Mathematical Bioeconomics: The Mathematics of Conservation, John Wiley & Sons, NJ, 2010.

[12]

G. Fabbri, S. Faggian and G. Freni, On competition for spatially distributed resources on networks, University Ca' Foscari of Venice, Dept. of Economics Research Paper Series No 7, (2020). doi: 10.2139/ssrn.3604337.

[13]

H. Föllmer and A. Schied, Convex measures of risk and trading constraints, Finance and Stochastics, 6 (2002), 429-447.  doi: 10.1007/s007800200072.

[14]

M. Fréchet, Les éléments aléatoires de nature quelconque dans un espace distancié, Annales de l'Institut de Henri Poincaré, 10 (1948), 215-310. 

[15]

M. Frittelli and E. R. Giannin, Putting order in risk measures, Journal of Banking & Finance, 26 (2002), 1473-1486.  doi: 10.1016/S0378-4266(02)00270-4.

[16]

L. Hansen and T. J. Sargent, Robust control and model uncertainty, American Economic Review, 91 (2001), 60-66.  doi: 10.1257/aer.91.2.60.

[17]

Z. LiZ. DuanL. Xie and X. Liu, Distributed robust control of linear multi-agent systems with parameter uncertainties, International Journal of Control, 85 (2012), 1039-1050.  doi: 10.1080/00207179.2012.674644.

[18]

F. MaccheroniM. Marinacci and A. Rustichini, Ambiguity aversion, robustness, and the variational representation of preferences, Econometrica, 74 (2006), 1447-1498.  doi: 10.1111/j.1468-0262.2006.00716.x.

[19]

G. I. Papayiannis and A. N. Yannacopoulos, A learning algorithm for source aggregation, Mathematical Methods in the Applied Sciences, 41 (2018), 1033-1039.  doi: 10.1002/mma.4086.

[20]

G. I. Papayiannis and A. N. Yannacopoulos, Convex risk measures for the aggregation of multiple information sources and applications in insurance, Scandinavian Actuarial Journal, 2018 (2018), 792-822.  doi: 10.1080/03461238.2018.1461129.

[21]

E. V. Petracou, A. Xepapadeas and A. N. Yannacopoulos, Decision Making Under Model Uncertainty: Fréchet–Wasserstein Mean Preferences, Management Science, (2021). doi: 10.1287/mnsc.2021.3961.

[22]

F. Santambrogio, Optimal Transport for Applied Mathematicians, Birkäuser, Cham, 2015. doi: 10.1007/978-3-319-20828-2.

[23]

R. M. Solow, Neoclassical growth theory, Handbook of macroeconomics, Vol. 1,637–667, Elsevier Science B.V., 1999. doi: 10.1016/S1574-0048(99)01012-5.

[24]

D. Tasche, Capital allocation to business units and sub-portfolios: The Euler principle, arXiv preprint, arXiv: 0708.2542, (2007).

[25]

C. Villani, Topics in Optimal Transportation, American Mathematical Soc., 2021. doi: 10.1090/gsm/058.

[26]

A. Xepapadeas and A. N. Yannacopoulos, Spatial growth with exogenous saving rates, Journal of Mathematical Economics, 67 (2016), 125-137.  doi: 10.1016/j.jmateco.2016.09.010.

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