The Stochastic Control Model for Use Conversion of Land

Abstract

In this paper, the authors investigate the optimal conversion rate at which land use is irreversibly converted from biodiversity conservation to agricultural production. This problem is formulated as a stochastic control model, then transformed into a HJB equation involving free boundary. Since the state equation has singularity, it is difficult to directly derive the boundary value condition for the HJB equation. They provide a new method to overcome the difficulty via constructing another auxiliary stochastic control problem, and impose a proper boundary value condition. Moreover, they establish the existence and uniqueness of the viscosity solution of the HJB equation. Finally, they propose a stable numerical method for the HJB equation involving free boundary, and show some numerical results.

Keywords

Optimal stochastic control / HJB equation / Free boundary / Land use

References

[1] Arrow K J, Fisher A C. Environmental preservation, uncertainty and irreversibility. Quarterly Journal of Economics, 1974, 88: 312-319

[2] Bulte E, Scoest D P, Kooten G C, Schipper R A. Forest conservation in Costa Rica when nonuse benefits are uncertain but rising. American Journal of Agriculturial Economics, 2002, 84: 150-160

[3] Clarke H R, Reed W J. Land development and wilderness conservation policies under uncertainty: A synthesis. Natural Resource Modelliing, 1990, 4: 11-37

[4] Dai M, Zhong Y. Penalty methods for continuous-time portfolio selection with proportional transaction costs. Journal of Computational Finance, 2010, 13: 1-31

[5] Dong G, Bian B. Uniqueness of viscosity solutions of fully nonlinear second order parabolic PDE’s. Chinese Annals of Mathematics Series B, 1990, 11(2): 156-170

[6] Fichera G. Sulle equazioni differenziali lineari eccittico-parabliche del secondo ordine. Atti. Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez., 1956, 5: 1-30

[7] Fisher A C, Krutilla J, Cicchetti C J. The economics of environmental preservation: A theoretical and empricial analysis. American Economic Review, 1972, 62: 605-619

[8] Fleming W H, Soner H M. Controlled Markov Processes and Viscosity Solutions, 2006, New York: Springer-Verlag

[9] Kushner H J, Dupuis P. Numerical Methods for Stochastic Control Problems in Continuous Time, 2001, New York: Springer-Verlag

[10] Leroux A D, Martin V L, Goeschl T. Optimal conservation, extinction debt, and the augmented quasi-option value. Journal of Environmental Economics and Management, 2009, 58: 43-57

[11] Yan H, Yi F, Yang Z, Liang G. Dynkin game of convertible bonds and their optimal strategy. Journal of Mathematical Analysis and Applications, 2015, 426: 64-88

[12] Yi F, Yang Z, Wang X. A variational inequality arising from European installment call options pricing. SIAM on Mathematical Analysis, 2008, 40: 306-326

[13] Yong J, Zhou X. Stochastic Controls Hamiltonian Systems and HJB Equations, 1999, New York: Springer-Verlag