OPTIMAL EXPLOITATION OF RENEWABLE RESOURCES BY THE VISCOSITY SOLUTION METHOD

We study the stochastic optimization problem of renewable resources to maximize the expected discounted utility of exploitation. We develop the viscosity solution method to the associated Hamilton–Jacobi–Bellman equation and further show the C 2-regularity of the viscosity solution under the strict concavity of the utility function. The optimal policy is shown to exist and given in a feedback form or a stochastic version of Hotelling's rule.

[1]  R. Vidal.  Dynamic optimization: The calculus of variations and optimal control in economics and management: Morton I. KAMIEN and Nancy L. SCHWARTZ Volume 4 in: Dynamic Economics: Theory and Applications, North-Holland, New York, 1981, xi + 331 pages, Dfl.90.00 , 1982 .

[2]  H. Soner.  Controlled markov processes, viscosity solutions and applications to mathematical finance , 1997 .

[3]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[4]  池田 信行,et al.  Stochastic differential equations and diffusion processes , 1981 .

[5]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[6]  M. Akian,et al.  On an Investment-Consumption Model With Transaction Costs , 1996 .

[7]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[8]  Alan Gleit.  Optimal harvesting in continuous time with stochastic growth , 1978 .

[9]  Thaleia Zariphopoulou,et al.  Consumption-investment models with constraints , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[10]  Leonard J. Mirman,et al.  Dynamic Programming Models of Fishing: Competition , 1981 .

[11]  W. Fleming,et al.  Hedging in incomplete markets with HARA utility , 1997 .

[12]  Jati K. Sengupta,et al.  Stochastic Control Theory , 1997 .

[13]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[14]  H. Hotelling.  The economics of exhaustible resources , 1931, Journal of Political Economy.

[15]  Donald Ludwig,et al.  Optimal Harvesting of a Randomly Fluctuating Resource. I: Application of Perturbation Methods , 1979 .

[16]  Robert S. Pindyck,et al.  Uncertainty in the Theory of Renewable Resource Markets , 1984 .