# A discrete optimality system for an optimal harvesting problem

- 101 Downloads

## Abstract

In this paper, we obtain the discrete optimality system of an optimal harvesting problem. While maximizing a combination of the total expected utility of the consumption and of the terminal size of a population, as a dynamic constraint, we assume that the density of the population is modeled by a stochastic quasi-linear heat equation. Finite-difference and symplectic partitioned Runge–Kutta (SPRK) schemes are used for space and time discretizations, respectively. It is the first time that a SPRK scheme is employed for the optimal control of stochastic partial differential equations. Monte-Carlo simulation is applied to handle expectation appearing in the cost functional. We present our results together with a numerical example. The paper ends with a conclusion and an outlook to future studies, on further research questions and applications.

## Keywords

Stochastic optimal control Optimal harvesting Stochastic partial differential equations Symplectic partitioned Runge–Kutta schemes## References

- Alvarez L, Shepp L (1998) Optimal harvesting of stochastically fluctuating populations. J Math Biol 37:155–177CrossRefGoogle Scholar
- Barth A, Lang A (2012) Simulation of stochastic partial differential equations using finite element methods. Stoch Int J Probab Stoch Process 84(2–3):217–231CrossRefGoogle Scholar
- Bismut JM (1973) Conjugate convex functions in optimal stochastic control. J Math Anal Appl 44:384–404CrossRefGoogle Scholar
- Bismut JM (1978) An introductory approach to duality in optimal stochastic control. SIAM Rev 20(1):61–78CrossRefGoogle Scholar
- Bonnans JF, Laurent-Varin J (2006) Computation of order conditions for symplectic partitioned Runge–Kutta schemes with application to optimal control. Numerische Mathematik 103(1):1–10CrossRefGoogle Scholar
- Burrage K, Burrage PM (1999) Order conditions of stochastic Runge–Kutta methods by B-series. SIAM J Numer Anal 38:1626–1646CrossRefGoogle Scholar
- Burrage PM (1999) Runge–Kutta methods for stochastic differential equations. PhD Thesis, Department of Mathematics, University of Queensland, AustraliaGoogle Scholar
- Butcher JC (2003) Numerical methods for ordinary differential equations. Wiley, West SussexCrossRefGoogle Scholar
- Clark CW (1931) Mathematical bioeconomics: the optimal management of renewable resources, 2nd edn. Wiley, New YorkGoogle Scholar
- Debrabant K, Kværnø A (2008/2009) B-series analysis of stochastic Runge-Kutta methods that use an iterative scheme to compute their internal stage values. SIAM J Numer Anal 47(1):181–203Google Scholar
- Debrabant K, Rößler A (2008) A Classification of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations. Math Comput Simul 77(4):408–420CrossRefGoogle Scholar
- Dontchev AL, Hager WW, Veliov VM (2001) Second-order Runge–Kutta approximations in control constrained optimal control. SIAM J Numer Anal 38(1):202–226CrossRefGoogle Scholar
- Fadhel SF, Abdulamear AA (2011) Explicit Runge–Kutta methods for solving stochastic differential equations. J Basrah Res Sci 37(4):300–313Google Scholar
- Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer, New YorkCrossRefGoogle Scholar
- Fleming WH, Soner HM (1993) Controlled Markov processes and viscosity solutions. Springer, New YorkGoogle Scholar
- Glasserman P (2004) Monte-Carlo methods in financial engineering. Springer, New YorkGoogle Scholar
- Hager WW (2000) Runge–Kutta methods in optimal control and the transformed adjoint system. Numerische Mathematik 87(2):247–282CrossRefGoogle Scholar
- Itô K (1944) Stochastic integral. Proc Imperial Acad Tokyo 20:519–524CrossRefGoogle Scholar
- Jentzen A, Kloeden PE (2009) The numerical approximation of stochastic partial differential equations. Milan J Math 77:205–244CrossRefGoogle Scholar
- Kaya CY (2010) Inexact restoration for Runge–Kutta discretization of optimal control problems. SIAM J Numer Anal 48(4):1492–1517CrossRefGoogle Scholar
- Kloeden PE, Platen E (1999) Numerical solution of stochastic differential equations, vol 21, 2nd edn. Springer, BerlinGoogle Scholar
- Korn R, Korn E, Kroisandt G (2010) Monte-Carlo methods and models in finance and insurance. Chapman and Hall/CRC Financial Mathematics Series. CRC Press, Boco Raton. ISBN 9781420076196Google Scholar
- Kroese DP, Taimre T, Botev ZI (2011) Handbook of Monte-Carlo methods. Wiley series in probability and statistics. Wiley, New YorkCrossRefGoogle Scholar
- Kushner HJ, Dupuis PG (1992) Numerical methods for stochastic control problems in continuous time. Springer, New YorkCrossRefGoogle Scholar
- Kunita H (1990) Stochastic flows and stochastic differential equations, vol 24. Studies in advanced mathematics. Cambridge University Press, CambridgeGoogle Scholar
- Lungu E, Øksendal B (1997) Optimal harvesting from a population in a stochastic crowded environment. Math Biosci 145:47–75CrossRefGoogle Scholar
- Lungu E, Øksendal B (2001) Optimal harvesting from interacting populations in a stochastic environment. BERNOULLI 7:527–539CrossRefGoogle Scholar
- Øksendal B (2005) Optimal control of stochastic partial differential equations. Stoch Anal Appl 23:165–179CrossRefGoogle Scholar
- Øksendal B, Våge G, Zhao HZ (2000) Asymptotic properties of the solutions to stochastic KPP equations. Proc R Soc Edinb 130(6):1363–1381Google Scholar
- Øksendal B, Våge G, Zhao HZ (2001) Two properties of stochastic KPP equations: ergodicity and pathwise property. Nonlinearity 14:639–662CrossRefGoogle Scholar
- Peng S, Wu Z (1999) Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J Control Optim 37:825–843CrossRefGoogle Scholar
- Peng S (1990) A general stochastic maximum principle for optimal control problems. SIAM J Control Optim 28:966979CrossRefGoogle Scholar
- Pinheiro S (2016) Optimal harvesting for a logistic growth model with predation and a constant elasticity of variance. Ann Oper Res :1–20Google Scholar
- Rößler A (2006) Rooted tree analysis for order conditions of stochastic Runge–Kutta methods for the weak approximation of stochastic differential equations. Stoch Anal Appl 24(1):97–134CrossRefGoogle Scholar
- Rößler A (2010) Runge–Kutta methods for the strong approximation of solutions of stochastic differential equations. SIAM J Numer Anal 48(3):922–952CrossRefGoogle Scholar
- Sanz-Serna JM (2016) Symplectic Runge–Kutta schemes for adjoint equations, automatic differentiation, optimal control, and more. SIAM Rev 58(1):3–33CrossRefGoogle Scholar
- Schulstok B (1998) Optimal income by harvesting under uncertainty. Cand Scient thesis, University of OsloGoogle Scholar
- Yılmaz F, Öz H, Weber GW (2015) Simulation of stochastic optimal control problems with symplectic partitioned Runge–Kutta scheme. Dyn Contin Discret Impuls Syst Ser B Appl Algoritm 22:425–440Google Scholar
- Yong J, Zhou XY (1999) Stochastic Controls: Hamiltonian Systems and HJB Equations. Applications of mathematics, Springer, New YorkCrossRefGoogle Scholar
- Yoshida H (1990) Construction of higher order symplectic integrators. Phys Lett A 150:262–268CrossRefGoogle Scholar