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An implicit method for the finite time horizon Hamilton-Jacobi-Bellman quasi-variational inequalities. (English) Zbl 1410.93148

Summary: We propose a new numerical method for solving the Hamilton-Jacobi-Bellman quasi-variational inequality associated with the combined impulse and stochastic optimal control problem over a finite time horizon. Our method corresponds to an implicit method in the field of numerical methods for partial differential equations, and thus it is advantageous in the sense that the stability condition is independent of the discretization parameters. We apply our method to the finite time horizon optimal forest harvesting problem, which considers exiting from the forestry business at a finite time. We show that the behavior of the obtained optimal harvesting strategy of the extended problem coincides with our intuition.

MSC:

93E20 Optimal stochastic control
65K15 Numerical methods for variational inequalities and related problems
49J40 Variational inequalities
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35Q93 PDEs in connection with control and optimization

Software:

PARDISO
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Pliska, S. R.; Suzuki, K., Optimal tracking for asset allocation with fixed and proportional transaction costs, Quant. Finance, 4, April (2), 233-243, (2004) · Zbl 1405.91568
[2] Palczewski, J.; Zabczyk, J., Portfolio diversification with Markovian prices, Probab. Math. Stat., 25, 1, 75-95, (2005) · Zbl 1152.93057
[3] Kharroubi, I.; Pham, H., Optimal portfolio liquidation with execution cost and risk, SIAM J. Financ. Math., 1, January (1), 897-931, (2010)
[4] G. Mundaca, B. Øksendal, Optimal stochastic intervention control with application to the exchange rate, March 1998. · Zbl 0943.91038
[5] Cadenillas, A.; Zapatero, F., Classical and impulse stochastic control of the exchange rate using interest rates and reserves, Math. Finance, 10, April (2), 141-156, (2000) · Zbl 1034.91036
[6] Korn, R., Some applications of impulse control in mathematical finance, Math. Methods Oper. Res., 50, December (3), 493-518, (1999) · Zbl 0942.91048
[7] Bensoussan, A.; Lions, J. L., Impulse control and quasi-variational inequalities, (1984), Gauthier-Villars
[8] Chancelier, J.-P.; Messaoud, M.; Sulem, A., A policy iteration algorithm for fixed point problems with nonexpansive operators, Math. Methods Oper. Res., 65, October (2), 239-259, (2006) · Zbl 1171.47051
[9] Chen, Z.; Forsyth, P. A., A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB), Numer. Math., 109, May (4), 535-569, (2008) · Zbl 1141.93066
[10] Willassen, Y., The stochastic rotation problem: a generalization of faustmann’s formula to stochastic forest growth, J. Econ. Dyn. Control, 22, April (4), 573-596, (1998) · Zbl 0899.90065
[11] Øksendal, B.; Sulem, A., Applied stochastic control of jump diffusions, (2007), Springer · Zbl 1116.93004
[12] Schenk, O.; Gärtner, K., Solving unsymmetric sparse systems of linear equations with PARDISO, Future Gen. Comput. Syst., 20, April (3), 475-487, (2004)
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