Geodesic
Optimal control of investments with random yield under
Matematičeskoe modelirovanie, Tome 19 (2007) no. 5, pp. 45-58.

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This paper solves the problem of optimal control of the level of investment in some asset whose price follows a geometric Brownian motion. Each transaction requires both fixed and proportional transaction costs. It is shown that the model can be generalized for a number of different assets. The general form of optimal control is found and a constructive algorithm for identefication of all parameters of the control is presented using quasi-variational inequalities. The algorithm yields a system of six nonlinear equations. It is shown that optimal control, that depends on four parameters in general, depends on two parameters if fixed transaction costs are zero, and depends on three parameters, if proportional transaction costs are zero. Numerical experiment is used to show how optimal control depends on all parameters of the model. Intergal representation of the value function is found, that may help to determine the optimal control. The case when the response to the control takes place after a constant time period is also studied. The property of the optimal control that it depends on only one state variable is proven. This fact is used to solve the problem with standard methods of quasi-variational inequalities. 
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     author = {V. V. Kitov},
     title = {Optimal control of investments with random yield under},
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     url = {http://geodesic.mathdoc.fr/item/MM_2007_19_5_a3/}
}
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V. V. Kitov. Optimal control of investments with random yield under. Matematičeskoe modelirovanie, Tome 19 (2007) no. 5, pp. 45-58. http://geodesic.mathdoc.fr/item/MM_2007_19_5_a3/

[1] A. Bensusan, Zh. Lions, Impulsnoe upravlenie i kvazivariatsionnye neravenstva, Nauka, M., 1987

[2] B. Perthame, “Continious and impulse control of diffusion proccesses in $R^n$”, Nonlinear Analysis, Theory, Methods and Applications, 8:10 (1984), 1227–1239 | DOI | MR | Zbl

[3] Gabriela Mundaca, Bernt Oksendal, “Optimal stochastic intervention control with application to the exchange rate”, Journal of Mathematical Economics, 1998, no. 29, 225–243 | MR | Zbl

[4] Abel Cadenillas, Fernando Zapatero, “Optimal central bank intervention in the foreign exchange market”, Journal of Economic Theory, 1999, no. 87, 218–242 | MR | Zbl

[5] Abel Cadenillas, Fernando Zapatero, “Classical and impulse stochastic control of the exchange rate using interest rates and reserves”, Mathematical Finance, 10:2 (2000), 141–156 | DOI | MR | Zbl

[6] Valeri I. Zakamouline, “European option pricing and hedging with both fixed and proportional transaction costs”, Journal of Economic Dynamics and Control, 30:1 (2006), 1–25 | DOI | MR | Zbl

[7] Avner Bar-Ilan, Agnes Sulem, Alessandro Zanello, “Time-to-build and capacity choice”, Journal of Economic Dynamics and Control, 26:1 (2002), 69–98 | DOI | MR

[8] Valeri I. Zakamouline, Optimal portfolio selection with both fixed and proportional transaction costs for a CRRA investor with finite horizon, 2002, Discussion papers of Department of Finance and Management Science, NHH

[9] B. Oksendal, Stokhasticheskie differentsialnye uravneniya, Mir, M., 2003

[10] P. L. Lions, “Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations”, Partial Differential Equations, 1983, no. 8, 1101–1174 | DOI | MR | Zbl

[11] B. Perthame, “On the regularity of the solutions of quasi-variational inequalities”, Journal of Functional Analysis, 1985, no. 64, 190–208 | DOI | MR

[12] A. Sulem, Quasi-variational inequalities and impulse control, MIT Press, 1993