Optimal control of a jump process. (English) Zbl 0349.60084
It is supposed that the behaviour of a process with a single random jump can be controlled by selecting Radon-Nikodym derivatives, that determine probability measures describing when the jump happens and where it goes. By observing that the minimum cost function is a “semi-martingale speciale” a dynamic programming minimum principle for the optimum control is obtained. An “adjoint variable” is introduced and shown to satisfy a certain differential equation, and the results are extended to multi-jump processes.
Reviewer: Robert J. Elliott (Edmonton)
MSC:
| 60J75 | Jump processes (MSC2010) |
| 93E20 | Optimal stochastic control |
References:
| [1] | Boel, R. K., Optimal control of jump processes (1974), Berkeley: University of California, Berkeley |
| [2] | Boel, R. K.; Varaiya, P., Optimal control of jump processes, SIAM J. Control and Optimization, 15, 92-119 (1977) · Zbl 0358.93047 |
| [3] | Davis, M. H.A., The representation of martingales of jump processes, SIAM J. Control and Optimization, 14, 623-638 (1976) · Zbl 0337.60048 |
| [4] | Davis, M. H.A., Absolutely continuous changes of measure for the elementary jump process (1975), London: Dept. of Computing and Control, Imperial College, London |
| [5] | Davis, M. H.A.; Varaiya, P., Dynamic programming conditions for partially observable stochastic system, SIAM J. Control and Optimization, 11, 226-261 (1973) · Zbl 0258.93029 |
| [6] | Elliott, R. J., Stochastic integrals for martingales of a jump process with partially accessible jump times, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 36, 213-226 (1976) · Zbl 0325.60055 |
| [7] | Elliott, R.J.: Levy systems and absolutely continuous changes of measure for a jump process. J. math. Analysis Appl. [To appear] · Zbl 0371.60064 |
| [8] | Elliott, R. J., Lévy functionals and jump process martingales, J. math. Analysis Appl., 57, 638-652 (1977) · Zbl 0349.60083 |
| [9] | Elliott, R. J., The optimal control of a stochastic system, SIAM J. Control and Optimization, 15, 756-778 (1977) · Zbl 0359.93046 |
| [10] | Feller, W., Introduction to probability theory and its applications. Vol. II (1966), New York: Wiley & Sons, New York · Zbl 0138.10207 |
| [11] | Jacod, J., Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 31, 235-253 (1975) · Zbl 0302.60032 |
| [12] | Loève, M., Probability Theory (1963), Princeton: Van Nostrand, Princeton · Zbl 0108.14202 |
| [13] | Meyer, P. A., Probability and potentials (1966), Waltham, Mass.: Blaisdell, Waltham, Mass. · Zbl 0138.10401 |
| [14] | Meyer, P. A., Un cours sur les integrales stochastiques, Sem. Prob. Univ. Strasbourg 1974/75 (1976), Berlin-New York: Springer, Berlin-New York · Zbl 0374.60070 |
| [15] | Pliska, S. R., Controlled jump processes, Stochastic Processes Appl., 3, 259-282 (1975) · Zbl 0313.60055 |
| [16] | Rishel, R., A minimum principle for controlled jump processes, Springer Lecture Notes in Economics and Mathematical Systems,107, 493-508 (1975), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0313.93065 |
| [17] | Rishel, R.: Controls optimal from timet onward and dynamic programming for systems of controlled jump processes. University of Kentucky, pre-print · Zbl 0381.49008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
