Abstract
The use of nonstandard techniques in stochastic optimal control theory is illustrated by discussing two controlled systems, the first with observations restricted to be a cumulative digital read-out, the second with complete observations and singular noise. The main ingredient in each case is the Loeb construction of standard measures from nonstandard measures, and the application of this to stochastic analysis as developed by Anderson and Keisler.
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References
Albeverio S., Fenstad J.-E., Hoegh-Krohn R. &Lindstrom, T.L., Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York, to appear.
Anderson, R.M., A non-standard representation for Browni an motion and Itô integration, Israel J. Math. 25 (1976), 15–46.
Anderson, R.M. and Rashid, S., A nonstandard characterisation of weak convergence, Proc. Amer. Math. Soc. 69 (1978), 327–332.
Benes V.E., Existence of optimal stochastic control laws, SIAM J. Control 9 (1971), 446–472.
Cutland N.J., Nonstandard measure theory and its applications, Bull. London Math. Soc. 15 (1983), 529–589.
Cutland N.J., Infinitesimal methods in control theory: deterministic and stochastic, Acta Appl. Math. 5 (1986), 105–135.
Cutland N.J., Optimal control for stochastic systems with singular noise, Systems& Control Letters 7 (1986), 55–59.
Keisler H.J., Foundations of Infinitesimal Calculus, Prindle, Webber & Schmidt, 1976.
Keisler H.J., An infinitesimal approach to stochastic analysis, Memoirs of the AMS 197, AMS, Providence 1984.
Hurd A.E.& Loeb P., An Introduction to Nonstandard Real Analysis, Academic Press, 1985.
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© 1988 D. Reidel Publishing Company
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Cutland, N.J. (1988). Nonstandard Techniques in Stochastic Optimal Control Theory. In: Albeverio, S., Blanchard, P., Hazewinkel, M., Streit, L. (eds) Stochastic Processes in Physics and Engineering. Mathematics and Its Applications, vol 42. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2893-0_4
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DOI: https://doi.org/10.1007/978-94-009-2893-0_4
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