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Some diffusion models for the mabinogion sheep problem of williams

Part of: Stochastic processes Stochastic systems and control

Published online by Cambridge University Press:  01 July 2016

Terence Chan
Terence Chan*
Affiliation:
Heriot-Watt University
*
∗ Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, U.K.
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Abstract

The ‘Mabinogion sheep’ problem, originally due to D. Williams, is a nice illustration in discrete time of the martingale optimality principle and the use of local time in stochastic control. The use of singular controls involving local time is even more strikingly highlighted in the context of continuous time. This paper considers a class of diffusion versions of the discrete-time Mabinogion sheep problem. The stochastic version of the Bellman dynamic programming approach leads to a free boundary problem in each case. The most surprising feature in the continuous-time context is the existence of diffusion versions of the original discrete-time problem for which the optimal boundary is different from that in the discrete-time case; even when the optimal boundary is the same, the value functions can be very different.

Keywords

MARTINGALE PRINCIPLE OF OPTIMALITYSTOCHASTIC CONTROLSINGULAR CONTROLFREE BOUNDARY PROBLEM

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 60G35: Signal detection and filtering 60G44: Martingales with continuous parameter
Type
General Applied Probablity
Information
Advances in Applied Probability , Volume 28 , Issue 3 , September 1996 , pp. 763 - 783
Copyright
Copyright © Applied Probability Trust 1996 

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References

Davis, M. H. A. and Norman, A. R. (1990) Portfolio selection with transaction costs. Math. Operat. Res. 15, 676–713.Google Scholar
Williams, D. (1991) Probability with Martingales. Cambridge University Press, Cambridge.Google Scholar