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Certain sequential decision problems involving normal random variables reduce to optimal stopping problems which can be related to the solution of corresponding free boundary problems for the heat equation. The numerical solution of these free boundary problems can then be approximated by calculating the solution of simpler optimal stopping problems by backward induction. This approach is not well adapted for very precise results but is surprisingly effective for rough pproximations. An estimate of the difference between the solutions of the related problems permits one to make continuity corrections which provide considerably improved accuracy. Further reductions in the necessary computational effort are possible by considering truncated procedures for one-sided boundaries and by exploiting monotone and symmetric boundaries.
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Chernoff et al. (1986) studied this question.
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