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When we solve Bellman equations, I normally would think of the Blanchard Kahn technique. But in the case that I have an optimal stopping problem, or where the decision that the agent has to take is to choose the maximum from two alternatives , I am not sure about how am I supposed to solve/proceed. There is no Euler equation because the problem is discrete. Any suggestions are welcome. Thanks!

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I recommend the manuscript here by Lawrence Evans. The related example is described as 'Rocket Railroad Car'. The first instance is on pages 9-12 where a geometric solution is provided. The choice set is not constrained to a discrete set in principle but the first part shows optimality when only the highest and lowest actions are chosen. On pages 35-36, Evans is proving that the geometric approach results in the correct solution, i.e. the discrete solution arises endogenously. Therefore, the remark is that you can consider a wider choice set and end up showing that the solution with discrete choices is also a solution when a larger choice set is considered.

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