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I am trying to figure out the Doob decomposition of an American put option in a discrete time binomial model.

I know how to price the American put, but I'm having trouble expressing it as the sum of a martingale and a predictable process. I thought the martingale would be the price of a European put (with the same strike and expiration), and the compensator process would be the difference between the price of the European put and American put (i.e. the premium the holder is paying for having the liberty of exercising the put at any time up to expiration) but it seems to be that that isn't strictly non-decreasing?

Am I on the right track? Thanks for any help, sorry this isn't the most well posed question!

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