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Since P(t)=k.N(t), we get: P(t) = Eq[ P(T)/N(T) ] .N(t) , i.e., P(t)/N(t) = Eq[ P(T)/N(T) ] In other words, P(.)/N(.) is martingale. What if some action is allowed in the term? Extend to the derivatives in which some choice of action is allowed.