How to choose mu to make an O-U process S(t) a martingale?mu is either a constant or function of t, S(t) satisfies 1 or 2:1. dS(t) = k (mu - S(t)) dt + b dW(t)2. dS(t) = S(t) k (mu - ln S(t))dt + S(t) b dW(t)How to choose mu or mu(t) to make S(t) a martingale?My solution is, for S(t) satisfies 2, by choosing mu(t) = ln S(0) +(1-exp(-2kt)) b^2/(4k)E[S(t)] = S(0)Is it enough to show that S(t) now is a martingale?I doubt my answer because I can not confirm it with Ito's Lemma.What is the appropriate mu? Are there any related website or papers?Thank you so much.I would appricate your input.

Sorry, your problem doesn't have a solution with deterministic mu:In case 1. you would need mu(t) = S(t), in case 2. mu(t)= ln S(t) : the dt term need to be zero.

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