Is there an analytical formulation for complex barrier option where from time zero to time 1 the barrier isknock-out on threshold 1, and from time 1 to time 2 the barrier is knock in?

Sure -- it's an integration (*) over known formulas, assuming GBM. Just do the integral numerically.Suppose [$]V_{KI}(\tau,x)[$] is the value of a knock-in with time-to-expiration [$]\tau[$], initial underlying [$]x[$], and whatever terminal payoff applies -- supressing the dependence on the KI barrier level. This is known under GBM. Separately, the knock-out Green function [$]G_{KO}(\tau,x,y)[$] is known.This is the probability density to arrive at [$]y[$] after time [$]\tau[$], starting at [$]x[$], and without getting knocked out --again, suppressing the dependence on the KO barrier level. Finally, with initial underlying price [$]x[$], your contract is worth(*) [$] e^{-r \tau_1} \int G_{KO}(\tau_1,x,y) \, V_{KI}(\tau_2 - \tau_1,y) \, dy [$] The integration range is [$]y \in (x, b)[$] for a KO barrier [$]b[$] above the initial price [$]x[$]. p.s. I suggest a check with a decent Monte Carlo -- am just thinking out loud, so haven't tried it.