W_t is a brownian motion with drift m and vol s, what is the process W_t | W_T for t <T?

W_t is a geometric, or an arithmetic b.m.? - the question seems to imply that it is geometric.As it is, I cannot understand the question very well. I know that the process z_t=W_t / W_T is used to simulate a brownian bridge. But z_t is not measurable with respect to the natural filtration F_t of W_t, it's only measurable w.r.t. F_T. Are you asking for its law of motion? Btw, the definition of brownian bridge that I know is:It's easy to derive its law of motion. It's more interesting to show that the Novikov condition is violated, so it may not be possible to apply Girsanov's Theorem.

Thanks for the input Vassili.The question is actually what's the process W_t conditional on W_T, W_t being an arithmetic bm. The conditional distribution at any fixed t is pretty easy to work out, the conditional process can be written as (t / T) W_T + z_t, where z_t is defined above, does anyone know how to show this?

What about the conditional distribution in case W_t is a mean reverting process? or a geometric brownian motion?can that be derived?

take a look atLyons, T. J., Zheng, W., A. On conditional diffusion processes. Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 3-4, 243-255what you need is also summarized inQian_Zheng

i cant find the first article , can someone help me?what i want to do ultimately is to perform a stochastic interpolation from semi-annual data to monthly data, for which the semi-annual model is e.g. a simple mean reverting model, or a log mean reverting model.