Suppose I have a GBM without drift [$]dS_t = S_t \sigma dW_t[$] and define [$]M = max_{(0\leq t\leq T)}S_t[$] as its maximum. I look for the expected spot conditional on [$]S_t[$] having never reached some barrier [$]B>S_0[$], i.e. [$]E(S_T|M < B)[$]. My thinking was that I integrate over the joint distribution of a Brownian Motion with drift [$]\hat{W}[$] (the drift coming from the convexity adjustment [$]-0.5\sigma^2t[$]) and its maximum M: [$]f_{M,\hat{W}}(m,w)[$]. I have set [$]b=\frac{log(B/S_0)}{\sigma}[$] and compute [$]E(S_T|M <B) = \int^b_0 \int^b_w S_0 e^{\sigma w} f_{M,\hat{W}}(m,w) dmdw + \int^0_{-\infty} \int^b_0 S_0 e^{\sigma w} f_{M,\hat{W}}(m,w) dmdw[$].The whole thing can e.g. be found in Shreve, Stochastic Calculus for Finance II, p. 295ff. While I can perfectly solve above integral, it gives me odd result that deviate from a simulated expectation.Many thanks for your help/comments.

Another approach to compute first hitting time by the powerful PDE approachhttp://homepages.ulb.ac.be/~ppatie/fpt_md.pdf

I dont think there is a need to fall back to numerical methods as there is a closed-form solution. I might be wrong though or overlook sth. important.

I think I now see the missing bit ... I need to integrate over the conditional distribution or equivalently, divide the integral by Prob(M < B) ... Thanks help anyway.

Strong guess. Using the GBM gives rather dull and obvious results, but it's a good model to start with.

Consider an up and out call, with barrier B and strike =0 (zero).It will simply be the (present value) of expected value of S(T), conditioned of not having crossed the barrier.