This paper analyzes the behavior of conditional forecast functions in stable systems. We study convergence of optimal forecast functions, of forecast functions obtained by conditioning on previous values, and conditional and joint densities.

Consider Brownian motion Bt and its maximum Mt = max0 [less-than-or-equals, slant] s [less-than-or-equals, slant] t Bs. We derive the joint distribution of (Ms, Bt) for all s and make a generalization to correlated BM. These distributions are applied to price barrier options.