Constrained Brownian processes and constrained Brownian bridges

We present a novel approach to describe Brownian paths that end at a given final point at a fixed time, and having the additional constrain of a fixed area under its curve. Both anticipative (in this case, knowledge of the future of the path is required) and non-anticipative solutions are exposed. In particular, for the non-anticipative representation a local stochastic differential equation is derived which leads to an effective Langevin equation. This representation allows us to generate statistically independent realizations of the constrained process in a very efficient way.