Embedding laws in diffusions by functions of time

We present a constructive probabilistic proof of the fact that if B=(B_t)_{t \ge 0} is standard Brownian motion started at 0 and mu is a given probability measure on R such that mu({0})=0 then there exists a unique left-continuous increasing function b and a unique left-continuous decreasing function c such that B stopped at tau_{b,c}=inf{t>0 : B_t \ge b(t) or B_t \le c(t)} has the law mu. The method of proof relies upon weak convergence arguments arising from Helly's selection theorem and makes use of the L\'evy metric which appears to be novel in the context of embedding theorems. We show that tau_{b,c} is minimal in the sense of Monroe so that the stopped process satisfies natural uniform integrability conditions expressed in terms of mu. We also show that tau_{b,c} has the smallest truncated expectation among all stopping times that embed mu into B. The main results extend from standard Brownian motion to all recurrent diffusion processes on the real line.