A Dubins type solution to the Skorokhod embedding problem

Dubins (1968) gave the first solution to the Skorokhod embedding problem (SEP) based solely on the underlying Brownian motion, and thus requiring no additional independent random variable. The Dubins solution to the SEP, can be expressed as τ≔sup{τn}τ≔sup{τn} with τn=inf{t≥τn−1:Wt∈ support of μn}τn=inf{t≥τn−1:Wt∈ support of μn}. Since the measures μnμn are defined recursively, in order to compute μnμn, each of μ0,…,μn−1μ0,…,μn−1 must first be computed. In this note, we give a new solution to the SEP by showing how to construct a different sequence of measures {μn}n∈N{μn}n∈N. The advantage of this solution is that for any given nn, the measure μnμn can be constructed directly without prior computation of the measures μ0,…,μn−1μ0,…,μn−1.