Completeness for a Brownian motion with polygonal stopping boundaries

Consider a Brownian motion X(t) with drift θ ∈ J, a real interval, and a polygonal stopping boundary with hitting time T, assumed finite a.s. The boundary consists of upper and/or lower boundary functions that are piecewise-linear and continuous, and possibly a final vertical boundary. The statistic S ≡ (T, X(T)) is known to be sufficient for the past history of the process, for θ ∈ J. We show that the family (in θ) of distributions of S is complete. Relevance to sequential analysis is briefly noted.