The Azéma–Yor embedding in non-singular diffusions

Abstract

Let ( X t ) t⩾0 be a non-singular (not necessarily recurrent) diffusion on R starting at zero, and let ν be a probability measure on R. Necessary and sufficient conditions are established for ν to admit the existence of a stopping time τ ∗ of ( X t ) solving the Skorokhod embedding problem, i.e. X τ ∗ has the law ν. Furthermore, an explicit construction of τ ∗ is carried out which reduces to the Azéma–Yor construction (Séminaire de Probabilités XIII, Lecture Notes in Mathematics, Vol. 721, Springer, Berlin, p. 90) when the process is a recurrent diffusion. In addition, this τ ∗ is characterized uniquely to be a pointwise smallest possible embedding that stochastically maximizes (minimizes) the maximum (minimum) process of ( X t ) up to the time of stopping.