Symmetry and Time Changed Brownian Motions
  In this paper we examine which Brownian Subordination with drift exhibits the symmetry property introduced by Fajardo and Mordecki (2006). We obtain that when the subordination results in a L\'evy process, a necessary and sufficient condition for the symmetry to hold is that drift must be equal to -1/2. 
Jos\'e Fajardo and Ernesto Mordecki