In the theory of martingales, the Dubins–Schwarz theorem (or Dambis–Dubins–Schwarz theorem) is a theorem that says all continuous local martingales and martingales are time-changed Brownian motions.
The theorem was proven in 1965 by Lester Dubins and Gideon E. Schwarz[1] and independently in the same year by K. E. Dambis, a doctorial student of Eugene Dynkin.
0 , loc
{\displaystyle M\in {\mathcal {M}}_{0,\operatorname {loc} }^{c}}
and define for all
the time-changes (i.e. stopping times)[4] Then
-Brownian motion and