Dubins–Schwarz theorem explained

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.^[2] ^[3]

Dubins-Schwarz theorem

Let

l{M}_{0,\operatorname{loc}

}^ be the space of

l{F}_t

-adapted continuous local martingales

M=(M_t)_t\geq

with

M₀₌₀

\langleM\rangle

be the quadratic variation.

Statement

Let

M\inl{M}_{0,\operatorname{loc}

}^ and

\langleM\rangle_infty=infty

and define for all

t\geq0

the time-changes (i.e. stopping times)^[4]

T_t=inf\{s:\langleM\rangle_s>t\}.

Then

B:=(B_t):=(M


	T_t

)

is a

l{F}
	T_t

-Brownian motion and

(M_t)=(B


	\langleM\rangle_t

)

Remarks

The condition

\langleM\rangle_infty=infty

guarantees that the underlying probability space is rich enough so that the Brownian motion exists. If one removes this conditions one might have to use enlargement of the filtered probability space.

is not a

l{F}_t

-Brownian motion.

(T_t)

are almost surely finite since

\langleM\rangle_infty=infty

References

10.1073/pnas.53.5.913. Lester E.. Dubins. Gideon. Schwarz. On Continuous Martingales . Proceedings of the National Academy of Sciences. 53 . 5. 913–916. 1965 . 16591279 . 301348 . 1965PNAS...53..913D . free .
K. E.. Dambis. On decomposition of continuous submartingales. Theory of Probability and Its Applications. 10. 1965. 3 . 401–410. 10.1137/1110048 .
On decomposition of continuous submartingales. Teor. Veroyatnost. I Primenen.. 10. 1965. 438–448. ru.
Book: Continuous Martingales and Brownian Motion. Daniel. Revuz. Marc. Yor. Grundlehren der mathematischen Wissenschaften . 1999 . 293 . Springer . 10.1007/978-3-662-06400-9 . 978-3-642-08400-3 .