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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

${\mathcal {M}}_{0,\operatorname {loc} }^{c}$ be the space of ${\mathcal {F}}_{t}$ -adapted continuous local martingales $M=(M_{t})_{t\geq 0}$ with $M_{0}=0$ .
$\langle M\rangle$ be the quadratic variation.

Statement

Let $M\in {\mathcal {M}}_{0,\operatorname {loc} }^{c}$ and $\langle M\rangle _{\infty }=\infty$ and define for all $t\geq 0$ the time-changes (i.e. stopping times)^[4]

T_{t}=\inf\{s:\langle M\rangle _{s}>t\}.

Then $B:=(B_{t}):=(M_{T_{t}})$ is a ${\mathcal {F}}_{T_{t}}$ -Brownian motion and $(M_{t})=(B_{\langle M\rangle _{t}})$ .

Remarks

The condition $\langle M\rangle _{\infty }=\infty$ guarantees that the underlying probability space is rich enough so that the Brownian motion exists. If one removes this conditions one might has to use enlargement of the filitered probability space.
$B$ is not a ${\mathcal {F}}_{t}$ -Brownian motion.
$(T_{t})$ are almost surely finite since $\langle M\rangle _{\infty }=\infty$ .

References

^ Dubins, Lester E.; Schwarz, Gideon (1965). "On Continuous Martingales". Proceedings of the National Academy of Sciences. 53 (5): 913–916. doi:10.1073/pnas.53.5.913. PMID 16591279.
^ Dambis, K. E. (1965). "On decomposition of continuous submartingales". Theory of Probability and Its Applications. 10 (3): 401–410. doi:10.1137/1110048.
^ "On decomposition of continuous submartingales". Teor. Veroyatnost. I Primenen. (in Russian). 10: 438–448. 1965.
^ Revuz, Daniel; Yor, Marc (1999). "Continuous Martingales and Brownian Motion". Grundlehren der Mathematischen Wissenschaften. 293. Springer. doi:10.1007/978-3-662-06400-9. ISBN 978-3-642-08400-3.

[1] Dubins, Lester E.; Schwarz, Gideon (1965). "On Continuous Martingales". Proceedings of the National Academy of Sciences. 53 (5): 913–916. doi:10.1073/pnas.53.5.913. PMID 16591279.

[2] Dambis, K. E. (1965). "On decomposition of continuous submartingales". Theory of Probability and Its Applications. 10 (3): 401–410. doi:10.1137/1110048.

[3] "On decomposition of continuous submartingales". Teor. Veroyatnost. I Primenen. (in Russian). 10: 438–448. 1965.

[4] Revuz, Daniel; Yor, Marc (1999). "Continuous Martingales and Brownian Motion". Grundlehren der Mathematischen Wissenschaften. 293. Springer. doi:10.1007/978-3-662-06400-9. ISBN 978-3-642-08400-3.

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@@ Line 1: / Line 1: @@
 In the theory of [[Martingale (probability theory)|martingale]]s, the '''Dubins-Schwarz theorem''' (or '''Dambis-Dubins-Schwarz theorem''') is a theorem that says all continuous [[local martingale]]s and martingales are time-changed [[Brownian motion]]s.
-The theorem was proven in [[1965]] by [[Lester Dubins]] and [[Gideon E. Schwarz]]<ref>{{cite journal|doi=10.1073/pnas.53.5.913|first1=Lester E.|last1=Dubins|first2=Gideon|last2=Schwarz|title=On Continuous Martingales |journal=Proceedings of the National Academy of Sciences|volume=53 |number=5|pages=913-916|date=1965 |url=https://www.pnas.org/doi/abs/10.1073/pnas.53.5.913}}</ref> and independently in the same year by [[K. E. Dambis]], a doctorial student of [[Eugene Dynkin]]<ref>{{cite journal|first=K. E.|last=Dambis|title=On decomposition of continuous submartingales|journal=Theory of Probability and its Applications|volume=10|date=1965|pages=401–410}}</ref><ref>{{cite journal|title=On decomposition of continuous submartingales|journal=Teor. Veroyatnost. i Primenen.|volume=10|date=1965|pages=438–448|lang=ru}}</ref>
+The theorem was proven in [[1965]] by [[Lester Dubins]] and [[Gideon E. Schwarz]]<ref>{{cite journal|doi=10.1073/pnas.53.5.913|first1=Lester E.|last1=Dubins|first2=Gideon|last2=Schwarz|title=On Continuous Martingales |journal=Proceedings of the National Academy of Sciences|volume=53 |number=5|pages=913–916|date=1965 |pmid=16591279 |doi-access=free }}</ref> and independently in the same year by [[K. E. Dambis]], a doctorial student of [[Eugene Dynkin]]<ref>{{cite journal|first=K. E.|last=Dambis|title=On decomposition of continuous submartingales|journal=Theory of Probability and Its Applications|volume=10|date=1965|issue=3 |pages=401–410|doi=10.1137/1110048 }}</ref><ref>{{cite journal|title=On decomposition of continuous submartingales|journal=Teor. Veroyatnost. I Primenen.|volume=10|date=1965|pages=438–448|lang=ru}}</ref>
 == Dubins-Schwarz theorem ==
@@ Line 9: / Line 9: @@
 === Statement ===
-Let <math>M\in \mathcal{M}_{0,\operatorname{loc}}^{c}</math> and <math>\langle M\rangle_{\infty}=\infty</math> and define for all <math>t\geq 0</math> the time-changes (i.e. [[stopping time]]s)<ref>{{cite book|title=Continuous Martingales and Brownian Motion|first1=Daniel|last1=Revuz|first2=Marc|last2=Yor|date=1999 |journal=Grundlehren der mathematischen Wissenschaften |volume=293 |publisher=Springer }}</ref>
+Let <math>M\in \mathcal{M}_{0,\operatorname{loc}}^{c}</math> and <math>\langle M\rangle_{\infty}=\infty</math> and define for all <math>t\geq 0</math> the time-changes (i.e. [[stopping time]]s)<ref>{{cite journal|title=Continuous Martingales and Brownian Motion|first1=Daniel|last1=Revuz|first2=Marc|last2=Yor|date=1999 |journal=Grundlehren der Mathematischen Wissenschaften |volume=293 |publisher=Springer |doi=10.1007/978-3-662-06400-9 |isbn=978-3-642-08400-3 }}</ref>
 :<math>T_t=\inf \{s:\langle M\rangle_s>t\}.</math>