Back to basics : the Dubins-Schwarz theorem

The Dubins-Schwarz theorem is an important result of stochastic calculus. It states essentially that continuous local martingales and in particular continuous martingales are time changed Brownian motion. It is named after the American mathematician Lester Dubins (1920-2010), and the Israeli mathematician and statistician Gideon E. Schwarz (1933-2007) who is also at the origin of the Bayesian information criterion (BIC) in statistics. He is neither the famous German mathematician Karl Hermann Amandus Schwarz (1841-1921) nor the famous French mathematician Laurent Schwartz (1915-2002). The Dubins-Schwarz theorem was also discovered independently by the Russian mathematician K.È. Dambis, who apparently published a single article, in Russian, in 1965, the same year as the paper by Dubins and Schwarz.

Dubins-Schwarz theorem. Let \( {M} \) be a continuous local martingale with respect to a filtration \( {{(\mathcal{F}_t)}_{t\geq0}} \), such that \( {M_0=0} \) and \( {\langle M\rangle_\infty=\infty} \) almost surely. For all \( {t\geq0} \), let

\[ T_t=\inf\{s\geq0:\langle M\rangle_s>t\}=\langle M\rangle_t^{-1} \]

be the generalized inverse of the non-decreasing process \( {\langle M\rangle} \) issued from \( {0} \). Then

1. \( {B={(M_{\langle M\rangle_t^{-1}})}_{t\geq0}} \) is a Brownian motion with respect to the filtration \( {{(\mathcal{F}_{T_t})}_{t\geq0}} \)
2. \( {{(B_{\langle M\rangle_t})}_{t\geq0}={(M_t)}_{t\geq0}} \).

For instance, if \( {M=\alpha W} \) where \( {\alpha>0} \) is a constant and \( {W} \) is a Brownian motion issued from the origin, then for all \( {t\geq0} \) we have \( {\langle M\rangle_t=\alpha^2t} \) and \( {T_t=\alpha^{-2}t} \), and the process

\[ B={(M_{T_t})}_{t\geq0}={(\alpha W_{\alpha^{-2}t})}_{t\geq0} \]

is a Brownian motion with respect to \( {{(\mathcal{F}_{\alpha^{-2}t})}_{t\geq0}} \). In this example, the change of time is deterministic, but in general, it is random, for instance if \( {M_t=\int_0^tW_s\mathrm{d}W_s} \) where \( {{(W_t)}_{t\geq0}} \) is a Brownian motion then \( {\langle M\rangle_t=\int_0^tW_s^2\mathrm{d}s} \) which is random.

Flatness lemma. Since \( {\langle M\rangle} \) can be flat on an interval, the map \( {t\mapsto T_t} \) can be discontinuous. But this does not contradict the continuity of \( {t\mapsto M_{T_t}} \). Indeed, the flatness lemma states that \( {M} \) and \( {\langle M\rangle} \) are constant on the same intervals in the sense that almost surely, for all \( {0\leq a<b} \),

\[ \forall t\in[a,b], M_t=M_a \quad\text{if and only if}\quad \langle M\rangle_b=\langle M\rangle_a. \]

Proof of the flatness lemma. Since \( {M} \) and \( {\langle M\rangle} \) are continuous, it suffices to show that for all \( {0\leq a\leq b} \), almost surely,

\[ \{\forall t\in[a,b]:M_t=M_a\}=\{\langle M\rangle_b=\langle M\rangle_a\}. \]

The inclusion \( {\subset} \) comes from the approximation of the quadratic variation \( {\langle M\rangle=[M]} \). Let us prove the converse. To this end, we consider the continuous local martingale \( {{(N_t)}_{t\geq0}={(M_t-M_{t\wedge a})}_{t\geq0}} \). We have

\[ \langle N\rangle =\langle M\rangle-2\langle M,M^a\rangle+\langle M^a\rangle =\langle M\rangle-2\langle M\rangle^a+\langle M\rangle^a =\langle M\rangle-\langle M\rangle^a. \]

For all \( {\varepsilon>0} \), we set the stopping time \( {T_\varepsilon=\inf\{t\geq0:\langle N\rangle_t>\varepsilon\}} \). The continuous semi-martingale \( {N^{T_\varepsilon}} \) satisfies \( {N^{T_\varepsilon}_0=0} \) and \( {\langle N^{T_\varepsilon}\rangle_\infty=\langle N\rangle_{T_\varepsilon}\leq\varepsilon} \). It follows that \( {N^{T_\varepsilon}} \) is a martingale bounded in \( {\mathrm{L}^2} \), and for all \( {t\geq0} \),

\[ \mathbb{E}(N^2_{t\wedge T_\varepsilon}) =\mathbb{E}(\langle N\rangle_{t\wedge T_\varepsilon}) \leq\varepsilon. \]

Let us define the event \( {A=\{\langle M\rangle_b=\langle M\rangle_a\}} \). Then \( {A\subset\{T_\varepsilon\geq b\}} \) and, for all \( {t\in[a,b]} \),

\[ \mathbb{E}(\mathbf{1}_AN^2_t) =\mathbb{E}(\mathbf{1}_AN^2_{t\wedge T_\varepsilon}) \leq\mathbb{E}(N^2_{t\wedge T_\varepsilon}) \leq\varepsilon. \]

By sending \( {\varepsilon} \) to \( {0} \) we obtain \( {\mathbb{E}(\mathbf{1}_AN^2_t)=0} \) and thus \( {N_t=0} \) almost surely on \( {A} \). This ends the proof of the flatness lemma, which is of independent interest.

Proof of the Dubins-Schwarz theorem. For all \( {t\geq0} \), the random variable \( {T_t} \) is a stopping time with respect to \( {{(\mathcal{F}_u)}_{u\geq0}} \), and \( {s\mapsto T_s} \) is non-decreasing. It follows that for all \( {0\leq s\leq t} \), \( {\mathcal{F}_{T_s}\subset\mathcal{F}_{T_t}} \), and thus \( {{(\mathcal{F}_{T_u})}_{u\geq0}} \) is a filtration. Moreover for all \( {t\geq0} \), \( {T_t} \) is a stopping time for the filtration \( {{(\mathcal{F}_{T_u})}_{u\geq0}} \). We have \( {T_t<\infty} \) for all \( {t\geq0} \) on the almost sure event \( {\{\langle M\rangle_\infty=\infty\}} \). By construction \( {{(T_t)}_{t\geq0}} \) is right continuous, non-decreasing (and thus with left limits), and adapted with respect to \( {{(\mathcal{F}_{T_t})}_{t\geq0}} \). Since \( {M} \) is continuous, \( {B={(M_{T_t})}_{t\geq0}} \) is right continuous with left limits. Moreover, for all \( {t\geq0} \),

\[ B_{t^-}=\lim_{s\underset{<}{\rightarrow}t}B_s=M_{T_{t^-}}. \]

By the flatness lemma, almost surely \( {B_{t^-}=B_t} \) for all \( {t\geq0} \), hence \( {B} \) is continuous.

Let us show that \( {B} \) is a Brownian motion for \( {{(\mathcal{F}_{T_t})}_{t\geq0}} \). For all \( {n\geq0} \), \( {M^{T_n}} \) is a continuous local martingale issued from the origin and \( {\langle M^{T_n}\rangle_\infty=\langle M\rangle_{T_n}=n} \) almost surely. It follows that for all \( {n\geq0} \), the processes

\[ M^{T_n} \quad\mbox{and}\quad (M^{T_n})^2-\langle M\rangle^{T_n} \]

are uniformly integrable martingales. Now, for all \( {0\leq s\leq t\leq n} \), and by the Doob stopping theorem for uniformly integrable martingales, using \( {T_s\leq T_t\leq T_n} \),

\[ \mathbb{E}(B_t\mid\mathcal{F}_{T_s}) =\mathbb{E}(M^{T_n}_{T_t}\mid\mathcal{F}_{T_s}) =M^{T_n}_{T_s} =M_{T_n\wedge T_s} =B_{s} \]

and similarly, using additionally the property \( {\langle M\rangle^{T_n}_{T_t}=\langle M\rangle_{T_n\wedge T_t}=\langle M\rangle_{T_t}=t} \),

\[ \mathbb{E}(B_t^2-t\mid\mathcal{F}_{T_s}) =\mathbb{E}((M^{T_n}_{T_t})^2-\langle M^{T_n}\rangle_{T_t}\mid\mathcal{F}_{T_s}) =(M^{T_n}_{T_s})^2-\langle M^{T_n}\rangle_{T_s} =B_{T_s}. \]

Thus \( {B} \) and \( {{(B_t^2-t)}_{t\geq0}} \) are martingales with respect to the filtration \( {{(\mathcal{F}_{T_t})}_{t\geq0}} \). It follows now from the Lévy characterization that \( {B} \) is a Brownian motion for \( {{(\mathcal{F}_{T_t})}_{t\geq0}} \).

Let us show that \( {M=B_{\langle M\rangle}} \). By definition of \( {B} \), almost surely, for all \( {t\geq0} \),

\[ B_{\langle M\rangle_t}=M_{T_{\langle M\rangle_t}}. \]

Now \( {T_{\langle M\rangle_t^-}\leq t\leq T_{\langle M\rangle_t}} \) and since \( {\langle M\rangle} \) takes the same value at \( {T_{\langle M\rangle_t^-}} \) and \( {T_{\langle M\rangle_t}} \), we get \( {t=T_{\langle M\rangle_t}} \) and the flatness lemma gives \( {M_t=M_{T_{\langle M\rangle_t}}} \) for all \( {t\geq0} \) almost surely. In other words, using the definition of \( {B} \), this means that almost surely, for all \( {t\geq0} \),

\[ M_t=M_{T_{\langle M\rangle_t}}=B_{\langle M\rangle_t}. \]

This ends the proof of the Dubins-Schwarz theorem.

Warnings about the Dubins-Schwarz theorem.

• The Dubins-Schwarz theorem does not state that \( {B_{\langle M\rangle}=M} \) for a Brownian motion \( {B} \) with respect to the filtration for which \( {M} \) is a local martingale.
• The Dubins-Schwarz theorem is not valid for semi-martingales.

Ornstein-Uhlenbeck process. For an arbitrary \( {x\in\mathbb{R}} \), let us consider the Ornstein-Uhlenbeck process \( {{(Z_t)}_{t\geq0}} \) issued from \( {x} \) and given for all \( {t\geq0} \) by

\[ Z_t=x\mathrm{e}^{-t}+\mathrm{e}^{-t}M_t \quad\mbox{where}\quad M_t=\sqrt{2}\int_0^t\mathrm{e}^s\mathrm{d}B_s \]

where \( {B={(B_t)}_{t\geq0}} \) is a Brownian motion in \( {\mathbb{R}} \) with respect to \( {{(\mathcal{F}_t)}_{t\geq0}} \). The process \( {{(Z_t)}_{t\geq0}} \) is the unique square integrable continuous semi-martingale solution of the stochastic differential equation \( {Z_t=x+\sqrt{2}B_t-\int_0^tZ_s\mathrm{d}s} \), \( {t\geq0} \).

The process \( {{(M_t)}_{t\geq0}} \) is Gaussian and for all \( {t\geq0} \), \( {M_t\sim\mathcal{N}(0,\langle M\rangle_t)} \) (Wiener integral) with \( {\langle M\rangle_t=\int_0^t(\sqrt{2}\mathrm{e}^{s})^2\mathrm{d}s=\mathrm{e}^{2t}-1} \). Hence, for all \( {t\geq0} \), we have the equality in law

\[ Z_t\overset{\mathrm{d}}{=}x\mathrm{e}^{-t}+\mathrm{e}^{-t}B_{\mathrm{e}^{2t}-1}. \]

The processes \( {{(Z_t)}_{t\geq0}} \) and \( {{(x\mathrm{e}^{-t}+\mathrm{e}^{-t}B_{\mathrm{e}^{2t}-1})}_{t\geq0}} \) have same one-dimensional marginal distributions, but they are not equal since the second is not measurable with respect to \( {{(\mathcal{F}_t)}_{t\geq0}} \).

However, since \( {{(M_t)}_{t\geq0}} \) is a continuous local martingale with respect to \( {{(\mathcal{F}_t)}_{t\geq0}} \) for which \( {M_0=0} \) and \( {\langle M\rangle_\infty=\infty} \), the Dubins-Schwarz theorem states that there exists a Brownian motion \( {{(W_t)}_{t\geq0}} \) with respect to \( {{(\mathcal{F}_{T_t})}_{t\geq0}} \) where

\[ T_t =\inf\{s\geq0:\langle M\rangle_s>t\} =\frac{\log(t+1)}{2} \]

such that

\[ {(Z_t)}_{t\geq0} = {(x\mathrm{e}^{-t}+\mathrm{e}^{-t}W_{\langle M\rangle_t})}_{t\geq0} = {(x\mathrm{e}^{-t}+\mathrm{e}^{-t}W_{\mathrm{e}^{2t}-1})}_{t\geq0}. \]

About Gideon E. Schwarz. Born 1933 in Salzburg, Austria. Escaped in 1938, after the Anschluss, to Palestine, today Israel. M.Sc. in Mathematics at the Hebrew University, Jerusalem in 1956. Ph.D. in Mathematical Statistics at Columbia University in 1961. Research fellowships: Miller Institute 1964-66, Institute for Advanced Studies on Mt. Scopus 1975-76. Visiting appointments: Stanford University, Tel Aviv University, University of California in Berkeley. Since 1961, Fellow of the Institute of Mathematical Statistics. Presently, Professor of Statistics at the Hebrew University. Taken from his paper The dark side of the Moebius strip, Amer. Math. Monthly 97 (1990), no. 10, 890-897.