Continuous Martingales as Time-Changed

5.3 A Few Consequences of Itô’s Formula

5.3.2 Continuous Martingales as Time-Changed

The next theorem shows that any continuous local martingaleMcan be written as a

“time-changed” Brownian motion (in fact, we prove this only whenhM;Mi1D 1, but see the remarks below). It follows that the sample paths ofM are Brownian sample paths run at a different (varying) speed, and certain almost sure properties of sample paths ofMcan be deduced from the corresponding properties of Brownian sample paths. For instance, under the conditionhM;Mi1 D 1, the sample paths ofM must oscillate betweenC1 and1 as t ! 1 (cf. the last assertion of Proposition2.14).

5.3 A Few Consequences of Itô’s Formula 121 Theorem 5.13 (Dambis–Dubins–Schwarz) Let M be a continuous local martin- gale such thathM;Mi1 D 1 a.s. There exists a Brownian motion.ˇs/s0 such that

a:s:8t0; MtDˇ<M;M>t: Remarks

(i) One can remove the assumptionhM;Mi₁ D 1, at the cost of enlarging the underlying probability space, see [70, Chapter V].

(ii) The Brownian motionˇ is not adapted with respect to the filtration.Ft/, but with respect to a “time-changed” filtration, as the following proof will show.

Proof We first assume thatM₀D0. For everyr0, we set rDinfft0W hM;Mi_trg:

Note thatr is a stopping time by Proposition3.9. Furthermore, we haver < 1 for everyr0, on the eventfhM;Mi₁D 1g. It will be convenient to redefine the variablesron the (negligible) eventN D fhM;Mi₁<1gby takingr.!/D0 for everyr 0if! 2 N . Since the filtration is complete,r remains a stopping time after this modification.

By construction, for every!2˝, the functionr7!r.!/is nondecreasing and left-continuous, and therefore has a right limit at everyr 0. This right limit is denoted byrCand we have

rC Dinfft0W hM;Mit>rg; except of course on the negligible setN, whererCD0.

We setˇr DMrfor everyr0. By Theorem3.7, the process.ˇr/r0is adapted with respect to the filtration.Gr/defined byGr DFr for everyr0, andG1 D F1. Note that the filtration.Gr/is complete since this property holds for.Ft/.

The sample pathsr7!ˇr.!/are left-continuous and have right limits given for everyr0by

ˇrCD lim

s##rˇsDM_rC:

In fact we haveˇrC Dˇr for everyr0, a.s., as a consequence of the following lemma.

Lemma 5.14 We have a.s. for every0a<b,

MtDMa; 8t2Œa;b ” hM;MibD hM;Mia:

Let us postpone the proof of the lemma. SincehM;Mi_r D hM;Mi_rCfor every r0, Lemma5.14implies thatM_r DM_rC, for everyr0, a.s. Hence the sample

122 5 Stochastic Integration paths ofˇare continuous (to be precise, we should redefineˇrD0, for everyr0, on the zero probability set where the property of Lemma5.14fails).

Let us verify thatˇs andˇs²sare martingales with respect to the filtration .Gs/. For every integern 1, the stopped continuous local martingalesMⁿ and .Mⁿ/²hM;Miⁿare uniformly integrable martingales (by Theorem4.13, recalling thatM₀ D 0 and noting that hMⁿ;Mⁿi1 D hM;Mi_n D n a.s.). The optional stopping theorem (Theorem3.22) then implies that, for every0rsn,

EŒˇsjGrDEŒM_sⁿ jFrDMⁿ_r Dˇr

and similarly

EŒˇ²ssjGrDEŒ.M_sⁿ/² hMⁿ;Mⁿi_sjFrD.M_rⁿ/² hMⁿ;Mⁿi_r Dˇ²rr: Then the cased D 1 of Theorem5.12shows thatˇ is a.Gr/-Brownian motion.

Finally, by the definition ofˇ, we have a.s. for everyt0, ˇ<M;M>t DM_<M;M>t:

But since_<M;M>t t_<M;M>tCand sincehM;Mitakes the same value at_<M;M>t

and at_<M;M>tC, Lemma5.14shows thatMt D M_<M;M>t for everyt 0, a.s. We conclude that we haveMtDˇ_<M;M>t for everyt0, a.s. This completes the proof whenM₀ D0.

IfM₀6D0, we writeMtDM₀CM⁰_t, and we apply the previous argument toM⁰, in order to get a Brownian motionˇ⁰withˇ₀⁰ D 0, such thatM_t⁰ D ˇ_<⁰_M0;M0>^t for everyt0a.s. Sinceˇ⁰is a.Gr/-Brownian motion,ˇ⁰is independent ofG₀ DF₀, hence ofM₀. Therefore,ˇs D M₀Cˇ_s⁰is also a Brownian motion, and we get the

desired representation forM. ut

Proof of Lemma5.14 Thanks to the continuity of sample paths ofMandhM;Mi, it is enough to verify that for any fixedaandbsuch that0a<b, we have

fM_tDMa; 8t2Œa;bg D fhM;Mi_bD hM;Mi_ag; a.s.

The fact that the event in the left-hand side is (a.s.) contained in the event in the right-hand side is easy from the approximations ofhM;Miin Theorem4.9.

Let us prove the converse. Consider the continuous local martingaleNt D Mt Mt^aand note that

hN;NitD hM;Mit hM;Mi_t^a: For every" > 0, introduce the stopping time

T_"Dinfft0W hN;Nit"g:

5.3 A Few Consequences of Itô’s Formula 123

ThenN^T"is a martingale inH²(sincehN^T";N^T"i1"). Fixt2Œa;b. We have EŒN_t^T² _"DEŒhN;Nit^T_"":

Hence, considering the eventAWD fhM;MibD hM;Miag fT_"bg, EŒ1AN_t²DEŒ1AN²_t^T"EŒN_t^T² _"":

By letting"go to0, we getEŒ1AN_t²D0and thusNtD0a.s. onA, which completes

the proof. ut

We can combine the arguments of the proof of Theorem5.13with Theorem5.12 to get the following technical result, which will be useful when we consider the image of planar Brownian motion under holomorphic transformations in Chap.7.

Proposition 5.15 Let M and N be two continuous local martingales such that M₀D N₀D0. Assume that

(i) hM;MitD hN;Nitfor every t0, a.s.

(ii) M and N are orthogonal (hM;NitD0for every t0, a.s.) (iii) hM;Mi1D hN;Ni1D 1, a.s.

LetˇD.ˇt/t0, resp.D.t/t0, be the real Brownian motion such MtDˇhM;Mit, resp. NtDhN;Nit, for every t0, a.s. Thenˇandare independent.

Proof We use the notation of the proof of Theorem5.13and note that we have ˇrDMrandrDNr, where

r Dinfft0W hM;Mitrg Dinfft0W hN;Nit rg:

We know thatˇandare.Gr/-Brownian motions. SinceM andNare orthogonal martingales, we also know that MtNt is a local martingale. As in the proof of Theorem5.13, and using now Proposition4.15(v), we get that, for everyn 1, MtⁿNtⁿ is a uniformly integrable martingale, and by applying the optional stopping theorem, we obtain that forrsn,

EŒˇssjGrDEŒM_sⁿN_sⁿ jFrDMⁿ_rN_rⁿ Dˇsr

so thatˇrris a.Gr/-martingale and the brackethˇ; i(evaluated in the filtration .Gr/) is identically zero. By Theorem 5.12, it follows that .ˇ; / is a two- dimensional Brownian motion and, sinceˇ0 D 0 D 0, this implies that ˇ and

are independent. ut

124 5 Stochastic Integration