132 5 Stochastic Integration
5.5 Girsanov’s Theorem 133 Then, ifTis a stopping time, the optional stopping theorem (Theorem3.22) gives for everyA2FT,
Q.A/DEQŒ1ADEPŒ1AD1DEPŒ1AEPŒD1jFTDEPŒ1ADT; and, sinceDTisFT-measurable, it follows that
DT D dQ dPjFT
:
Let us prove the last assertion. For every" > 0, set T"Dinfft0WDt< "g
and note thatT"is a stopping time as the first hitting time of an open set by a càdlàg process (recall Proposition3.9and the fact that the filtration is right-continuous).
Then, noting that the eventfT"<1gisFT"-measurable, Q.T"<1/DEPŒ1fT"<1gDT""
sinceDT""onfT"<1gby the right-continuity of sample paths. It immediately follows that
Q \1
nD1
fT1=n<1g D0
and sincePis absolutely continuous with respect toQwe have also P
\1
nD1
fT1=n<1g D0:
But this exactly means that,Pa.s., there exists an integern1such thatT1=n D 1,
giving the last assertion of the proposition. ut
Proposition 5.21 Let D be a continuous local martingale taking (strictly) positive values. There exists a unique continuous local martingale L such that
DtDexp
Lt1 2hL;Lit
DE.L/t:
Moreover, L is given by the formula LtDlogD0C
Z t
0 Ds1dDs:
134 5 Stochastic Integration Proof Uniqueness is an easy consequence of Theorem4.8. Then, since D takes positive values, we can apply Itô’s formula to logDt (see the remark before Proposition5.11), and we get
logDtDlogD0C Z t
0
dDs
Ds
1 2
Z t 0
dhD;Dis
D2s DLt1 2hL;Lit;
whereLis as in the statement. ut
We now state the main theorem of this section, which explains the relation between continuous local martingales under P and continuous local martingales underQ.
Theorem 5.22 (Girsanov) Assume that the probability measures P and Q are mutually absolutely continuous onF1. Let.Dt/t0be the martingale with càdlàg sample paths such that, for every t0,
DtD dQ dPjFt
:
Assume that D has continuous sample paths, and let L be the unique continuous local martingale such that DtDE.L/t. Then, if M is a continuous local martingale under P, the process
MQ DM hM;Li is a continuous local martingale under Q.
Remark By consequences of the martingale representation theorem explained at the end of the previous section, the continuity assumption for the sample paths of Dalways holds when.Ft/is the (completed) canonical filtration of a Brownian motion. In applications of Theorem5.22, one often starts from the martingale.Dt/ to define the probability measureQ, so that the continuity assumption is satisfied by construction (see the examples in the next section).
Proof The fact that Dt can be written in the form Dt D E.L/t follows from Proposition5.21(we are assuming thatDhas continuous sample paths, and we also know from Proposition5.20thatDtakes positive values). Then, letT be a stopping time and letXbe an adapted process with continuous sample paths. We claim that, if.XD/Tis a martingale underP, thenXTis a martingale underQ. Let us verify the claim. By Proposition5.20,EQŒjXT^tjD EPŒjXT^tDT^tj <1, and it follows that XtT 2L1.Q/. Then, letA2Fsands<t. SinceA\ fT >sg 2Fs, we have, using the fact that.XD/Tis a martingale underP,
EPŒ1A\fT>sgXT^tDT^tDEPŒ1A\fT>sgXT^sDT^s:
5.5 Girsanov’s Theorem 135
By Proposition5.20,
DT^tD dQ dPjFT^t
; DT^sD dQ dPjFT^s
;
and thus, sinceA\ fT>sg 2FT^sFT^t, it follows that EQŒ1A\fT>sgXT^tDEQŒ1A\fT>sgXT^s: On the other hand, it is immediate that
EQŒ1A\fTsgXT^tDEQŒ1A\fTsgXT^s:
By combining with the previous display, we haveEQŒ1AXT^tDEQŒ1AXT^s;giving our claim. As a consequence of the claim, we get that, ifXDis a continuous local martingale underP, thenXis a continuous local martingale underQ.
Next letM be a continuous local martingale under P, and let MQ be as in the statement of the theorem. We apply the preceding observation toX D QM, noting that, by Itô’s formula,
MQtDt DM0D0C Z t
0
MQsdDsC Z t
0 DsdMs Z t
0 DsdhM;LisC hM;Dit
DM0D0C Z t
0
MQsdDsC Z t
0 DsdMs
since dhM;LisDDs1dhM;Disby Proposition5.21. We get thatMDQ is a continuous local martingale underP, and thusMQ is a continuous local martingale underQ. ut Consequences
(a) A process M which is a continuous local martingale under P remains a semimartingale underQ, and its canonical decomposition under Q is M D MChMQ ;Li(recall that the notion of a finite variation process does not depend on the underlying probability measure). It follows that the class of semimartingales underPis contained in the class of semimartingales underQ.
In fact these two classes are equal. Indeed, under the assumptions of Theorem 5.22, P and Q play symmetric roles, since the Radon–Nikodym derivative ofPwith respect toQon the-fieldFtisDt 1, which has continuous sample paths ifDdoes.
We may furthermore notice that Dt1Dexp
LtC hL;Lit1 2hL;Lit
Dexp
QLt1 2hQL;LiQ t
DE.QL/t;
136 5 Stochastic Integration
whereLQ D L hL;Liis a continuous local martingale underQ, andhQL;Li DQ hL;Li. So, under the assumptions of Theorem5.22, the roles ofPandQcan be interchanged providedDis replaced byD1andLis replaced byQL.
(b) LetXandYbe two semimartingales (underPor underQ). The brackethX;Yi is the same underPand underQ. In fact this bracket is given in both cases by the approximation of Proposition4.21(this observation was used implicitly in (a) above).
Similarly, if H is a locally bounded progressive process, the stochastic integralHX is the same underP and underQ. To see this it is enough to consider the case whenX D M is a continuous local martingale (underP).
Write.HM/P for the stochastic integral underPand.HM/Q for the one underQ. By linearity,
.H QM/PD.HM/PH hM;Li D.HM/P h.HM/P;Li; and Theorem5.22shows that.H QM/Pis a continuous local martingale underQ.
Furthermore the bracket of this continuous local martingale with any continuous local martingaleNunderQis equal toH hM;Ni DH h QM;Ni, and it follows from Theorem5.6that.H QM/PD.H QM/Qhence also.HM/P D.HM/Q.
With the notation of Theorem5.22, set MQ D GQP.M/. ThenGQP maps the set of allP-continuous local martingales onto the set of allQ-continuous local martingales. One easily verifies, using the remarks in (a) above, thatGPQıGQP D Id. Furthermore, the mappingGQPcommutes with the stochastic integral: ifHis a locally bounded progressive process,HGQP.M/DGQP.HM/.
(c) Suppose thatMDBis an.Ft/-Brownian motion underP, thenBQ DB hB;Li is a continuous local martingale underQ, with quadratic variationh QB;BiQ t D hB;Bit D t. By Theorem5.12, it follows thatBQ is an.Ft/-Brownian motion underQ.
In most applications of Girsanov’s theorem, one constructs the probability measureQin the following way. Start from a continuous local martingaleLsuch thatL0D0andhL;Li1<1a.s. The latter condition implies that the limitL1WD limt!1Lt exists a.s. (see Exercise4.24). ThenE.L/t is a nonnegative continuous local martingale hence a supermartingale (Proposition4.7), which converges a.s.
toE.L/1 D exp.L1 12hL;Li1/, andEŒE.L/1 1by Fatou’s lemma. If the property
EŒE.L/1D1 (5.18)
holds, thenE.L/is a uniformly integrable martingale (by Fatou’s lemma again, one hasE.L/t EŒE.L/1 j Ft, but (5.18) implies that EŒE.L/1 D EŒE.L/0 D EŒE.L/t for every t 0). If we let Qbe the probability measure with density E.L/1with respect toP, we are in the setting of Theorem5.22, withDt DE.L/t. It is therefore very important to give conditions that ensure that (5.18) holds.
5.5 Girsanov’s Theorem 137 Theorem 5.23 Let L be a continuous local martingale such that L0 D0. Consider the following properties:
(i) EŒexp12hL;Li1 <1(Novikov’s criterion);
(ii) L is a uniformly integrable martingale, and EŒexp12L1 < 1(Kazamaki’s criterion);
(iii) E.L/is a uniformly integrable martingale.
Then,(i))(ii))(iii).
Proof (i))(ii) Property (i) implies thatEŒhL;Li1 < 1 hence also thatLis a continuous martingale bounded inL2(Theorem4.13). Then,
exp1
2L1D.E.L/1/1=2.exp.1
2hL;Li1//1=2 so that, by the Cauchy–Schwarz inequality,
EŒexp1
2L1 .EŒE.L/1/1=2.EŒexp.1
2hL;Li1//1=2 .EŒexp.1
2hL;Li1//1=2<1:
(ii))(iii) SinceLis a uniformly integrable martingale, Theorem3.22shows that, for any stopping timeT, we haveLT DEŒL1 jFT. Jensen’s inequality then gives
exp1
2LT EŒexp1
2L1jFT:
By assumption,EŒexp12L1 <1, which implies that the collection of all variables of the formEŒexp12L1 j FT, for any stopping timeT, is uniformly integrable.
The preceding bound then shows that the collection of all variables exp12LT, for any stopping timeT, is also uniformly integrable.
For0 <a< 1, setZ.ta/Dexp.1aLCat /. Then, one easily verifies that E.aL/tD.E.L/t/a2.Zt.a//1a2:
If 2F andT is a stopping time, Hölder’s inequality gives
EŒ1 E.aL/TEŒE.L/Ta2EŒ1 ZT.a/1a2EŒ1 ZT.a/1a2EŒ1 exp1
2LT2a.1a/: In the second inequality, we used the propertyEŒE.L/T 1, which holds by Proposition3.25becauseE.L/is a nonnegative supermartingale andE.L/0 D 1.
In the third inequality, we use Jensen’s inequality, noting that 12Caa > 1. Since the
138 5 Stochastic Integration
collection of all variables of the form exp12LT, for any stopping timeT, is uniformly integrable, the preceding display shows that so is the collection of all variables E.aL/Tfor any stopping timeT. By the definition of a continuous local martingale, there is an increasing sequenceTn " 1of stopping times, such that, for everyn, E.aL/t^Tnis a martingale. If0st, we can use uniform integrability to pass to the limitn ! 1in the equalityEŒE.aL/t^Tn jFs D E.aL/s^Tn and we get that E.aL/is a uniformly integrable martingale. It follows that
1DEŒE.aL/1EŒE.L/1a2EŒZ1.a/1a2 EŒE.L/1a2EŒexp1
2L12a.1a/; using again Jensen’s inequality as above. Whena !1, this givesEŒE.L/1 1
henceEŒE.L/1D1. ut