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
5.6 A Few Applications of Girsanov’s Theorem 139 We can restate the latter property by saying that, under the probability measure Q, there exists an.Ft/-Brownian motionˇsuch that the processX DBsolves the stochastic differential equation
dXtDdˇtCb.t;Xt/dt:
This equation is of the type that will be considered in Chap.7below, but in contrast with the statements of this chapter, we are not making any regularity assumption on the functionb. It is remarkable that Girsanov’s theorem allows one to construct solutions of stochastic differential equations without regularity conditions on the coefficients.
The Cameron–Martin formula We now specialize the preceding discussion to the case whereb.t;x/does not depend onx. We assume thatb.t;x/D g.t/, where g2L2.RC;B.RC/;dt/, and we also set, for everyt0,
h.t/D Z t
0 g.s/ds:
The setH of all functionshthat can be written in this form is called the Cameron–
Martin space. Ifh 2H, we sometimes writehP D gfor the associated function in L2.RC;B.RC/;dt/(this is the derivative ofhin the sense of distributions).
As a special case of the previous discussion, under the probability measure QWDD1PDexp
Z 1
0 g.s/dBs1 2
Z 1 0 g.s/2ds
P;
the processˇt WD Bth.t/is a Brownian motion. Hence, for every nonnegative measurable function˚onC.RC;R/,
EPŒD1˚..Bt/t0/DEQŒ˚..Bt/t0/DEQŒ˚..ˇtCh.t//t0/ DEPŒ˚..BtCh.t//t0/:
The equality between the two ends of the last display is the Cameron–Martin formula. In the next proposition, we write this formula in the special case of the canonical construction of Brownian motion on the Wiener space (see the end of Sect.2.2).
Proposition 5.24 (Cameron–Martin formula) Let W.dw/be the Wiener measure on C.RC;R/, and let h be a function in the Cameron–Martin spaceH. Then, for every nonnegative measurable function˚on C.RC;R/,
Z
W.dw/ ˚.wCh/D Z
W.dw/exp Z 1
0
h.s/P dw.s/1 2
Z 1 0
h.s/P 2ds ˚.w/:
140 5 Stochastic Integration
Remark The integralR1
0 hP.s/dw.s/is a stochastic integral with respect to w.s/ (which is a Brownian motion underW.dw/), but it can also be viewed as a Wiener integral since the functionhP.s/is deterministic. The Cameron–Martin formula can be established by Gaussian calculations that do not involve stochastic integrals or Girsanov’s theorem (see e.g. Chapter 1 of [62]). Still it is instructive to derive this formula as a special case of Girsanov’s theorem.
The Cameron–Martin formula gives a “quasi-invariance” property of Wiener measure under the translations by functions of the Cameron–Martin space: The image of Wiener measureW.dw/under the mapping w7!wChhas a density with respect toW.dw/and this density is the terminal value of the exponential martingale associated with the martingaleRt
0hP.s/dw.s/.
Law of hitting times for Brownian motion with drift LetBbe a real Brownian motion withB0D0, and for everya> 0, letTa WDinfft0WBtDag. Ifc2Ris given, we aim at computing the law of the stopping time
UaWDinfft0WBtCctDag:
Of course, if c D 0, we haveUa D Ta, and the desired distribution is given by Corollary2.22. Girsanov’s theorem (or rather the Cameron–Martin formula) will allow us to derive the case wherecis arbitrary from the special casecD0.
Fixt> 0and apply the Cameron–Martin formula with h.s/P Dc1fstg ; h.s/Dc.s^t/ ; and, for every w2C.RC;R/,
˚.w/D1fmaxŒ0;tw.s/ag: It follows that
P.Uat/DEŒ˚.BCh/
DE
h˚.B/exp Z 1
0
h.s/P dBs1 2
Z 1 0
h.s/P 2ds i
DEŒ1fTatgexp.cBtc2 2t//
DEŒ1fTatgexp.cBt^Tac2
2.t^Ta//
DEŒ1fTatgexp.cac2 2Ta/
Exercises 141
D Z t
0 ds a
p2s3ea22s ecac22s D
Z t
0 ds a
p2s3e21s.acs/2;
where, in the fourth equality, we used the optional stopping theorem (Corol- lary3.23) to write
EŒexp.cBtc2
2t/jFt^TaDexp.cBt^Tac2
2.t^Ta//;
and we also made use of the explicit density ofTa given in Corollary 2.22. This calculation shows that the variableUahas a density onRCgiven by
.s/D a
p2s3e21s.acs/2: By integrating this density, we can verify that
P.Ua<1/D
1 ifc0;
e2caifc0;
which may also be checked more easily by applying the optional stopping theorem to the continuous martingale exp.2c.BtCct//.
Exercises
In the following exercises, processes are defined on a probability space.˝;F;P/ equipped with a complete filtration.Ft/t2Œ0;1.
Exercise 5.25 LetBbe an.Ft/-Brownian motion withB0 D 0, and letHbe an adapted process with continuous sample paths. Show thatB1
t
Rt
0HsdBsconverges in probability whent!0and determine the limit.
Exercise 5.26
1. LetBbe a one-dimensional.Ft/-Brownian motion withB0 D0. Letf be a twice continuously differentiable function onR, and letgbe a continuous function on R. Verify that the process
Xt Df.Bt/exp
Z t
0 g.Bs/ds
142 5 Stochastic Integration is a semimartingale, and give its decomposition as the sum of a continuous local martingale and a finite variation process.
2. Prove thatXis a continuous local martingale if and only if the functionf satisfies the differential equation
f00D2g f:
3. From now on, we suppose in addition thatgis nonnegative and vanishes outside a compact subinterval of.0;1/. Justify the existence and uniqueness of a solution f1of the equationf00 D 2g f such thatf1.0/D 1andf10.0/D 0. Leta > 0and TaDinfft0WBtDag. Prove that
E h
exp
Z Ta
0 g.Bs/ds
iD 1 f1.a/:
Exercise 5.27 (Stochastic calculus with the supremum) Preliminary question. Let mWRC !Rbe a continuous function such thatm.0/D0, and letsWRC !R be the monotone increasing function defined by
s.t/D sup
0rtm.r/:
Show that, for every bounded Borel functionhonRand everyt> 0, Z t
0 .s.r/m.r//h.r/ds.r/D0:
(One may first observe thatR
1I.r/ds.r/D0for every open intervalIthat does not intersectfr0Ws.r/Dm.r/g.)
1. LetMbe a continuous local martingale such thatM0 D 0, and for everyt 0, let
StD sup
0rtMr:
Let' W RC ! Rbe a twice continuously differentiable function. Justify the equality
'.St/D'.0/C Z t
0 '0.Ss/dSs:
Exercises 143
2. Show that
.StMt/ '.St/D˚.St/ Z t
0 '.Ss/dMs
where˚.x/DRx
0 '.y/dyfor everyx2R.
3. Infer that, for every > 0,
eStC.StMt/eSt is a continuous local martingale.
4. Leta> 0andT Dinfft0WStMtDag. We assume thathM;Mi1D 1a.s.
Show thatT <1a.s. andSTis exponentially distributed with parameter1=a.
Exercise 5.28 LetBbe an.Ft/-Brownian motion started from1. We fix"2.0; 1/
and setT"Dinfft0WBtD"g. We also let > 0and˛2Rnf0g.
1. Show thatZt D.Bt^T"/˛is a semimartingale and give its canonical decomposi- tion as the sum of a continuous local martingale and a finite variation process.
2. Show that the process
ZtD.Bt^T"/˛ exp
Z t^T"
0
ds B2s
is a continuous local martingale if˛andsatisfy a polynomial equation to be determined.
3. Compute
E h
exp
Z T"
0
ds B2s
i:
Exercise 5.29 Let .Xt/t0 be a semimartingale. We assume that there exists an .Ft/-Brownian motion.Bt/t0started from0and a continuous functionbWR! R, such that
XtDBtC Z t
0 b.Xs/ds:
1. LetF W R ! Rbe a twice continuously differentiable function onR. Show that, forF.Xt/to be a continuous local martingale, it suffices thatFsatisfies a second-order differential equation to be determined.
2. Give the solution of this differential equation which is such thatF.0/D 0and F0.0/ D 1. In what follows,Fstands for this particular solution, which can be written in the formF.x/ D Rx
0 exp.2ˇ.y//dy, with a functionˇ that will be determined in terms ofb.
144 5 Stochastic Integration 3. In this question only, we assume thatbis integrable, i.e.R
Rjb.x/jdx<1.
(a) Show that the continuous local martingaleMtDF.Xt/is a martingale.
(b) Show thathM;Mi1D 1a.s.
(c) Infer that
lim sup
t!1 XtD C1; lim inf
t!1 XtD 1; a.s.
4. We come back to the general case. Letc< 0andd> 0, and TcDinfft0WXtcg; TdDinfft0WXtdg:
Show that, on the eventfTc^Td D 1g, the random variablesjBnC1Bnj, for integersn0, are bounded above by a (deterministic) constant which does not depend onn. Infer thatP.Tc^TdD 1/D0.
5. ComputeP.Tc<Td/in terms ofF.c/ofF.d/.
6. We assume thatbvanishes on.1; 0and that there exists a constant˛ > 1=2 such thatb.x/˛=xfor everyx1. Show that, for every" > 0, one can choose c< 0such that
P.Tn<Tc; for everyn1/1":
Infer thatXt ! C1ast ! 1, a.s. (Hint: Observe that the continuous local martingaleMt^Tcis bounded.)
7. Suppose nowb.x/D1=.2x/for everyx1. Show that lim inf
t!1 XtD 1; a.s.
Exercise 5.30 (Lévy area) Let .Xt;Yt/t0 be a two-dimensional.Ft/-Brownian motion started from0. We set, for everyt0:
AtD Z t
0 XsdYs Z t
0 YsdXs (Lévy’s area):
1. ComputehA;Aitand infer that.At/t0is a square-integrable (true) martingale.
2. Let > 0. Justify the equality
EŒeiAtDEŒcos.At/:
Exercises 145 3. Letf be a twice continuously differentiable function onRC. Give the canonical
decomposition of the semimartingales ZtDcos.At/;
WtD f0.t/
2 .Xt2CYt2/Cf.t/:
Verify thathZ;WitD0.
4. Show that, for the processZteWt to be a continuous local martingale, it suffices thatf solves the differential equation
f00.t/Df0.t/22: 5. Letr> 0. Verify that the function
f.t/D log cosh..rt//
solves the differential equation of question4.and derive the formula EŒeiArD 1
cosh.r/:
Exercise 5.31 (Squared Bessel processes) Let B be an .Ft/-Brownian motion started from0, and letXbe a continuous semimartingale. We assume thatXtakes values inRC, and is such that, for everyt0,
XtDxC2 Z t
0
pXsdBsC˛t
wherexand˛are nonnegative real numbers.
1. Letf WRC !RCbe a continuous function, and let' be a twice continuously differentiable function onRC, takingstrictly positivevalues, which solves the differential equation
'00D2f'
and satisfies'.0/D1and'0.1/D0. Observe that the function'must then be decreasing over the intervalŒ0; 1.
We set
u.t/D '0.t/
2'.t/
146 5 Stochastic Integration
for everyt0. Verify that we have, for everyt0, u0.t/C2u.t/2Df.t/;
then show that, for everyt0, u.t/Xt
Z t
0 f.s/XsdsDu.0/xC Z t
0 u.s/dXs2 Z t
0 u.s/2Xsds:
We set
YtDu.t/Xt Z t
0 f.s/Xsds: 2. Show that, for everyt0,
'.t/˛=2eYt DE.N/t
whereE.N/t D exp.Nt12hN;Nit/denotes the exponential martingale associ- ated with the continuous local martingale
NtDu.0/xC2 Z t
0 u.s/p XsdBs: 3. Infer from the previous question that
E h
exp
Z 1
0 f.s/Xsds
iD'.1/˛=2exp.x 2'0.0//:
4. Let > 0. Show that E
h exp
Z 1
0 Xsds
iD.cosh.p
2//˛=2 exp.x 2
p2tanh.p 2//:
5. Show that, ifˇD.ˇt/t0is a real Brownian motion started fromy, one has, for every > 0,
E h
exp
Z 1
0 ˇs2ds
iD.cosh.p
2//1=2 exp.y2 2
p2 tanh.p 2//:
Exercise 5.32 (Tanaka’s formula and local time) Let B be an .Ft/-Brownian motion started from0. For every" > 0, we define a function g" W R ! Rby settingg".x/Dp
"Cx2.
Exercises 147
1. Show that
g".Bt/Dg".0/CMt"CA"t
whereM"is a square integrable continuous martingale that will be identified in the form of a stochastic integral, andA"is an increasing process.
2. We set sgn.x/D1fx>0g1fx<0gfor everyx2R. Show that, for everyt0, Mt"!L2
"!0
Z t
0 sgn.Bs/dBs:
Infer that there exists an increasing processLsuch that, for everyt0, jBtj D
Z t
0 sgn.Bs/dBsCLt:
3. Observing thatA"t ! Ltwhen " !0, show that, for everyı > 0, for every choice of0 <u < v, the condition (jBtj ı for everyt 2 Œu; v) a.s. implies thatLv D Lu. Infer that the functiont 7!Ltis a.s. constant on every connected component of the open setft0WBt6D0g.
4. We set ˇt D Rt
0sgn.Bs/dBs for every t 0. Show that .ˇt/t0 is an .Ft/- Brownian motion started from0.
5. Show thatLtDsup
st.ˇs/, a.s. (In order to derive the boundLtsup
st.ˇs/, one may consider the last zero ofBbefore timet, and use question3.) Give the law ofLt.
6. For every" > 0, we define two sequences of stopping times.S"n/n1and.Tn"/n1, by setting
S"1D0 ; T1"Dinfft0W jBtj D"g and then, by induction,
S"nC1DinfftTn"WBtD0g; TnC" 1DinfftS"nC1W jBtj D"g: For everyt0, we setNt"Dsupfn1WTn"tg, where sup¿D0. Show that
"N"t L2
!"!0Lt: (One may observe that
LtC Z t
0
X1
nD1
1ŒS"n;T"n.s/
sgn.Bs/dBsD"Nt"Crt"
where the “remainder”rt"satisfiesjr"tj ".)
148 5 Stochastic Integration
7. Show that Nt1=p
t converges in law as t ! 1 tojUj, where U is N .0; 1/- distributed.
(Many results of Exercise5.32are reproved and generalized in Chap.8.)
Exercise 5.33 (Study of multidimensional Brownian motion) Let Bt D .B1t;B2t; : : : ;BNt / be an N-dimensional .Ft/-Brownian motion started from xD.x1; : : : ;xN/2RN. We suppose thatN2.
1. Verify thatjBtj2is a continuous semimartingale, and that the martingale part of jBtj2is a true martingale.
2. We set
ˇtD XN
iD1
Z t 0
Bis jBsjdBis with the convention that jBBis
sj D 0 if jBsj D 0. Justify the definition of the stochastic integrals appearing in the definition ofˇt, then show that the process .ˇt/t0is an.Ft/-Brownian motion started from0.
3. Show that
jBtj2D jxj2C2 Z t
0 jBsjdˇsCN t:
4. From now on, we assume thatx6D0. Let"2.0;jxj/andT"Dinfft0W jBtj
"g. We setf.a/D logaifN D 2, andf.a/D a2N ifN 3, for everya> 0.
Verify thatf.jBt^T"j/is a continuous local martingale.
5. LetR>jxjandSRDinfft0W jBtj Rg. Show that P.T"<SR/D f.R/f.jxj/ f.R/f."/ :
Observing thatP.T"<SR/!0when"!0, show thatBt6D0for everyt0, a.s.
6. Show that, a.s., for everyt0,
jBtj D jxj CˇtC N1 2
Z t 0
ds jBsj:
7. We assume thatN3. Show thatjBtj ! 1whent! 1, a.s. (Hint: Observe thatjBtj2Nis a nonnegative supermartingale.)
8. We assume N D 3. Using the form of the Gaussian density, verify that the collection of random variables.jBtj1/t0is bounded inL2. Show that.jBtj1/t0
is a continuous local martingale but is not a (true) martingale.
Notes and Comments 149 (Chapter7presents a slightly different approach to the results of this exercise, see in particular Proposition7.16.)
Exercise 5.34 (Application of the Cameron–Martin formula) Let B be an .Ft/- Brownian motion started from0. We setBt DsupfjBsj Wstgfor everyt0. 1. SetU1 D infft 0 W jBtj D 1g andV1 D infft U1 W Bt D 0g. Justify
the equalityP.BV1 < 2/D 1=2, and then show that one can find two constants
˛ > 0and > 0such that
P.V1 ˛;BV1 < 2/D > 0:
2. Show that, for every integern1,P.Bn˛ < 2/n.Hint: Construct a suitable sequenceV1;V2; : : :of stopping times such that, for everyn2,
P.Vnn˛ ;BVn < 2/P.Vn1.n1/˛ ;BVn1 < 2/:
Conclude that, for every" > 0andt0,P.Bt "/ > 0.
3. Lethbe a twice continuously differentiable function onRC such thath.0/D0, and letK > 0. Via a suitable application of Itô’s formula, show that there exists a constantAsuch that, for every" > 0,
ˇˇˇ Z K
0 h0.s/dBsˇˇˇA" a.s. on the eventfBK "g:
4. We setXtDBth.t/andXtDsupfjXsj Wstg. Infer from question3.that lim"#0
P.XK"/
P.BK"/ Dexp 1
2 Z K
0 h0.s/2ds :
Notes and Comments
The reader who wishes to learn more about the topics of this chapter is strongly advised to look at the excellent books by Karatzas and Shreve [49], Revuz and Yor [70] and Rogers and Williams [72]. A more concise introduction to stochastic integration can also be found in Chung and Williams [10].
Stochastic integrals with respect to Brownian motion were introduced by Itô [36]
in 1944. His motivation was to give a rigorous approach to the stochastic differential equations that govern diffusion processes. Doob [15] suggested to study stochastic integrals as martingales. Several authors then contributed to the theory, including Kunita and Watanabe [50] and Meyer [60]. We have chosen to restrict our attention to stochastic integration with respect to continuous semimartingales. The reader interested in the more general case of semimartingales with jumps can consult the
150 5 Stochastic Integration treatise of Dellacherie and Meyer [14] and the more recent books of Protter [63]
or Jacod and Shiryaev [44]. Itô’s formula was derived in [40] for processes that are stochastic integrals with respect to Brownian motions, and in our general context it appeared in the work of Kunita and Watanabe [50]. Theorem5.12, at least in the case dD1, is usually attributed to Lévy, although it seems difficult to find this statement in Lévy’s work (see however [54, Chapitre III]). Theorem5.13 showing that any continuous martingale can be written as a time-changed Brownian motion is due to Dambis [11] and Dubins–Schwarz [17]. The Burkholder–Davis–Gundy inequalities appear in [7], see also the expository article of Burkholder [6] for the history of these famous inequalities. Theorem5.18 goes back to Itô [39] – in the different form of the chaos decomposition of Wiener functionals – and was a great success of the theory of stochastic integration. This theorem and its numerous extensions have found many applications in the area of mathematical finance. Girsanov’s theorem appears in [29] in 1960, whereas the Cameron–Martin formula goes back to [8] in 1944. Applications of Girsanov’s theorem to stochastic differential equations are developed in the book [77] of Stroock and Varadhan. Exercise5.30 is concerned with the so-called Lévy area of planar Brownian motion, which was studied by Lévy [53,54] with a different definition. Exercise5.31 is inspired by Pitman and Yor [67].
Chapter 6
General Theory of Markov Processes
Our goal in this chapter is to give a concise introduction to the main ideas of the theory of continuous time Markov processes. Markov processes form a fundamental class of stochastic processes, with many applications in real life problems outside mathematics. The reason why Markov processes are so important comes from the so-called Markov property, which enables many explicit calculations that would be intractable for more general random processes. Although the theory of Markov processes is by no means the central topic of this book, it will play a significant role in the next chapters, in particular in our discussion of stochastic differential equations. In fact the whole invention of Itô’s stochastic calculus was motivated by the study of the Markov processes obtained as solutions of stochastic differential equations, which are also called diffusion processes.
This chapter is mostly independent of the previous ones, even though Brownian motion is used as a basic example, and the martingale theory developed in Chap.3 plays an important role. After a section dealing with the general definitions and the problem of existence, we focus on the particular case of Feller processes, and in that framework we introduce the key notion of the generator. We establish regularity properties of Feller processes as consequences of the analogous results for supermartingales. We then discuss the strong Markov property, and we conclude the chapter by presenting three important classes of Markov processes.