90 4 Continuous Semimartingales
Exercises 91
By Theorem4.9,
n!1lim
pn
X
iD1
.Mtni Mtin1/2D hM;MitD hX;Xit;
in probability. On the other hand,
pn
X
iD1
.Atni Atni1/2
1ipsupn
jAtni Atin1jXpn
iD1
jAtni Atni1j
Z t
0
jdAsj
1ipsupn
jAtni Atni1j;
which tends to0a.s. whenn! 1by the continuity of sample paths ofA. The same argument shows that
ˇˇˇ
pn
X
iD1
.AtinAtni1/.Mtni Mtni1/ˇˇˇ Z t
0
jdAsj
1ipsupn
jMtni Mtni1j
tends to0a.s. ut
Exercises
In the following exercises, processes are defined on a probability space.˝;F;P/
equipped with a complete filtration.Ft/t2Œ0;1.
Exercise 4.22 Let Ube an F0-measurable real random variable, and let M be a continuous local martingale. Show that the processNt DUMtis a continuous local martingale. (This result was used in the construction of the quadratic variation of a continuous local martingale.)
Exercise 4.23
1. LetMbe a (true) martingale with continuous sample paths, such thatM0D0. We assume that.Mt/t0 is also a Gaussian process. Show that, for everyt 0and everys> 0, the random variableMtCsMtis independent of.Mr; 0rt/.
2. Under the assumptions of question 1., show that there exists a continuous monotone nondecreasing functionf W RC ! RC such that hM;Mit D f.t/
for everyt0.
92 4 Continuous Semimartingales
Exercise 4.24 LetMbe a continuous local martingale withM0 D0. 1. For every integern1, we setTnDinfft0W jMtj Dng. Show that, a.s.
n
t!1limMtexists and is finite oD
[1 nD1
fTnD 1g fhM;Mi1<1g:
2. We setSnDinfft0W hM;MitDngfor everyn1. Show that, a.s., fhM;Mi1<1g D
[1 nD1
fSn D 1g n
t!1lim Mtexists and is finite o;
and conclude that n
t!1lim Mtexists and is finite
oD fhM;Mi1<1g ; a.s.
Exercise 4.25 For every integern 1, letMn D .Mtn/t0 be a continuous local martingale withM0D0. We assume that
n!1limhMn;Mni1D0 in probability.
1. Let" > 0, and, for everyn1, let
T"n Dinfft0W hMn;Mnit"g:
Justify the fact thatT"nis a stopping time, then prove that the stopped continuous local martingale
Mnt;"DMt^Tn "n ; 8t0 ; is a true martingale bounded inL2.
2. Show that
E h
sup
t0jMtn;"j2i 4 "2:
3. Writing, for everya> 0, P
sup
t0jMtnj a P
sup
t0jMtn;"j a
CP.T"n<1/;
Exercises 93
show that
n!1lim
sup
t0jMntj D0
in probability.
Exercise 4.26
1. LetAbe an increasing process (adapted, with continuous sample paths and such thatA0D0) such thatA1<1a.s., and letZbe an integrable random variable.
We assume that, for every stopping timeT,
EŒA1ATEŒZ1fT<1g:
Show, by introducing an appropriate stopping time, that, for every > 0, EŒ.A1/1fA1>gEŒZ1fA1>g:
2. Letf WRC!Rbe a continuously differentiable monotone increasing function such thatf.0/D 0and setF.x/D Rx
0 f.t/dtfor everyx 0. Show that, under the assumptions of question 1., one has
EŒF.A1/EŒZ f.A1/:
(Hint: It may be useful to observe thatF.x/D xf.x/Rx
0 f0./dfor every x0.)
3. LetM be a (true) martingale with continuous sample paths and bounded inL2 such thatM0 D 0, and letM1be the almost sure limit ofMt ast! 1. Show that the assumptions of question 1. hold whenAtD hM;MitandZDM12 . Infer that, for every realq1,
EŒ.hM;Mi1/qC1.qC1/EŒ.hM;Mi1/qM12 : 4. Letp2be a real number such thatEŒ.hM;Mi1/p <1. Show that
EŒ.hM;Mi1/pppEŒjM1j2p:
5. LetNbe a continuous local martingale such thatN0D0, and letTbe a stopping time such that the stopped martingaleNTis uniformly integrable. Show that, for every realp2,
EŒ.hN;NiT/pppEŒjNTj2p:
Give an example showing that this result may fail if NT is not uniformly integrable.
94 4 Continuous Semimartingales Exercise 4.27 Let .Xt/t0 be an adapted process with continuous sample paths and taking nonnegative values. Let .At/t0 be an increasing process (adapted, with continuous sample paths and such thatA0 D 0). We consider the following condition:
(D) For every bounded stopping timeT, we haveEŒXTEŒAT.
1. Show that, ifMis a square integrable martingale with continuous sample paths andM0 D0, the condition (D) holds forXtDMt2andAtD hM;Mit.
2. Show that the conclusion of the previous question still holds if one only assumes thatMis a continuous local martingale withM0D0.
3. We setXt DsupstXs. Show that, under the condition (D), we have, for every bounded stopping timeSand everyc> 0,
P.XS c/ 1 cEŒAS:
(Hint: One may apply (D) toT DS^R, whereRDinfft0WXtcg.) 4. Infer that, still under the condition (D), one has, for every (finite or not) stopping
timeS,
P.XS>c/ 1 cEŒAS
(whenStakes the value1, we of course defineX1 Dsups0Xs).
5. Letc> 0andd > 0, andS D infft 0 W At dg. LetT be a stopping time.
Noting that
fXT>cg
fXT^S>cg [ fAT dg show that, under the condition (D), one has
P.XT>c/ 1
cEŒAT^dCP.ATd/:
6. Use questions (2) and (5) to verify that, if M.n/ is a sequence of continuous local martingales andT is a stopping time such thathM.n/;M.n/iT converges in probability to0asn! 1, then,
n!1lim
supsTjM.sn/j
D0 ; in probability.
Notes and Comments 95
Notes and Comments
The book [14] of Dellacherie and Meyer is again an excellent reference for the topics of this chapter, in the more general setting of local martingales and semimartingales with càdlàg sample paths. See also [72] and [49] (in particular, a discussion of the elementary theory of finite variation processes can be found in [72]). The notion of a local martingale appeared in Itô and Watanabe [43] in 1965.
The notion of a semimartingale seems to be due to Fisk [25] in 1965, who used the name “quasimartingales”. See also Meyer [60]. The classical approach to the quadratic variation of a continuous (local) martingale is based on the Doob–Meyer decomposition theorem [58], see e.g. [49]. Our more elementary presentation is inspired by [70].
Chapter 5
Stochastic Integration
This chapter is at the core of the present book. We start by defining the stochastic integral with respect to a square-integrable continuous martingale, considering first the integral of elementary processes (which play a role analogous to step functions in the theory of the Riemann integral) and then using an isometry between Hilbert spaces to deal with the general case. It is easy to extend the definition of stochastic integrals to continuous local martingales and semimartingales. We then derive the celebrated Itô’s formula, which shows that the image of one or several continuous semimartingales under a smooth function is still a continuous semimartingale, whose canonical decomposition is given in terms of stochastic integrals. Itô’s formula is the main technical tool of stochastic calculus, and we discuss several important applications of this formula, including Lévy’s theorem characterizing Brownian motion as a continuous local martingale with quadratic variation process equal to t, the Burkholder–Davis–Gundy inequalities and the representation of martingales as stochastic integrals in a Brownian filtration. The end of the chapter is devoted to Girsanov’s theorem, which deals with the stability of the notions of a martingale and a semimartingale under an absolutely continuous change of probability measure. As an application of Girsanov’s theorem, we establish the famous Cameron–Martin formula giving the image of the Wiener measure under a translation by a deterministic function.