4.1 Finite Variation Processes
4.1.1 Functions with Finite Variation
Chapter 4
Continuous Semimartingales
Continuous semimartingales provide the general class of processes with continuous sample paths for which we will develop the theory of stochastic integration in the next chapter. By definition, a continuous semimartingale is the sum of a continuous local martingale and a (continuous) finite variation process. In the present chapter, we study separately these two classes of processes. We start with some preliminaries about deterministic functions with finite variation, before considering the corresponding random processes. We then define (continuous) local martingales and we construct the quadratic variation of a local martingale, which will play a fundamental role in the construction of stochastic integrals. We explain how properties of a local martingale are related to those of its quadratic variation.
Finally, we introduce continuous semimartingales and their quadratic variation processes.
70 4 Continuous Semimartingales Definition 4.1 LetT 0. A continuous functionaWŒ0;T!Rsuch thata.0/D0 is said to havefinite variationif there exists a signed measureonŒ0;Tsuch that a.t/D.Œ0;t/for everyt2Œ0;T.
The measure is then determined uniquely by a. Since a is continuous and a.0/D0, it follows thathas no atoms.
Remark The general definition of a function with finite variation does not require continuity nor the condition a.0/ D 0. We impose these two conditions for convenience.
The decomposition ofas a difference of two finite positive measures onŒ0;T is not unique, but there exists a unique decomposition D C such that C and are supported on disjoint Borel sets. To get the existence of such a decomposition, start from an arbitrary decompositionD12, set D1C2
and then use the Radon–Nikodym theorem to find two nonnegative Borel functions h1andh2onŒ0;Tsuch that
1.dt/Dh1.t/.dt/; 2.dt/Dh2.t/.dt/:
Then, ifh.t/Dh1.t/h2.t/, we have
.dt/Dh.t/.dt/Dh.t/C.dt/h.t/.dt/;
which gives the decompositionDCwithC.dt/Dh.t/C.dt/,.dt/D h.t/.dt/, and the measuresCandare supported respectively on the disjoint Borel setsDC D ft W h.t/ > 0gandD D ftW h.t/ < 0g. The uniqueness of this decomposition D C follows from the fact that we have necessarily, for everyA2B.Œ0;T/,
C.A/Dsupf.C/WC2B.Œ0;T/; CAg:
We writejjfor the (finite) positive measurejj DCC. The measurejj is called thetotal variationofa. We havej.A/j jj.A/for everyA2B.Œ0;T/.
Moreover, the Radon–Nikodym derivative ofwith respect tojjis d
djj D1DC1D:
The fact that a.t/ D C.Œ0;t/ .Œ0;t/ shows that a is the difference of two monotone nondecreasing continuous functions that vanish at 0 (since has no atoms, the same holds for C of). Conversely, the difference of two monotone nondecreasing continuous functions that vanish at0has finite variation in the sense of the previous definition. Indeed, this follows from the well-known fact that the formulag.t/D .Œ0;t/,t 2 Œ0;Tinduces a bijection between monotone nondecreasing right-continuous functionsg W Œ0;T ! RC and finite positive measuresonŒ0;T.
4.1 Finite Variation Processes 71 Letf WŒ0;T!Rbe a measurable function such thatR
Œ0;Tjf.s/j jj.ds/ <1.
We set
Z T
0 f.s/da.s/D Z
Œ0;Tf.s/ .ds/;
Z T
0 f.s/jda.s/j D Z
Œ0;Tf.s/jj.ds/:
Then the bound
ˇˇˇˇZ T
0 f.s/da.s/ˇˇ ˇˇ
Z T 0
jf.s/j jda.s/j
holds. By restricting a to Œ0;t (which amounts to restricting , C, ), we can define Rt
0f.s/da.s/ for every t 2 Œ0;T, and we observe that the function t 7!Rt
0f.s/da.s/also has finite variation onŒ0;T(the associated measure is just 0.ds/Df.s/.ds/).
Proposition 4.2 For every t2.0;T, Z t
0 jda.s/j Dsup ( p
X
iD1
ja.ti/a.ti1/j )
;
where the supremum is over all subdivisions0 D t0 < t1 < < tp D t ofŒ0;t.
More precisely, for any increasing sequence0 D tn0 < tn1 < < tpnn D t of subdivisions ofŒ0;twhose mesh tends to0, we have
n!1lim
pn
X
iD1
ja.tni/a.tin 1/j D Z t
0 jda.s/j:
Remark In the usual presentation of functions with finite variation, one starts from the property that the supremum in the first display of the proposition is finite.
Proof Clearly, it is enough to treat the caset D T. The inequality in the first assertion is very easy since, for any subdivision0 D t0 < t1 < < tp D T of Œ0;T,
ja.ti/a.ti1/j D j..ti1;ti/j jj..ti1;ti/; 8i2 f1; : : : ;pg; and
Xp iD1
jj..ti1;ti/D jj.Œ0;t/D Z T
0 jda.s/j:
72 4 Continuous Semimartingales In order to get the reverse inequality, it suffices to prove the second assertion. So we consider an increasing sequence0 D tn0 < tn1 < < tnpn D T of subdivisions ofŒ0;T, whose mesh maxftintni1 W 1 i pngtends to 0. Although we are proving a “deterministic” result, we will use a martingale argument. Leaving aside the trivial case wherejj D0, we introduce the probability space˝DŒ0;T, which is equipped with the Borel -field B D B.Œ0;T/ and the probability measure P.ds/D.jj.Œ0;T//1jj.ds/. On this probability space, we consider the discrete filtration.Bn/n0such that, for every integern0,Bnis the-field generated by the intervals.tin 1;tni,1ipn. We then set
X.s/D1DC.s/1D.s/D d djj.s/;
and, for everyn0,
XnDEŒXjBn:
Properties of conditional expectation show that Xn is constant on every interval .tni1;tinand takes the value
..tni1;tni/
jj..tni1;tni/ D a.tni/a.tin 1/ jj..tni1;tni/
on this interval. On the other hand, the sequence.Xn/is a closed martingale, with respect to the discrete filtration.Bn/. SinceX is measurable with respect toB D W
nBn, this martingale converges toXinL1, by the convergence theorem for closed discrete martingales (see AppendixA2). In particular,
n!1lim EŒjXnjDEŒjXjD1;
where the last equality is clear sincejX.s/j D 1,jj.ds/a.e. The desired result follows by noting that
EŒjXnjD.jj.Œ0;T//1
pn
X
iD1
ja.tni/a.tni1/j;
and recalling thatjj.Œ0;T/DRT
0 jda.s/j. ut
We now give a useful approximation lemma for the integral of a continuous function with respect to a function with finite variation.
4.1 Finite Variation Processes 73 Lemma 4.3 If f WŒ0;T!Ris a continuous function, and if0Dtn0<t1n< <
tnpn DT is a sequence of subdivisions ofŒ0;Twhose mesh tends to0, we have Z T
0 f.s/da.s/D lim
n!1 pn
X
iD1
f.tni1/ .a.tin/a.tni1//:
Proof Letfnbe defined onŒ0;Tbyfn.s/Df.tni1/ifs2.tni1;tni,1ipn, and fn.0/Df.0/. Then,
pn
X
iD1
f.tin 1/ .a.tin/a.tni1//D Z
Œ0;Tfn.s/ .ds/;
and the desired result follows by dominated convergence sincefn.s/ ! f.s/as n! 1, for everys2Œ0;T. ut We say that a functionaWRC !Ris a finite variation function onRC if the restriction ofatoŒ0;Thas finite variation onŒ0;T, for everyT > 0. Then there is a unique-finite (positive) measure onRCwhose restriction to every intervalŒ0;T is the total variation measure of the restriction ofatoŒ0;T, and we write
Z 1
0 f.s/jda.s/j
for the integral of a nonnegative Borel functionf onRCwith respect to this-finite measure. Furthermore, we can define
Z 1
0 f.s/da.s/D lim
T!1
Z T
0 f.s/da.s/2 .1;1/ for any real Borel functionf onRCsuch thatR1
0 jf.s/jjda.s/j<1.