Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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510 9 General Processes: Mathematical Facts
Lemma 9.1.1.1 let X be a c`ad process, such that X
q
=0, P-a.s., ∀q ∈ Q
+
.
Then, the process (X
t
,t≥ 0) is indistinguishable from 0.
In contrast, here is a non-c`adl`ag process, with restriction to Q
+
equal to 0,
which is not indistinguishable from 0: take X
t
= N
t
− N
t−
,whereN is a
Poisson process.
9.1.2 Finite Variation Processes, Pure Jump Processes
When a process X is c`adl`ag, (i.e., continuous on the right and with limits
on the left), we denote by X
t
−
the left limit of X
s
when s → t, s < t,as
previously. The jump of X at time t is denoted by ΔX
t
= X
t
− X
t
−
,and
we shall call (ΔX
t
,t ≥ 0) the jump process of X. We recall (see Definition
1.1.10.2)thatanincreasingprocessA is a c`adl`ag process (A
t
,t≥ 0) equal to
0 at time 0, such that A
s
≤ A
t
for s ≤ t.IfA is an increasing process, there
exists a continuous increasing process A
c
and a purely discontinuous process
A
d
(i.e., A
d
is equal to the sum of its jumps: A
d
t
=
0<s≤t
ΔA
s
) such that
A = A
c
+ A
d
. This decomposition is obviously unique.
If U is a c`adl`ag function, the set {t : |U(t) − U(t−)| >a} is discrete for
each a>0andtheset{t : U (t) = U(t−)} is at most countable. Obviously,
the same property holds (almost surely) for c`adl`ag processes.
If moreover the c`adl`ag process U has finite variation on finite intervals
(see Definition 1.1.10.8), then for every t>0theseriesU
d
t
=
0<s≤t
ΔU
s
is
absolutely convergent, U
d
is c`adl`ag, it has finite variation on finite intervals
and U
c
= U −U
d
is continuous.
Definition 9.1.2.1 If U
c
=0, that is U
t
=
0<s≤t
ΔU
s
, the process U
is said to be a pure jump process. In general a finite variation U admits a
unique decomposition as U
t
= U
c
t
+
s≤t
ΔU
s
with U
c
continuous with finite
variation.
Let U be a c`adl`ag process with integrable variation. The Stieltjes
integral
∞
0
θ
s
dU
s
is defined for elementary processes θ, i.e., processes of
the form θ
s
= ϑ
a
1
]a,b]
(s) with ϑ
a
a r.v., by U
θ
t
:=
∞
0
θ
s
dU
s
= ϑ
a
(U
b
− U
a
)
and for θ such that
∞
0
|θ
s
||dU
s
| < ∞ by linearity and passage to the limit.
(Hence, the integral is defined path-by-path.) Then, one defines the integral
U
θ
t
=
t
0
θ
s
dU
s
=
]0,t]
θ
s
dU
s
=
∞
0
1
{]0,t]}
θ
s
dU
s
.
Note that the jump processes of U
θ
and U are related by:
ΔU
θ
t
= θ
t
ΔU
t
.
9.1 Some Basic Facts about c`adl`ag Processes 511
If U and V are two finite variation processes, the Stieltjes’ integration by
parts formula can be written as follows:
U
t
V
t
= U
0
V
0
+
]0,t]
V
s
dU
s
+
]0,t]
U
s
−
dV
s
(9.1.1)
= U
0
V
0
+
]0,t]
V
s
−
dU
s
+
]0,t]
U
s
−
dV
s
+
s≤t
ΔU
s
ΔV
s
.
As a partial check, one can verify that the jump process of the left-hand side,
i.e., U
t
V
t
− U
t
−
V
t
−
, is equal to the jump process of the right-hand side, i.e.,
V
t
−
ΔU
t
+ U
t
−
ΔV
t
+ ΔU
t
ΔV
t
.
We shall often write
t
0
V
s
dU
s
for
]0,t]
V
s
dU
s
.
Proposition 9.1.2.2 Let F be a C
2
function and U ac`adl`ag process with
finite variation. Then, the process F (U) is also c`adl`ag with finite variation
and
F (U
t
)=F (U
0
)+
t
0
F
(U
s
−
)dU
s
+
s≤t
[F (U
s
) − F (U
s
−
) − F
(U
s
−
)ΔU
s
]
= F (U
0
)+
t
0
F
(U
s
−
)dU
c
s
+
s≤t
[F (U
s
) − F (U
s
−
)] (9.1.2)
= F (U
0
)+
t
0
F
(U
s
)dU
c
s
+
s≤t
[F (U
s
) − F (U
s
−
)] ,
where U = U
c
+ U
d
is the decomposition of U into its continuous and
discontinuous parts.
Proof: TheresultistrueforF (x)=x and if the result holds for F ,by
integration by parts, it holds for xF . Hence, the result is true for polynomials
and for C
1
functions by Weierstrass approximation. The C
2
assumption on F
(a little bit too strong) is needed to ensure the convergence of the series.
As for continuous path semi-martingales, we introduce, besides the
ordinary exponential, the notion of the stochastic exponential in the form
of the solution to a linear stochastic equation.
Lemma 9.1.2.3 Let U be a c`adl`ag process with finite variation. The unique
solution of
dY
t
= Y
t
−
dU
t
,Y
0
= y
is the stochastic exponential of U (the Dol´eans-Dade exponential of U )equal
to
512 9 General Processes: Mathematical Facts
Y
t
= y exp(U
c
t
− U
c
0
)
s≤t
(1 + ΔU
s
) (9.1.3)
= y exp(U
t
− U
0
)
s≤t
(1 + ΔU
s
)e
−ΔU
s
. (9.1.4)
Proof: Applying the integration by parts formula to the right-hand side
of (9.1.3) shows that it is a solution to the equation dY
t
= Y
t
−
dU
t
.Asfor
the uniqueness, if Y
i
,i =1, 2 are two solutions, then, setting Z = Y
1
− Y
2
we get Z
t
=
t
0
Z
s−
dU
s
. Let us denote by M the running maximum of Z,
i.e.,M
t
:= sup
s≤t
|Z
s
|, then, if V
t
is the variation process of U
t
|Z
t
|≤M
t
V
t
which implies that
|Z
t
|≤M
t
t
0
V
s−
dV
s
= M
t
V
2
t
2
.
Iterating, we obtain |Z
t
|≤M
t
V
n
t
n!
and the uniqueness follows by letting
n →∞.
We will generalize later (see, e.g., Exercise 9.4.3.5 and Proposition
10.2.4.2) the stochastic exponential to semi-martingales.
Comment 9.1.2.4 At this point, for this class of processes, the second
formula (9.1.4) may not be so useful, but later on, in Subsection 9.4.3,
when U is a semi-martingale, we shall need this expression.
9.1.3 Some σ-algebras
Definition 9.1.3.1 Let F be a given filtration (called the reference filtration).
• The optional σ-algebra O is the σ-algebra on R
+
×Ω generated by c`adl`ag
F-adapted processes (considered as mappings on R
+
× Ω).
• The predictable σ-algebra P is generated by the F-adapted c`ag (or
continuous) processes. The inclusion P⊂Oholds.
These two σ-algebras P and O are equal if all F-martingales are continuous,
in particular if F is a strong (or even weak) Brownian filtration. In general
O = P∨σ(ΔM, M describing the set of martingales) .
The optional and predictable σ-algebras were defined with the help of stopping
times in Subsection 1.2.3.
A process is said to be predictable (resp. optional) if it is measurable
with respect to the predictable (resp. optional) σ-field. If X is a predictable
9.2 Stochastic Integration for Square Integrable Martingales 513
(resp. optional) process and T a stopping time, then the stopped process X
T
is also predictable (resp. optional). Every process which is c`ag and adapted
is predictable, every process which is c`ad and adapted is optional. If X is a
c`adl`ag adapted process, then (X
t
−
,t≥ 0) is a predictable process.
A stopping time T is predictable (see Definition 1.2.3.1) if and only if
the process (1
{t<T }
=1− 1
{T ≤t}
,t ≥ 0) is predictable, that is if and only if
the stochastic interval [[0,T[[ = {(ω, t): 0≤ t<T(ω)} is predictable. Note
that O = P if and only if any stopping time is predictable (this is the case if
the reference filtration is a weakly Brownian filtration; see Subsection 5.8.1).
A stopping time T is totally inaccessible if P(T = S<∞) = 0 for all
predictable stopping times S. See Dellacherie [240], Dellacherie and Meyer
[242] and Elliott [313] for related results.
Often, when dealing with point processes, there is a space of marks (E,E)
besides the filtered probability space. In such a setting, the following definition
makes sense: a predictable function is a map H : Ω ×R
+
×E → R which
is P×Emeasurable.
9.2 Stochastic Integration for Square Integrable
Martingales
We present, in this section the notion of stochastic integration with respect to
a square integrable martingale. We follow closely the presentation of Meyer
[647]. We shall extend this notion to local martingales and semi-martingales
in the next section.
9.2.1 Square Integrable Martingales
We recall that H
2
is the set of square integrable martingales, i.e.,
martingales such that sup
t<∞
E(M
2
t
) < ∞ (see Subsection 1.2.2). If M is
a square integrable martingale, then M
τ
= E(M
∞
|F
τ
) for any stopping time
τ where M
∞
= lim
t→∞
M
t
and M
2
2
:= E(M
2
∞
)=sup
t<∞
E(M
2
t
). Let
M
∗
∞
=sup
t
|M
t
|. Then, (Doob’s inequality)
M
∗
∞
2
2
≤ 4M
∞
2
2
.
Definition 9.2.1.1 Two martingales in H
2
are orthogonal if their product
is a martingale.
Definition 9.2.1.2 We denote by H
2,c
the space of continuous square
integrable martingales and by H
2,d
the set of square integrable martingales
orthogonal to H
2,c
. A martingale in H
2,d
is called a purely discontinuous
martingale.
514 9 General Processes: Mathematical Facts
Warning 9.2.1.3 We stress that a purely discontinuous martingale is not
necessarily of bounded variation (see Az´ema’s martingale in Example 9.3.3.6).
Very often, a martingale in H
2,d
is said to be a compensated sum of
jumps. We explain here the meaning of this definition. If M ∈ H
2,d
there
exists a sequence of stopping times (T
n
,n≥ 1) such that
{(t, ω):ΔM
t
(ω) =0}⊂∪
n
{(t, ω):t = T
n
(ω)},
i.e., the sequence (T
n
,n ≥ 1) exhausts the set of jump times of M. Define
A
n
t
= ΔM
T
n
1
{T
n
≤t}
and let M
n
= A
n
−A
n,(p)
be the compensated martingale
associated with A
n
,whereA
n,(p)
is the dual predictable projection of A
n
(defined in Section 5.2)). It can be proved that, as k →∞,
k
n=1
M
n
t
converges in L
2
to M
t
.
Example 9.2.1.4 If M is the martingale associated with a Poisson process,
i.e., M
t
= N
t
− λt,thenM is purely discontinuous and A
n
t
= 1
{T
n
≤t}
.It
follows that
k
n=1
M
n
t
= N
t∧T
k
− λ(t ∧ T
k
).
Theorem 9.2.1.5 Any martingale M in H
2,d
is orthogonal to any square
integrable martingale N with no jumps in common with M. For any square
integrable martingale M,
E
s≤t
(ΔM
s
)
2
≤ E(M
2
t
) .
For a purely discontinuous martingale E(
s≤t
(ΔM
s
)
2
)=E(M
2
t
).
Proof: We refer the reader to Meyer [647].
Comment 9.2.1.6 Note that the sum
s≤t
(ΔM
s
)
2
is convergent and that,
in general, the sum
s≤t
|ΔM
s
| is not convergent (see Example of Az´ema’s
martingale in Example 9.3.3.6).
Proposition 9.2.1.7 (a) Let M ∈ H
2,d
. Then the process M
2
t
−
s≤t
(ΔM
s
)
2
is a martingale.
(b) Let M ∈ H
2,d
and N ∈ H
2
. Then, the process M
t
N
t
−
s≤t
ΔM
s
ΔN
s
is a martingale and E(M
∞
N
∞
)=E(
s
ΔM
s
ΔN
s
).
Proof: The proof of a) is an application of Proposition 1.2.3.7.First,ifτ
is a stopping time, the stopped process M
τ
is a martingale, hence E(M
2
τ
−
s≤τ
(ΔM
s
)
2
) = 0. This implies that the process M
2
t
−
s≤t
(ΔM
s
)
2
is a
martingale.
9.2 Stochastic Integration for Square Integrable Martingales 515
Definition 9.2.1.8 For any martingale M ∈ H
2
, we denote by M
c
its
projection on H
2,c
and by M
d
its projection on H
2,d
.Then,M = M
c
+ M
d
is the decomposition of any martingale in H
2
into its continuous and purely
discontinuous parts.
The process M
2
is a submartingale bounded by (M
∗
)
2
, and the process
E(M
2
∞
|F
t
) − M
2
t
is a supermartingale of class (D), hence the Doob-Meyer
supermartingale decomposition Theorem 1.2.1.6 applies: there exists a unique
predictable increasing process M (called the predictable bracket)such
that M
2
−M is a martingale equal to 0 at time 0.
The predictable bracket satisfies
M
t
= P − lim
i
E((M
t
(n)
i
− M
t
(n)
i−1
)
2
|F
t
(n)
i−1
)
where 0 = t
(n)
0
<t
(n)
1
< ···<t
(n)
p(n)
= t and sup
i=1,...,p(n)
(t
(n)
i
− t
(n)
i−1
)goesto
0. Furthermore, if M is continuous, then
M
t
= P − lim
i
(M
t
(n)
i
− M
t
(n)
i−1
)
2
.
Definition 9.2.1.9 Let M
c
denote the continuous part of M. We define the
quadratic variation process of M as
[M]
t
= M
c
t
+
0≤s≤t
(ΔM
s
)
2
.
For M ∈ H
2
, the process [M] is an increasing integrable process, and the
process M
2
− [M] is a martingale. Indeed, M = M
c
+ M
d
, hence
M
2
t
− [M]
t
=(M
c
t
)
2
−M
c
t
+(M
d
t
)
2
−
0≤s≤t
(ΔM
s
)
2
+2M
c
t
M
d
t
,
is the sum of the 3 martingales
(M
c
t
)
2
−M
c
t
, (M
d
t
)
2
−
0≤s≤t
(ΔM
s
)
2
, 2M
c
t
M
d
t
.
By polarisation, one defines for M and N in H
2
, the quadratic covariation
process:
M,N =
1
2
(M + N −M −N )
which is the unique predictable process with integrable variation such that
MN −M,N is a martingale and
[M,N]=
1
2
([M + N ] − [M] − [N])
which is the unique optional process with integrable variation such that the
process MN − [M,N] is a martingale and Δ[M,N]
t
= ΔM
t
ΔN
t
.
The processes MN − [M,N]and[M, N] −M,N are martingales, and
the martingales M and N are orthogonal if and only if M,N is null.
516 9 General Processes: Mathematical Facts
Kunita-Watanabe Inequalities
Let H and K be two measurable processes on Ω × R
+
.Then
∞
0
|H
s
||K
s
||dM,N
s
|≤
∞
0
H
2
s
dM
s
1/2
∞
0
K
2
s
dN
s
1/2
∞
0
|H
s
||K
s
||d[M,N
s
]|≤
∞
0
H
2
s
d[M]
s
1/2
∞
0
K
2
s
d[N]
s
1/2
.
The idea of the proof is to establish the result for elementary processes, using
the Cauchy-Schwarz inequality, and to pass to the limit using the monotone
class theorem.
9.2.2 Stochastic Integral
Now let H be a predictable elementary bounded process, i.e., a process of the
form
H
t
= H
0
+
n
i=1
h
i
1
]t
i
,t
i+1
]
(t)
for a finite sequence of real numbers t
0
=0<t
1
< ··· <t
n
<t
n+1
= ∞,
where the random variables h
i
∈F
t
i
are bounded. We call Λ the vector space
of these elementary processes. We define, for H ∈ Λ:
∞
0
H
s
dM
s
= H
0
M
0
+
n
i=1
h
i
(M
t
i+1
− M
t
i
)
and (HM)
t
:=
t
0
H
s
dM
s
=
∞
0
1
[0,t]
(s)H
s
dM
s
.Itiseasytocheckthatthe
process HM is a square integrable martingale and that
E((HM)
2
∞
)=E
[0,∞[
H
2
s
dM
s
= E
[0,∞[
H
2
s
d[M]
s
.
If H is a predictable process such that E
[0,∞[
H
2
s
dM
s
< ∞, then, for
any N ∈ H
2
, E
[0,∞[
|H
s
||dM,N
s
|
< ∞.
Theorem 9.2.2.1 Let L
2
(M) be the set of predictable processes H such that
H
L
2
(M)
:=
E
[0,∞[
H
2
s
dM
s
1/2
< ∞.
The linear mapping H → HM defined on Λ admits a unique extension as
a continuous linear mapping from L
2
(M) to H
2
. Furthermore, the processes
Δ(HM)
t
and H
t
ΔM
t
are indistinguishable.
9.3 Stochastic Integration for Semi-martingales 517
The stochastic integral I = HM is the unique martingale in H
2
such
that, for any N ∈ H
2
E(I
∞
N
∞
)=E
[0,∞[
H
s
dM,N
s
.
Moreover, one has
I,N
t
=
t
0
H
s
dM,N
s
, [I,N]
t
=
t
0
H
s
d[M,N]
s
.
If H ∈ L
2
(M), the continuous and the discontinuous parts of HM are HM
c
and HM
d
. It is important to note that if M is a martingale with integrable
variation, and H a predictable process such that E
∞
0
H
2
s
d[M]
s
< ∞ and
E
∞
0
|H
s
||dM
s
|
< ∞, then the stochastic integral and the Stieltjes integral
are equal.
9.3 Stochastic Integration for Semi-martingales
We now present the extension of stochastic integrals for a local martingale,
then for a semi-martingale.
9.3.1 Local Martingales
In order to extend the definition of stochastic integrals to local martingales
(see Subsection 1.2.4), we recall a result that will reduce the problem to the
case of H
2
martingales and integrable quadratic variation processes.
Lemma 9.3.1.1 Let M be a local martingale. There exists a sequence T
n
of
stopping times such that, for any n the stopped martingale M
T
n
is of the form
N
n
+ U
n
,whereN
n
∈ H
2
and U
n
is a process with integrable variation.
Sketch of the Proof: In a first step, one establishes that there exists a
sequence of stopping times T
n
such that the stopped processes M
n
= M
T
n
are martingales and E(|M
T
n
||F
s
)1
{s<T
n
}
is bounded. Then, one denotes
C
t
= M
n
t
1
{T
n
≤t}
= M
n
T
n
1
{T
n
≤t}
and X
t
= M
n
t
1
{T
n
≤t}
= M
n
t
− C
t
.Let
C
(p)
denote the dual predictable projection of the integrable variation process
C,andsetU = C − C
(p)
and N = X + C
(p)
= M − U.Itremainstoprove
that the local martingale N is in H
2
(see [647]).
One starts by a definition of the quadratic variation of a local martingale.
This is extremely important, as it is the main tool for defining stochastic
integrals with respect to local martingales.
If M is a local martingale, its continuous part, defined in Definition 9.2.1.8
admits a predictable (in fact a continuous) bracket, denoted M
c
,M
c
.
518 9 General Processes: Mathematical Facts
Definition 9.3.1.2 The quadratic variation of a local martingale M is
[M,M]
t
= M
c
,M
c
t
+
s≤t
(ΔM
s
)
2
.
It is important to emphasize that, for any local martingale M ,thequantity
0≤s≤t
(ΔM
s
)
2
is well defined. Note that in general, [M,M] is optional and not
predictable.
The process [M,M] is an increasing process such that, by definition
Δ[M,M]
t
=(ΔM
t
)
2
. The quadratic variation of M is the limit in probability
of the sum of the square of the increments, i.e.,
[M,M]
t
= P − lim
p(n)
i=1
(M
t
n
i
− M
t
n
i−1
)
2
, (9.3.1)
where 0 = t
0
≤ t
n
1
···≤t
n
p(n)
= t and sup
i
(t
n
i
− t
n
i−1
)goesto0.
The continuous part of the increasing process [M,M] is denoted by
[M,M]
c
, hence
[M,M]
t
=[M,M]
c
t
+
0≤s≤t
(ΔM
s
)
2
and it follows that [M,M]
c
t
= M
c
,M
c
t
.
We now define the stochastic integral of a locally bounded predictable
process with respect to a local martingale. Recall that a process is locally
bounded if there exists a sequence of stopping times T
n
and constants K
n
such that |H
t
|1
{t≤T
n
}
≤ K
n
.
Theorem 9.3.1.3 Let M be a local martingale and H a locally bounded pre-
dictable process. There exists a (unique) local martingale HM =
·
0
H
s
dM
s
such that
[HM,N]=H[M,N]
for any bounded martingale N.
Sketch of the Proof: One defines the martingale HM
n
for M
n
= M
T
n
where (T
n
; n ≥ 1) is a sequence of stopping times such that the decomposition
given in Lemma 9.3.1.1 of the local martingale holds true, and H is bounded.
It remains to check that there exists a process denoted HM such that
HM
n
=(HM)
T
n
(see [647]).
Example 9.3.1.4 Let M
t
= N
t
− λt where N is a Poisson process with
parameter λ. From the equality (8.2.2), the predictable quadratic variation
M,M (the predictable bracket) of M is λt. The quadratic variation [M,M]
of M is N. Indeed, the process M
2
−N is a martingale, hence, the quadratic
variation of M is N,sinceΔN =(ΔN)
2
.
9.3 Stochastic Integration for Semi-martingales 519
9.3.2 Quadratic Covariation and Predictable Bracket of Two
Local Martingales
Proposition 9.3.2.1 Let M be a local martingale. The quadratic varia-
tion process ([M,M]
t
,t≥ 0) satisfies
[M,M]
t
= M
2
t
− M
2
0
− 2
t
0
M
s−
dM
s
. (9.3.2)
The process M
2
t
− [M, M]
t
is a local martingale.
Proof: We assume that M
0
=0.Lett
n
i
,i ≤ p(n) be an increasing finite
sequence of real numbers with t
n
0
=0andt
n
p(n)+1
= t.Forfixedn, write (with
t
i
= t
n
i
)
M
2
t
=
p(n)
i=0
M
2
t
i+1
− M
2
t
i
=
p(n)
i=0
(M
t
i
+ ΔM
t
i
,t
i+1
)
2
− M
2
t
i
=2
p(n)
i=0
M
t
i
ΔM
t
i
,t
i+1
+
p(n)
i=0
(ΔM
t
i
,t
i+1
)
2
where ΔM
t
i
,t
i+1
= M
t
i+1
− M
t
i
and pass to the limit when n goes to infinity
and sup(t
n
i
− t
n
i−1
)goesto0.
Let M and N be two local martingales. The quadratic covariation of two
local martingales is defined via the polarization identity
[M + N, M + N]=[M,M]+[N, N]+2[M,N] .
By polarisation of formula (9.3.2) the quadratic covariation of M and N
satisfies
[M,N]
t
= M
t
N
t
− M
0
N
0
−
t
0
M
s−
dN
s
−
t
0
N
s−
dM
s
. (9.3.3)
It follows that
Δ[M,N]
t
= Δ(MN)
t
− M
t
−
ΔN
t
− N
t
−
ΔM
t
= M
t
N
t
− M
t
−
N
t
−
− M
t
−
(N
t
− N
t
−
) − N
t
−
(M
t
− M
t
−
)
= ΔM
t
ΔN
t
.
Proposition 9.3.2.2 The quadratic covariation of the local martingales M
and N is the unique c`adl`ag process [M,N] with finite variation, such that
(i) MN − [M,N] is a local martingale,
(ii) Δ[M, N]
t
= ΔM
t
ΔN
t
.