Stochastic integration lecture 12
The dominated convergence theorem
Limit characterisations of H · X and [X , Y ].
1/6
The dominated convergence theorem
Theorem 3.3.1 (dominated convergence theorem for stochastic
integrals). Let X ∈ cS and let t ≥ 0. Assume that (H n ) is a sequence of
progressive processes and that H is another progressive process. If H n
almost surely converges pointwise to H on [[0, t]] and |H n | and |H| are
bounded by K ∈ I on [[0, t]], then H n and H are in I as well, and
P
sup |(H n · X )s − (H · X )s | −→ 0.
s≤t
Proof. For the case where K is bounded, the result follows by repeated
applications of the ordinary dominated convergence theorem as well as the
“continuity property” of the quadratic variation given in Lemma 2.4.11. 2/6
Limit characterisations of H · X and [X , Y ]
Definition. Fix t ≥ 0 and let a partition 0 = t0 < t1 < · · · < tn = t of
[0, t] be given. We refer to maxk≤n |tk − tk−1 | as the mesh of the partition.
Theorem 3.3.2. Let X ∈ cS and let H be adapted and continuous. Let
t ≥ 0 and assume that (tkn )k≤Kn is a sequence of partitions of [0, t] with
P n
P
n
n
n
mesh tending to zero. Then K
k=1 Htk−1 (Xtk − Xtk−1 ) −→ (H · X )t .
Proof. Apply Lemma 3.2.8 on integration with simple integrands and the
dominated convergence result from Theorem 3.3.1.
Theorem 3.3.2 shows that the abstract definition of the integral actually
yields an integral corresponding to our intuitive expectations.
3/6
Limit characterisations of H · X and [X , Y ]
Theorem 3.3.3 (the integration-by-parts formula). Let X , Y ∈ cS.
Let (tkn )k≤Kn be a sequence of partitions of [0, t] with mesh tending to
zero. It then holds that
Kn
X
P
n
n
(Xtkn − Xtk−1
)(Ytkn − Ytk−1
) −→ [X , Y ]t
k=1
Xt Yt = X0 Y0 + (Y · X )t + (X · Y )t + [X , Y ]t .
The first result of Theorem 3.3.3 shows that the names “quadratic
variation” and “quadratic covariation” are well-chosen, in the sense that
these processes in fact are limits of sums of quadratic increments and
cross-increments.
4/6
Limit characterisations of H · X and [X , Y ]
Proof. By polarization, as when defining the quadratic covariation from
the quadratic variation, it suffices to consider a single continuous
semimartingale X and prove
Kn
X
P
n
(Xtkn − Xtk−1
)2 −→ [X ]t
k=1
Xt2 =
X02 + 2(X · X )t + [X ]t .
Central idea. For the case X = M, M ∈ cM` , apply the identity:
Mt2
=2
Kn
X
k=1
M
n
tk−1
(M − M
tkn
n
tk−1
)+
Kn
X
n
(Mtkn − Mtk−1
)2 .
k=1
The general case follows by applying the sample path properties of A,
where X = X0 + M + A.
5/6
Limit characterisations of H · X and [X , Y ]
Lemma 3.3.4. Let X and Y be in cS and let H be adapted and
continuous. Let t ≥ 0 and let (tkn )k≤Kn be a sequence of partitions of [0, t]
with mesh tending to zero. Then
Kn
X
P
n
n
n
Htk−1
(Xtkn − Xtk−1
)(Ytkn − Ytk−1
) −→ (H · [X , Y ])t .
k=1
Proof. Apply polarization to reduce to the case of a single martingale.
Use the integration-by-parts formula of Theorem 3.3.3 and the result on
approximations in probability of the stochastic integral from Theorem
3.3.2.
6/6