2Semi-martingales
III Stochastic Calculus and Applications
2.6 Semi-martingale
Definition
(Semi-martingale)
.
A (continuous) adapted process
X
is a (contin-
uous) semi-martingale if
X = X
0
+ M + A,
where
X
0
∈ F
0
,
M
is a continuous local martingale with
M
0
= 0, and
A
is a
continuous finite variation process with A
0
= 0.
This decomposition is unique up to indistinguishables.
Definition
(Quadratic variation)
.
Let
X
=
X
0
+
M
+
A
and
X
0
=
X
0
0
+
M
0
+
A
0
be (continuous) semi-martingales. Set
hXi = hMi, hX, X
0
i = hM, M
0
i.
This definition makes sense, because continuous finite variation processes do
not have quadratic variation.
Exercise. We have
hX, Y i
(n)
t
=
d2
n
te
X
i=1
(X
i2
−n
− X
(i−1)2
−n
)(Y
i2
−n
− Y
(i−1)2
−n
) → hX, Y i u.c.p.