In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process.
Suppose that
Xt
(\Omega,l{F},P)
t
[X]t
[X]t=\lim\Vert
n(X | |
\sum | |
tk |
-X | |
tk-1 |
)2
P
[0,t]
P
t>0
More generally, the covariation (or cross-variance) of two processes
X
Y
[X,Y]t=\lim\Vert
n | |
\sum | |
k=1 |
\left(X | |
tk |
-X | |
tk-1 |
\right)\left(Y | |
tk |
-Y | |
tk-1 |
\right).
[X,Y]t=\tfrac{1}{2}([X+Y]t-[X]t-[Y]t).
Notation: the quadratic variation is also notated as
\langleX\ranglet
\langleX,X\ranglet
A process
X
This statement can be generalized to non-continuous processes. Any càdlàg finite variation process
X
X
Xt
t
Xt-
X
t
\DeltaXt=Xt-Xt-
[X]t=\sum0<s\le(\Delta
2. | |
X | |
s) |
The proof that continuous finite variation processes have zero quadratic variation follows from the following inequality. Here,
P
[0,t]
Vt(X)
X
[0,t]
n(X | |
\begin{align} \sum | |
tk |
-X | |
tk-1 |
2&\lemax | |
) | |
k\len |
|X | |
tk |
-X | |
tk-1 |
n|X | |
|\sum | |
tk |
-X | |
tk-1 |
|\\ &\lemax|u-v|\le\Vert|Xu-Xv|Vt(X). \end{align}
X
\VertP\Vert
B
[B]t=t
L2
\begin{align} Xt&=X0+
t\sigma | |
\int | |
sdB |
s+
t\mu | |
\int | |
sd[B] |
s\\ &=X0+
t\sigma | |
\int | |
sdB |
s+
t\mu | |
\int | |
sds,\end{align} |
B
[X]t=\int
2ds. | |
s |
Quadratic variations and covariations of all semimartingales can be shown to exist. They form an important part of the theory of stochastic calculus, appearing in Itô's lemma, which is the generalization of the chain rule to the Itô integral. The quadratic covariation also appears in the integration by parts formula
XtYt=X0Y0+\int
tX | |
s- |
dYs+
tY | |
\int | |
s- |
dXs+[X,Y]t,
[X,Y]
Alternatively this can be written as a stochastic differential equation:
d(XtYt)=Xt-dYt+Yt-dXt+dXtdYt,
dXtdYt=d[X,Y]t.
All càdlàg martingales, and local martingales have well defined quadratic variation, which follows from the fact that such processes are examples of semimartingales.It can be shown that the quadratic variation
[M]
M
\Delta[M]=\DeltaM2
M2-[M]
M
A useful result for square integrable martingales is the Itô isometry, which can be used to calculate the variance of Itô integrals,
t | |
\operatorname{E}\left(\left(\int | |
0 |
HdM\right)2\right)=
tH | |
\operatorname{E}\left(\int | |
0 |
2d[M]\right).
M
H
Another important result is the Burkholder–Davis–Gundy inequality. This gives bounds for the maximum of a martingale in terms of the quadratic variation. For a local martingale
M
Mt*=\operatorname{sup}s\in[0,t]|Ms|
p\geq1
cp\operatorname{E}([M]
p/2 | |
t |
)\le
p)\le | |
\operatorname{E}((M | |
t) |
Cp\operatorname{E}([M]
p/2 | |
t |
).
cp<Cp
p
M
t
M
p>0
An alternative process, the predictable quadratic variation is sometimes used for locally square integrable martingales. This is written as
\langleMt\rangle
M2-\langleM\rangle