Springer Finance

3.4 Quadratic Variation 103

computed) converges to T as IIJIII ---+ 0. We shall show that it has expected

value T, and its variance converges to zero. Hence, it converges to its expected

value T, regardless of the path along which we are doing the computation.^2

The sampled quadratic variation is the sum of independent random vari­

ables. Therefore, its mean and variance are the sums of the means and vari­

ances of these random variables. We have

IE [ (W(ti+I )-W(ti))^2 ] = Var [W(ti+I)-W(ti)] = ti+l-ti, (3.4. 6 )

which implies

n-^1 n-^1

IEQrr =LIE [(w(tj+I)-W(ti))^2 ] = L(ti+I-ti) = T,

j=O j=O

as desired. Moreover,

Var [ (W(tj+I)-W(tj))^2 ]

=IE [ ( (W(tj+I)-W(ti))^2 - (ti +l-ti)r]

=IE [ (W(tj+I)-W(ti ))4] - 2(ti + I -ti)IE [ (W(ti + I)-W(ti))^2 ]

+(tj+1 -tj ) 2 •

The fourth moment of a normal random variable with zero mean is three times
its variance squared (see Exercise 3.3). Therefore,

IE [ (W(tJ+1)-W(tj))4] = 3 (ti +l -ti)^2 ,

Var [ (W(ti+I)-W(ti))^2 ] = 3(ti+l -ti)^2 - 2(tj+l-ti)^2 + (ti+l-ti)^2

and
n- 1

= 2(tj+l -tj)^2 , (3.4.7)

n- 1

Var(Qrr) = L:var [(w(tj+1)-W(ti))^2 ] L: 2 (tj+l-tj)^2

j =O j=O

n- 1

:::; L^2 II1III(tj+1-tj) =^2 II1IIIT.

j=O

In particular, limiiiiii-tO Var(Qrr) = 0, and we conclude that limiiiiii-tO Qrr =
IEQrr = T.^0

(^2) The convergence we prove is actually convergence in mean square, also called
L^2 -convergence. When this convergence takes place, there is a subsequence along
which the convergence is almost sure (i.e., the convergence takes place for all
paths except for a set of paths having probability zero). We shall not dwell on
subtle differences among types of convergence of random variables.