Why Brownian Motion Has Quadratic Variation Equal to Time

# Why does [dz(t)]^2 converge to dt over infinitesimally short time periods?


# Why does [dz(t)]^2 converge to dt over infinitesimally short time periods?












I have some trouble understanding a chapter in George Pennacchi textbook "Asset Pricing". Here the author shows that the square of a Wiener Process $[dz(t)]^2$ converges to $dt$ for infinitesimally short time periods.

By the definition of the Itô-Integral for $\int_0^T[dz(t)]^2$: \begin{equation} \lim_{n\rightarrow\infty}E_0[(\sum_{i=1}^n [\Delta z_i^2]-\int_0^T[dz(t)]^2)^2]=0 \end{equation}

Then he states that \begin{equation} E_0[(\sum_{i=1}^n [\Delta z_i^2] -T)^2]=2T\Delta t \end{equation}, which is the equation I cannot derive. The limit for this expression as $\Delta t \rightarrow 0$ is 0 which is obvious. But what I am also not able to grasp is why this result together with the definition of the Itô Integral finally shows that \begin{equation} \int_0^T[dz(t)]^2=\int_0^T dt \end{equation}

Can someone give me some assistance in understanding this important outcome?

## Answer by vonjd (score 5, accepted)

https://quant.stackexchange.com/a/12883

Beware, oversimplification ahead! (This means that the following is technically not correct, in fact it is false! But: It gives an intuition what is going on!)

If you toss a coin and calculate heads as $-1$ and tails as $1$ you get a mean of $0$ with a variance of $1$. When you add up multiple coin tosses, i.e. create a random process $dz(t)$, the mean stays the same, i.e. $0$ -> this process is a martingale.

Now you square this process: On average you will get $\frac{(-1)^2+1^2}{2}=1$, which means that your mean gets one unit bigger per timestep -> $[dz(t)]^2=dt$

So here you see how a martingale could become a process which grows with time on average when you square it - or in more general terms: how a symmetric process could become asymmetric through a non-linear transformation.

The following is a very nice exposition showing these ideas in a very intuitive manner: Stochastic Calculus and the Nobel Prize Winning Black-Scholes Equation by Frank Morgan

Addendum To address the question why this discrete intuition might make sense even in the continuous case you have to know that the abovementioned stochastic process results in a binomial tree. The basis of this tree is, as the name suggests, the binomial distribution. When you simultaneously reduce the size of and increase the number of time steps this binomial distribution becomes (under some technical conditions) the normal distribution. This is the result of the de Moivre–Laplace theorem. The corresponding continuous stochastic process is a Wiener process (also often called standard Brownian motion), which closes the circle.

## Answer by LazyCat (score 2)

https://quant.stackexchange.com/a/12882

Intuitively, because of the central limit theorem: wiener process is a limit of a random walk, and after n steps a random walk moves away from the origin by ~ $\sqrt{n}$

Edit: here is a complete answer. First the formula for the sum. The trick is the following simple observation: if $X_1,.. X_n$ are independent zero mean, then $E(\sum X_i)^2 = \sum{EX_i^2}$. In the case of the formula, independent zero mean incremnents are $d_i=\Delta z_i^2 - T/N,$ since z(t) is Brownian motion.

The sum in question is $$E(\sum{\Delta z_i}^2-T)^2 = E(\sum{d_i})^2 = \sum{E(d_i^2)} = \sum(E\Delta z_i^4 - 2 T / N * E(\Delta z_i^2) + T^2/N^2) = 2 T^2 / N = 2 T * \Delta t$$

Second, for the relation to the last equality: the right hand side is T. The left hand side is in some (precise) sense is the limit of $\sum{\Delta z_i^2}$ One standard way to prove convergences in probability, is to estimate the second moment of the difference. This is the formula we just proved.

## Answer by Kumar (score -4)

https://quant.stackexchange.com/a/12876

Intuitively you are measuring the distance between two random variables which under limiting case turns out to be very small. So you can use one instead of another.