quadratic variation

The presence of quadratic variation makes stochastic calculus fundamentally different from classical calculus.

Motivation §

Suppose $f(t)$ represents the position a particle travelling on a 1D smooth path. Then $f(b)-f(a)$ is the displacement, and the total distance travelled is

$${\int }_a^b|f^{\prime }(t)|dx$$

How do we approximate this quantity?

If we divide $[0,t]$ into infinitely many partitions $P_k$ with ${\lim }_{k\to \infty }\text{mesh}(P_k)=0$, where $\text{mesh}(\mathcal{P}):={\max }_i(t_i-t_{i-1})$ is the mesh size of the partition, then we can approximate the total distance with the following sum:

$${\int }_a^b|f^{\prime }(t)|dx=\underset{k\to \infty }{\lim }\sum _i^n|f(t_{k,i})-f(t_{k,i-1})|$$

This is the first order variation of f. The quadratic variation is just

$$\underset{k\to \infty }{\lim }\sum _i^n|f(t_{k,i})-f(t_{k,i-1}){|}^2=0$$

Proof §

By the mean value theorem, we may find in each time interval $[t_{k,i-1},t_{k,i}]$ a number $t_{k,i}^{\ast }$ s.t.

$$f\left(t_{k,i}\right)-f\left(t_{k,i-1}\right)=f^{\prime }\left(t_{k,i}^{\ast }\right)\left(t_{k,i}-t_{k,i-1}\right).$$

Squaring it, we have

$$\begin{array}{rl}{\left|f\left(t_{k,i}\right)-f\left(t_{k,i-1}\right)\right|}^2& ={\left|f^{\prime }\left(t_{k,i}^{\ast }\right)\right|}^2{\left|t_{k,i}-t_{k,i-1}\right|}^2\\ & \le \mathrm{mesh}\left({\mathcal{P}}_k\right){\left|f^{\prime }\left(t_{k,i}^{\ast }\right)\right|}^2\left(t_{k,i}-t_{k,i-1}\right).\end{array}$$

Summing over $i$, we have the bound

$$\sum _i{\left|f\left(t_{k,i}\right)-f\left(t_{k,i-1}\right)\right|}^2\le \mathrm{mesh}\left({\mathcal{P}}_k\right)\sum _i{\left|f^{\prime }\left(t_{k,i}^{\ast }\right)\right|}^2\left(t_{k,i}-t_{k,i-1}\right).$$

As $k\to \infty ,{\sum }_i{\left|f^{\prime }\left(t_{k,i}^{\ast }\right)\right|}^2\left(t_{k,i}-t_{k,i-1}\right)$ converges to ${\int }_a^b{\left|f^{\prime }(t)\right|}^{2}dt$ but $\mathrm{mesh}\left({\mathcal{P}}_k\right)\to 0$, so the limit evaluates to zero.

Formal definition §

The quadratic variation of a stochastic process ${\left\{X_t\right\}}_{t\ge 0}$ is

$$⟨X{⟩}_t:=\underset{\text{mesh}(\mathcal{P})\to 0}{\lim }\sum _i{\left(X_{t_{i+1}}-X_{t_i}\right)}^2,t\ge 0$$

We can also consider the covariation between two processes $\left\{X_t\right\}$ and $\left\{Y_t\right\}$:

$$⟨X,Y{⟩}_t:=\underset{\mathrm{mesh}(\mathcal{P})\to 0}{\lim }\sum _i\left(X_{t_{i+1}}-X_{t_i}\right)\left(Y_{t_{i+1}}-Y_{t_i}\right),t\ge 0$$

Examples §

Brownian motion §

We have

$$\mathbb{E}\left[\sum _i{\left|B\left(t_{k,i}\right)-B\left(t_{k,i-1}\right)\right|}^2\right]=t$$

We can show that the sum (of the squared differences) converges to t in L2 space, and hence in probability:

$$\underset{k\to \infty }{\lim }\mathbb{E}\left[{\left(\sum _i{\left|B\left(t_{k,i}\right)-B\left(t_{k,i-1}\right)\right|}^2-t\right)}^2\right]=0$$

Note that the quadratic variation, if exists, is itself a stochastic process. We denote the quadratic variation process of the Brownian motion with $⟨B{⟩}_t=t$.

Ito process §

If $\left\{X_t\right\}$ is an Itô process with dynamics $d{X}_t=a_{t}dt+b_{t}d{B}_t$, then its quadratic variation exists and is given by

$$⟨X{⟩}_t={\int }_0^t{\left|b_s\right|}^{2}ds$$

This is used to derive Ito’s lemma.