# Quadratic Variation of Ito Integrals

Given a stochastic process  $$f_t$$ and $$g_t$$ adapted to a filtration $$\mathcal F_t$$ satisfying

$\int_0^T\mathbf E f_t^2 dt < \infty\quad\text{and}\quad \int_0^T\mathbf E g_t^2 dt < \infty$

define

$M_t =\int_0^t f_s dW_s \quad \text{and}\quad N_t =\int_0^t g_s dW_s$

for some standard Brownian Motion also adapted to the  filtration $$\mathcal F_t$$ . Though it is not necessary, assume that  there exists a $$K>0$$ so that  $$|f_t|$$ and $$|g_t|$$  are less than some $$K$$  for all $$t$$ almost surely.

Let $$\{ t_i^{(n)} : i=0,\dots,N(n)\}$$ be sequence of partitions of $$[0,T]$$ of the form

$0 =t_0^{(n)} < t_1^{(n)} <\cdots<t_N^{(n)}=T$

such that

$\lim_{n \rightarrow \infty} \sup_i |t_{i+1}^{(n)} – t_i^{(n)}| = 0$

Defining

$V_n[M]=\sum_{i=1}^{N(n)} \big(M_{t_i} -M_{t_{i-1}}\big)^2$

and

$Q_n[M,N]= \sum_{i=1}^{N(n)} \big(M_{t_i} -M_{t_{i-1}}\big)\big(N_{t_i} -N_{t_{i-1}}\big)$

Clearly $$V_n[M]= Q_n[M,M]$$. Show  that the following points hold.

1. The “polarization equality” holds:
$4 Q_n[M,N] =V_n[M+N] -V_n[M-N]$
Hence it is enough to understand the limit of $$n \rightarrow \infty$$ of $$Q_n$$ or $$V_n$$.
2. $\mathbf E V_n[M]= \int_0^T \mathbf E f_t^2 dt$
3. * $$V_n[M]\rightarrow \int_0^T f_t^2 dt$$ as $$n \rightarrow \infty$$ in $$L^2$$. That is to say
$\lim_{n \rightarrow \infty}\mathbf E \Big[ \big( V_n[M] – \int_0^T f_t^2 dt \big)^2 \Big]=0$
This limit is called the Quadratic Variation of the Martingale $$M$$.
4. Using the results above, show that $$Q_n[M,N]\rightarrow \int_0^T f_t g_t dt$$ as $$n \rightarrow \infty$$ in $$L^2$$. This is called the cross-quadratic variation of $$M$$ and $$N$$.
5. * Prove by direct calculation that  in the spirit of 3) from above that   $$Q_n[M,N]\rightarrow \int_0^T f_t g_t dt$$ as $$n \rightarrow \infty$$ in $$L^2$$.

In this context, one writes $$\langle M \rangle_T$$ for the limit of the $$V_n[M]$$  which is called the quadratic variation process of $$M_T$$. Similarly  one writes  $$\langle M,N \rangle_T$$ for the  limit of $$Q_n[M,N]$$  which is called the cross-quadratic variation process of $$M_T$$ and $$N_T$$. Clearly $$\langle M \rangle_T = \langle M,M \rangle_T$$ and $$\langle M+N,M \rangle_T = \langle M, M+N\rangle_T= \langle M \rangle_T + \langle M, N\rangle_T$$.