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.

- 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 \(V_n\).
- \[\mathbf E V_n[M]= \int_0^T \mathbf E f_t^2 dt\]
- * \(V_n[M]\rightarrow \int_0^T f_t^2 dt\) 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 \(M\).
- 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 \(N\).
- * 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 \( \langle M+N,M \rangle_T = \langle M, M+N\rangle_T= \langle M \rangle_T + \langle M, N\rangle_T\).