这是一份Sydney悉尼大学MATH4511的成功案例
This process is the predictable covariation of $X^{p}$ and $X^{q}$ and is denoted by
$$
\left\langle X^{p}, X^{q}\right\rangle_{t}=\sum_{r=1}^{m} \int_{0}^{t} H_{s}^{p r} H_{s}^{q r} d s
$$
We note that $\left\langle X^{p}, X^{q}\right\rangle$ is symmetric and bilinear as a function on Itô processes.
Taking
$$
Y_{t}=Y_{0}+\int_{0}^{t} K_{s}^{\prime} d s
$$
and
$$
X_{t}=X_{0}+\int_{0}^{t} K_{s} d s+\sum_{j=1}^{m} H_{s}^{j} d W_{s}^{j}
$$
we see $\langle X, Y\rangle_{t}=0$.
Furthermore, considering special cases, formula gives
$$
\left\langle\int_{0}^{t} H_{s}^{p i} d W_{s}^{i}, \int_{0}^{t} H_{s}^{q j} d W_{s}^{j}\right\rangle=0 \quad \text { if } \quad i \neq j
$$
and
$$
\left\langle\int_{0}^{t} H_{s}^{p i} d W_{s}^{i}, \int_{0}^{t} H_{s}^{q i} d W_{s}^{i}\right\rangle=\int_{0}^{t} H_{s}^{p i} H_{s}^{q i} d s .
$$
MATH4511 COURSE NOTES ：
$$
E\left|X_{t}^{n+1}-X_{t}^{n}\right|^{2} \leq L^{n} \int_{0}^{t} \frac{(t-s)^{n-1}}{(n-1) !} E\left|X_{s}^{1}-\xi\right|^{2} d s
$$
and
$$
E\left|X_{s}^{1}-\xi\right|^{2} \leq \operatorname{LTK}^{2}\left(1+E|\xi|^{2}\right)
$$
Therefore,
$$
E\left|X_{t}^{n+1}-X_{t}^{n}\right|^{2} \leq C \frac{T^{n}}{n !} .
$$