(Stratonovich integral). Let $W(t), t \geq 0$, be a Brownian motion. Let $T$ be a fixed positive number and let $\Pi=\left\{t_0, t_1, \ldots, t_n\right\}$ be a partition of $[0, T]$ (i.e., $0=t_0<t_1<\cdots<t_n=T$ ). For each $j$, define $t_j^*=\frac{t_j+t_{2+1}}{2}$ to be the midpoint of the interval $\left[t_j, t_{j+1}\right]$.
(i) Define the half-sample quadratic variation corresponding to $\Pi$ to be
$$
Q_{n / 2}=\sum_{j=0}^{n-1}\left(W\left(t_j^*\right)-W\left(t_j\right)\right)^2
$$
Show that $Q_{\Pi / 2}$ has limit $\frac{1}{2} T$ as $\|\Pi\| \rightarrow 0$. (Hint: It suffices to show that $\mathbf{E} Q_{\Pi / 2}=\frac{1}{2} T$ and $\lim _{\|\Pi\| \rightarrow 0} \operatorname{Var}\left(Q_{\Pi / 2}\right)=0$.)
4.10 Exercises
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(ii) Define the Stratonovich integral of $W(t)$ with respect to $W(t)$ to be
$$
\int_0^T W(t) \circ d W(t)=\lim _{\|\Pi\| \rightarrow 0} \sum_{j=0}^{n-1} W\left(t_j^*\right)\left(W\left(t_{j+1}\right)-W\left(t_j\right)\right) .
$$
In contrast to the Itô integral $\int_0^T W(t) d W(t)=\frac{1}{2} W^2(T)-\frac{1}{2} T$ of (4.3.4), which evaluates the integrand at the left endpoint of each subinterval $\left[t_j, t_{j+1}\right]$, here we evaluate the integrand at the midpoint $t_j^*$. Show that
$$
\int_0^T W(t) \circ d W(t)=\frac{1}{2} W^2(T) .
$$
(Hint: Write the approximating sum in (4.10.1) as the sum of an approximating sum for the Itô integral $\int_0^T W(t) d W(t)$ and $Q_{\Pi / 2}$. The approximating sum for the Itô integral is the one corresponding to the partition $0=t_0<t_0^*<t_1<t_1^*<\cdots<t_{n-1}^*<t_n=T$, not the partition $\Pi$. .)