. Let $W(t), 0 \leq t \leq T$, be a Brownian motion, and let $\mathcal{F}(t)$, $0 \leq t \leq T$, be an associated filtration. Let $\Delta(t), 0 \leq t \leq T$, be a nonrandom simple process (i.e., there is a partition $\Pi=\left\{t_0, t_1, \ldots, t_n\right\}$ of $[0, T]$ such that
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4 Stochastic Calculus
for every $j, \Delta\left(t_j\right)$ is a nonrandom quantity and $\Delta(t)=\Delta\left(t_j\right)$ is constant in $t$ on the subinterval $\left[t_j, t_{j+1}\right)$ ). For $t \in\left[t_k, t_{k+1}\right]$, define the stochastic integral
$$
I(t)=\sum_{j=0}^{k-1} \Delta\left(t_j\right)\left[W\left(t_{j+1}\right)-W\left(t_j\right)\right]+\Delta\left(t_k\right)\left[W(t)-W\left(t_k\right)\right] .
$$
(i) Show that whenever $0 \leq s<t \leq T$, the increment $I(t)-I(s)$ is independent of $\mathcal{F}(s)$. (Simplification: If $s$ is between two partition points, we can always insert $s$ as an extra partition point. Then we can relabel the partition points so that they are still called $t_0, t_1, \ldots, t_n$, but with a larger value of $n$ and now with $s=t_k$ for some value of $k$. Of course, we must set $\Delta(s)=\Delta\left(t_{k-1}\right)$ so that $\Delta$ takes the same value on the interval $\left[s, t_{k+1}\right)$ as on the interval $\left[t_{k-1}, s\right)$. Similarly, we can insert $t$ as an extra partition point if it is not already one. Consequently, to show that $I(t)-I(s)$ is independent of $\mathcal{F}(s)$ for all $0 \leq s<t \leq T$, it suffices to show that $I\left(t_k\right)-I\left(t_{\ell}\right)$ is independent of $\mathcal{F}\left(t_{\ell}\right)$ whenever $t_k$ and $t_{\ell}$ are two partition points with $t_{\ell}<t_k$. This is all you need to do.)
(ii) Show that whenever $0 \leq s<t \leq T$, the increment $I(t)-I(s)$ is a normally distributed random variable with mean zero and variance $\int_s^t \Delta^2(u) d u$.
(iii) Use (i) and (ii) to show that $I(t), 0 \leq t \leq T$, is a martingale.
(iv) Show that $I^2(t)-\int_0^t \Delta^2(u) d u, 0 \leq t \leq T$, is a martingale.