The following question is from my professor, I have no clue about it at all.
Let $(W (t))_{t\geq0}$ denote a standard Brownian motion. Consider a model of asset price given as
$$\mathrm{d} S(t)=\mu S(t) \mathrm{d} t+\sigma(t) S(t) \mathrm{d} W(t), \quad S(0)=s_{0}$$
where $\mu > 0$ is a constant and $\sigma(t)>0$ is a deterministic (non-random) function of time. Ito's isometry tell us that $\mathbb{E}\left[\left(\int_{0}^{t} f(s, W(s)) \mathrm{d} W(s)\right)^{2}\right]=\int_{0}^{t} \mathbb{E}\left[f^{2}(s,W(s))\right] \mathrm{d} s$ for a square integrable function f.
a). Using Itô’s isometry, compute the mean and variance of $\int_{0}^{t} \sigma(s) \mathrm{d} W(s)$.
b). Compute the mean of S(T).
For part (a), can I use the following process:
Let $Z(t)=\sigma(s)dW(s)$, $Z(t)$ is a martingale so $$\mathbb{E}(\int_{0}^{t} \sigma(s) \mathrm{d} W(s))=0$$
Moreover, based on the Ito's isometry, $$Var(Z(t))=\mathbb{E}\left(\int_{0}^{t} \sigma(s) \mathrm{d} W(s)\right)^2=\int_{0}^{t} \mathbb{E}\left[\sigma(s)^2\right] \mathrm{d} s=?$$
I am stuck here.