I have attempted to answer the following multi-part question but am especially having trouble with part iv) of the question. Any feedback would be greatly appreciated!.
Consider the Ito process $X(t) = \int_{0}^{t} sdB(s), t\leq1$
i) Compute the mean of X
If the Ito process is a martingale and thus has mean 0 if $\mathbb{E}\left( \int_{0}^{t} sdB(s) \right) < \infty $. This is satisfied so the mean is 0.
ii) What is the generator of X?
$$=\frac{1}{2}t^2 \frac{\partial^2}{\partial x^2}$$
iii) Let $L$ be the generator of $X$ and set $f(x,t)=e^{ux}$. Determine $Lf$
$$= \frac{1}{2}t^2 u^2e^{ux}$$
iv) Let $h(t) = \mathbb{E}[e^{uX(t)}]$. Use Dynkin's Formula to show:
$$h'(t) = \frac{1}{2}u^2t^2h(t), h(0)=1$$
v) Solve the ODE in iv) to obtain the mgf $h(t)$
$$h = exp\left\{\frac{1}{6}u^2t^3\right\}$$
