Consider an Ito process
$$ dX_t = \sigma_t dB_t $$
where $\sigma_t$ is a two-state continuous-time Markov chain with state space $\{ \sigma_1, \sigma_2 \}$ that switches state with Poisson intensity $\lambda$ and $B_t$ is standard Brownian motion.
Questions:
What does Ito's formula for $X_t$ looks like? Is there a formula to compute the quadratic variation $[X, X]_t$ when $X$ is an integral against a semi-martingale...?
What is the infinitesmal generator $A$, defined for sufficiently nice $f$ by
$$ Af(x) = \lim_{s \rightarrow 0} \frac{E^x[f(X_{t+s})] - f(x)}{s}? $$
- Is the correlation property between $\sigma_t$ and $B_t$ relevant for these questions?