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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:

  1. 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...?

  2. What is the infinitesmal generator $A$, defined for sufficiently nice $f$ by

$$ Af(x) = \lim_{s \rightarrow 0} \frac{E^x[f(X_{t+s})] - f(x)}{s}? $$

  1. Is the correlation property between $\sigma_t$ and $B_t$ relevant for these questions?
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  • $\begingroup$ ad 2.: $X$ is not going to be a Markov process. Anyway, $Af(x) = \frac{\sigma_0}{2} f''(x)$, $x \in \mathbb{R}$ for any $f \in C_0^\infty(\mathbb{R})$. $\endgroup$ – Thomas Rippl Jul 24 '15 at 14:30
  • $\begingroup$ @thomas Is that easy to argue rigorously? $\endgroup$ – Michael Jul 24 '15 at 19:52
  • $\begingroup$ No. Only if $\sigma_0$ is not random. Then it is "trivial". Otherwise you get $E[\sigma_0]/2 f''(x)$. ad 1.: Theorem I.4.40 in Jacod&Shiryaev, 2003, is a rather general formula to compute $\langle \cdot, \cdot \rangle$. ad 3: what is "correlation property"? $\endgroup$ – Thomas Rippl Jul 25 '15 at 7:44
  • $\begingroup$ @thomas Thank you. By "correlation property" I just mean whether, a clean answer is possible only if e.g., $\sigma_t$ and $B_t$ are independent, etc. $\endgroup$ – Michael Jul 25 '15 at 12:23

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