Hi,

On my probability space, I want to define a stochastic process, $\displaystyle I_t$ (indexed by time and time is continuous $\displaystyle t \in [0,\infty)$) that takes two values 1 or 0. There is some transition matrix that describes the probability of switching from one state back to another in some instant of time 'h'. So some kind of on off thing.

In addition to this there is another process,$\displaystyle H_t$ , on the same space that follows GBM (geometric Brownian motion). I want to derive the stochastic differential equation (using ito's lemma) of a function whose value is conditional on the value the other two processes, so something like $\displaystyle df(H_t, I_t) = $.

What I want to know is how I can write the stochastic differential of the two state markov process? what will $\displaystyle dI_t$ look like?

Thanks a lot,

Amatya