where is the transition rate for going from state to state and is the transition rate for going from going from state to state . The process is also known under the names Kac process (after mathematician Mark Kac),[1] and dichotomous random process.[2]
Solution
The master equation is compactly written in a matrix form by introducing a vector ,
where
is the transition rate matrix. The formal solution is constructed from the initial condition (that defines that at , the state is ) by
where is the identity matrix and is the average transition rate. As , the solution approaches a stationary distribution given by
Properties
Knowledge of an initial state decays exponentially. Therefore, for a time , the process will reach the following stationary values, denoted by subscript s:
1 2 Bondarenko, YV (2000). "Probabilistic Model for Description of Evolution of Financial Indices". Cybernetics and Systems Analysis. 36 (5): 738–742. doi:10.1023/A:1009437108439. S2CID115293176.
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