Question

Let \(O(t)\) be the occupation time for state 0 in the two-state continuous-time Markov chain. Find \(E[O(t) \mid X(0)=1]\).

Short answer

\(E[O(t) \mid X(0)=1] = t + \frac{1}{q_{01}}(e^{-q_{01} t} - 1).\)

Step-by-step solution

Define the Transition Probabilities and Rates

For a two-state continuous-time Markov chain, we have the following transition probabilities: 1. \(P_{01}(t) = P(X(t) = 0 \mid X(0) = 1)\): The probability of transitioning from state 1 to state 0 over a time interval of length \(t\). 2. \(P_{11}(t) = P(X(t) = 1 \mid X(0) = 1)\): The probability of remaining in state 1 over a time interval of length \(t\). We will also use the following transition rates: 1. \(q_{01}\): The rate of transitioning from state 1 to state 0. 2. \(q_{11} = -q_{01}\): The rate of remaining in state 1.

Solve the Kolmogorov Forward Equations for \(P_{01}(t)\) and \(P_{11}(t)\)

We have the following system of Kolmogorov forward equations: \(\frac{d}{dt}P_{01}(t) = q_{01} P_{11}(t)\\ \frac{d}{dt}P_{11}(t) = q_{11} P_{11}(t)\) Now, solve the equations with the given initial conditions, \(P_{01}(0) = 0\) and \(P_{11}(0) = 1\). We obtain: \(P_{11}(t) = e^{q_{11}t} = e^{-q_{01}t},\\ P_{01}(t) = 1 - e^{-q_{01}t}\).

Calculate the Expected Occupation Time for State 0

Since the occupation time of state 0, \(O(t)\), is a continuous function, we can find the expected value by integrating the occupation time over the time interval \([0, t]\): \[E[O(t) \mid X(0)=1] = \int_{0}^{t}E[O(u) \mid X(0)=1] \, du = \int_{0}^{t}P_{01}(u) \, du.\] Now, substitute the expression of \(P_{01}(t)\) from Step 2: \(E[O(t) \mid X(0)=1] = \int_{0}^{t}(1 - e^{-q_{01} u})du\). Integrate to get the expression for the expected occupation time: \(E[O(t) \mid X(0)=1] = \left. [u + \frac{1}{q_{01}}(e^{-q_{01} u} - 1) \right]_{0}^{t} = t + \frac{1}{q_{01}}(e^{-q_{01} t} - 1).\) Thus, the expected occupation time for state 0 is given by: \[E[O(t) \mid X(0)=1] = t + \frac{1}{q_{01}}(e^{-q_{01} t} - 1).\]