Consider a time-inhomogeneous continuous-time Markov chain with transition probabilities given by Prob(x(t) = j | x(s) = i) = Pij(s,t) for t ≥ s. Assume that P(s,t) = [Pij(s,t)] is a very smooth function of t and s (i.e., continuously differentiable). Define JP(s,t).
Show that aP(s,t)/at = Q(t). [Hint: Use P(s,t) = I + (t - s)Q(t) + o(t - s)].
Show that the following partial differential equation is satisfied by the transition probability matrix, JP(s,t): Q(s)P(s,t).
This equation is called the backward equation. [Hint: Use the Chapman-Kolmogorov equation P(s,t) = P(s,0)P(0,t)].
Similarly, show that the following equation (called the forward equation) is also satisfied: aP(s,t)/at = P(s,t)Q(t).
If x(t) is the vector of occupation probabilities at time t, show that 7'(t) = x(t)Q(t).