Question

If \(\\{X(t)\\}\) and \(\\{Y(t)\\}\) are independent continuous-time Markov chains, both of which are time reversible, show that the process \(\\{X(t), Y(t)\\}\) is also a time reversible Markov chain.

Short answer

Since both \(X(t))\) and \(Y(t)\) are Markov chains, their joint process \(\\{X(t), Y(t)\\}\) is also a Markov chain by proving the Markov property for any three times \(t_1<t_2<t_3\) and any three pairs \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\). As both \(X(t)\) and \(Y(t)\) are time reversible, the joint process satisfies time reversibility property too \(P((X(t+s), Y(t+s)) = (x_1, y_1) | (X(t), Y(t)) = (x_2, y_2)) = P((X(s), Y(s)) = (x_2, y_2) | (X(0), Y(0)) = (x_1, y_1))\). Thus, the joint process \(\\{X(t), Y(t)\\}\) is a time reversible Markov chain.

Step-by-step solution

Prove that the joint process is a Markov chain

To prove that the joint process \(\\{X(t), Y(t)\\}\) is a Markov chain, we need to show that for any three times \(t_1 < t_2 < t_3\) and any three pairs \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\), we have the Markov property: \(P((X(t_3), Y(t_3)) = (x_3, y_3) | (X(t_2), Y(t_2)) = (x_2, y_2), (X(t_1), Y(t_1)) = (x_1, y_1)) = P((X(t_3), Y(t_3)) = (x_3, y_3) | (X(t_2), Y(t_2)) = (x_2, y_2))\) Since we are given that \(X(t)\) and \(Y(t)\) are independent, we have \(P((X(t_3), Y(t_3)) = (x_3, y_3) | (X(t_2), Y(t_2)) = (x_2, y_2), (X(t_1), Y(t_1)) = (x_1, y_1)) = P(X(t_3) = x_3 | X(t_2) = x_2, X(t_1) = x_1)P(Y(t_3) = y_3 | Y(t_2) = y_2, Y(t_1) = y_1)\) Since both \(X(t)\) and \(Y(t)\) are Markov chains, we have \(P(X(t_3) = x_3 | X(t_2) = x_2, X(t_1) = x_1) = P(X(t_3) = x_3 | X(t_2) = x_2)\) and \(P(Y(t_3) = y_3 | Y(t_2) = y_2, Y(t_1) = y_1) = P(Y(t_3) = y_3 | Y(t_2) = y_2)\) Therefore, \(P((X(t_3), Y(t_3)) = (x_3, y_3) | (X(t_2), Y(t_2)) = (x_2, y_2), (X(t_1), Y(t_1)) = (x_1, y_1)) = P((X(t_3), Y(t_3)) = (x_3, y_3) | (X(t_2), Y(t_2)) = (x_2, y_2))\) Hence, the joint process \(\\{X(t), Y(t)\\}\) is a Markov chain.

Prove that the joint process is time reversible

Now, we need to show that if \(X(t)\) and \(Y(t)\) are time reversible, then so is their joint process. The joint process will be time reversible if, for any two pairs \((x_1, y_1)\) and \((x_2, y_2)\) and any time \(t\), we have \(P((X(t+s), Y(t+s)) = (x_1, y_1) | (X(t), Y(t)) = (x_2, y_2)) = P((X(s), Y(s)) = (x_2, y_2) | (X(0), Y(0)) = (x_1, y_1))\) We know that \(X(t)\) and \(Y(t)\) are time reversible, so we have \(P(X(t+s) = x_1 | X(t) = x_2) = P(X(s) = x_2 | X(0) = x_1)\) and \(P(Y(t+s) = y_1 | Y(t) = y_2) = P(Y(s) = y_2 | Y(0) = y_1)\) Using the independence of \(X(t)\) and \(Y(t)\), we have \(P((X(t+s), Y(t+s)) = (x_1, y_1) | (X(t), Y(t)) = (x_2, y_2)) = P(X(t+s) = x_1 | X(t) = x_2)P(Y(t+s) = y_1 | Y(t) = y_2) = P(X(s) = x_2 | X(0) = x_1)P(Y(s) = y_2 | Y(0) = y_1) = P((X(s), Y(s)) = (x_2, y_2) | (X(0), Y(0)) = (x_1, y_1))\) Thus, the joint process \(\\{X(t), Y(t)\\}\) is time reversible, proving the given exercise.