Question

Consider the general two-state continuous-time Markov chain with transition rates overline{Q} = egin{pmatrix} 0 & mu \ lambda & 0 end{pmatrix} The states are labelled 1 and 2. (a) Write down the backward equations. (b) Show that for any $t ge 0$, $lambda p_{11}(t) + mu p_{21}(t) = lambda$. (c) Solve the backward equations. You may use the fact that the solution of the ordinary differential equation $x'(t) = ax(t) + b(t)$ is $x(t) = x(0)e^{at} + e^{at} int_0^t e^{-as}b(s)ds.$

          Consider the general two-state continuous-time Markov chain with transition rates
overline{Q} = egin{pmatrix} 0 & mu \ lambda & 0 end{pmatrix}
The states are labelled 1 and 2.
(a) Write down the backward equations.
(b) Show that for any $t ge 0$, $lambda p_{11}(t) + mu p_{21}(t) = lambda$.
(c) Solve the backward equations. You may use the fact that the solution of the ordinary
differential equation $x'(t) = ax(t) + b(t)$ is
$x(t) = x(0)e^{at} + e^{at} int_0^t e^{-as}b(s)ds.$

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Consider the general two-state continuous-time Markov chain with transition rates
overlineQ = eginpmatrix 0 mu lambda 0 endpmatrix
The states are labelled 1 and 2.
(a) Write down the backward equations.
(b) Show that for any t ge 0, lambda p11(t) + mu p21(t) = lambda.
(c) Solve the backward equations. You may use the fact that the solution of the ordinary
differential equation x'(t) = ax(t) + b(t) is
x(t) = x(0)e^at + e^at int0^t e^-asb(s)ds.

Added by Samantha L.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Sri K

02/26/2023

Step 1

(a) The backward equations for a two-state continuous-time Markov chain with transition rates $q_{12}$ and $q_{21}$ are given by: $$ \frac{d}{dt}p_{11}(t) = -q_{12}p_{11}(t) + q_{21}p_{21}(t) $$ $$ \frac{d}{dt}p_{21}(t) = q_{12}p_{11}(t) - q_{21}p_{21}(t) $$ Show more…

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Consider the general two-state continuous-time Markov chain with transition rates Q = (0 μ; λ 0) The states are labelled 1 and 2. (a) Write down the backward equations. (b) Show that for any t ≥ 0, λp11(t) + μp21(t) = λ. (c) Solve the backward equations. You may use the fact that the solution of the ordinary differential equation x'(t) = ax(t) + b(t) is x(t) = x(0)e^(at) + e^(at) ∫ e^(-as)b(s)ds.

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