Apply and reverse a state recurrence with external change

An adjusted state recurrence

An adjusted recurrence first moves the current state with T and then applies the signed external-change column B. Positive and negative entries in B are kept as written, and the full rule is repeated for each period

1. Transition before adjustment Calculate the usual transition product before adding the signed external-change column \(\displaystyle S_{n+1}=T S_n+B\)
2. Apply the adjustment each period Use the same external-change column after every transition when the rule is repeated \(\displaystyle S_2=T S_1+B\)
3. Keep the signs in B A positive entry adds to a state and a negative entry removes from that state \(\displaystyle B=\begin{pmatrix}4\\-2\end{pmatrix}\)
4. Undo in reverse order To recover an earlier state, subtract B first and then multiply on the left by T inverse \(\displaystyle S_n=T^{-1}\left(S_{n+1}-B\right)\)

Worked example

A two-state model has transition matrix T, state S₀ and signed external change B. Find S₁

1. \(\displaystyle \frac{2}{3}\times9=6,\quad \frac{1}{3}\times6=2,\quad \frac{1}{3}\times9=3,\quad \frac{2}{3}\times6=4\)The first row of T gives the two contributions to the first component after the transition
2. \(\displaystyle T S_0=\begin{pmatrix}6+2\\3+4\end{pmatrix}=\begin{pmatrix}8\\7\end{pmatrix}\)Adding the contributions in each row gives the transitioned state
3. \(\displaystyle S_1=T S_0+B=\begin{pmatrix}8\\7\end{pmatrix}+\begin{pmatrix}3\\-2\end{pmatrix}=\begin{pmatrix}11\\5\end{pmatrix}\)Apply the signed external change after the transition to obtain S₁

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