ExERCISE: $($SEMI$-)$ MARKOV

Consider an insurance contract with the states alive $(1),$ dead $(2),$ and surrender

$(3),$ and the only non$-$zero intensities ${}_{12}(t)$ and ${}_{13}(t,u).$

A$.$ Write the Kolmogorov backward equations for this contract.

B$.$ Assume that the following payments are non$-$zero $b^{12}(t),b^1(t),b^{13}(t,u),$ and

$b^2(t).$ Interpret the payment structure in terms of an insurance contract, write

Thiele equations for the prospective reserves and determine all relevant risk sums.

C$.$ How do the above expressions simplify if you instead assume a pure Markov

model?