Advanced Mortality Models

A multi-state model has states (e.g., active, disabled, dead) with given transition intensities, and you must compute t_p_{x}^{ij} or related expectations.

Write Kolmogorov forward equations for the transition probabilities. For models with constant intensities, solve the ODE system explicitly. For non-constant intensities, use numerical integration. For benefit valuation, the APV is the integral over t of the relevant transition probability times the discounting factor and the benefit at time t.

d/dt t_p_x^{ij} = Σ_{k ≠ j} t_p_x^{ik} μ_{x+t}^{kj} - Σ_{k ≠ j} t_p_x^{ij} μ_{x+t}^{jk}.

You are given integer-age mortality values and must compute a mid-year survival or expected life under UDD or constant force.

Under UDD: t_q_x = t · q_x for 0 ≤ t ≤ 1, giving a linear interpolation in deaths. Under constant force: t_p_x = (p_x)^t, an exponential interpolation in survivors. Under Balducci: 1-t_q_{x+t} = (1-t) q_x. Choose the assumption that matches the question (and is consistent with anything else stated in the problem).

UDD: t_q_x = t q_x; CF: t_p_x = p_x^t; Balducci: 1-t_q_{x+t} = (1-t) q_x.

A select and ultimate table with a non-trivial select period is given and you must compute APVs for a select issue.

Track the select duration alongside the attained age until you exit the select period. Within the select period, use q_{[x]+s}; beyond, use q_{x+s}. Build the cumulative survival product across the trajectory. For reserves on a select policy, recognize that the reserve at duration t reads off the table at age x + t with select duration t.

t_p_{[x]} = Π_{k=0}^{t-1} p_{[x]+k}; switch to ultimate when select period ends.