Reserve derivatives for the disability model with recovery

Computes the right-hand sides of the coupled Thiele differential equations in Equations (14.25) and (14.26) for the healthy-life reserve \({}_{t}\overline{V}^{(0)}\) and the disabled-life reserve \({}_{t}\overline{V}^{(1)}\).

Usage

thiele_dVdt_01(t, V0, V1, delta, Pbar, B, R, mu01, mu02, mu10, mu12)

Arguments

t

Time.

V0

Value of \({}_{t}\overline{V}^{(0)}\).

V1

Value of \({}_{t}\overline{V}^{(1)}\).

delta

Force of interest.

Pbar

Continuous premium rate.

B

Death benefit.

R

Continuous disability income rate.

mu01

Function of time returning \(\mu_{x+t}^{01}\).

mu02

Function of time returning \(\mu_{x+t}^{02}\).

mu10

Function of time returning \(\mu_{x+t}^{10}\).

mu12

Function of time returning \(\mu_{x+t}^{12}\).

Value

A named numeric vector with components dV0 and dV1.

Details

The equations are $$ \frac{d}{dt}{}_{t}\overline{V}^{(0)} = \overline{P} + \delta {}_{t}\overline{V}^{(0)} - \mu_{x+t}^{02}(B-{}_{t}\overline{V}^{(0)}) - \mu_{x+t}^{01}({}_{t}\overline{V}^{(1)}-{}_{t}\overline{V}^{(0)}) $$ and $$ \frac{d}{dt}{}_{t}\overline{V}^{(1)} = \delta {}_{t}\overline{V}^{(1)} - R - \mu_{x+t}^{12}(B-{}_{t}\overline{V}^{(1)}) - \mu_{x+t}^{10}({}_{t}\overline{V}^{(0)}-{}_{t}\overline{V}^{(1)}) $$

Examples

mu01 <- function(t) 0.10 * t + 0.20
mu02 <- function(t) 0.20
mu10 <- function(t) 0.50
mu12 <- function(t) 0.125 * t + 0.20

thiele_dVdt_01(
  t = 2.0, V0 = 0, V1 = 0,
  delta = 0.04, Pbar = 446.95,
  B = 1000, R = 1000,
  mu01 = mu01, mu02 = mu02, mu10 = mu10, mu12 = mu12
)
#>      dV0      dV1 
#>   246.95 -1450.00