# Kolmogorov's Forward equations in BUS3024S

+1 vote
137 views

Hi guys, I am a bit confused about the Kolmogorov forward equations that we derive in Contingencies:

$$\frac{d}{dt}\ _tp_x = \Sigma_{k \neq j} (_tp_x^{ik}\mu_{x+t}^{kj} - _tp_x^{ij}\mu_{x+t}^{jk})$$

Because they look a bit different to those I have derived before in other stats courses and Models, which are:

$$\frac{d}{dt} \ _tp_x = \Sigma_{k \neq j } (_tp_x^{ik}\mu_{x+t}^{kj})$$

$$\sum_{k \neq j} (_tp_x^{ik}\mu_{x+t}^{kj} - _tp_x^{ij}\mu_{x+t}^{jk}) = \sum_{k \neq j} (_tp_x^{ik} \mu_{x+t}^{kj}) - \sum_{k \neq j} (_tp_x^{ij} \mu_{x+t}^{jk})$$
$$\sum_{k \neq j} (_tp_x^{ik}\mu_{x+t}^{kj} - _tp_x^{ij}\mu_{x+t}^{jk}) = \sum_{k \neq j} (_tp_x^{ik} \mu_{x+t}^{kj}) - _tp_x^{ij} \sum_{k \neq j} \mu_{x+t}^{jk}$$
$$\sum_{k \neq j} (_tp_x^{ik}\mu_{x+t}^{kj} - _tp_x^{ij}\mu_{x+t}^{jk}) = \sum_{k \neq j} (_tp_x^{ik} \mu_{x+t}^{kj}) + _tp_x^{ij}\mu_{x+t}^{jj}$$  since $$\mu_{x+t}^{jj} = - \sum_{k \neq j} \mu_{x+t}^{jk}$$. Therefore:
$$\sum_{k \neq j} (_tp_x^{ik}\mu_{x+t}^{kj} - _tp_x^{ij}\mu_{x+t}^{jk}) = \sum_{k} (_tp_x^{ik} \mu_{x+t}^{kj}) = \frac{d}{dt} \ _tp_x^{ij}$$