## The Kolmogorov Equations

The Kolmogorov equations are used during to model birth and death processes.

Suppose the population at timeiswith probabilityso we can writeSuppose also that the probability that the population changes fromto in the time intervalis written

At timewe can write

or(1)

If the birth rate when the population isis writtenand the death rate isthen

(2)

(3)

(3)

Substituting (2), (3) and (4) into (1) gives

so that

Thereforefor

These are the Kolmogorov forward equations.