Consider the change of state in a very short time interval h, either Y(t) stays at its present state or it changes to another state. If the state changes from y1 to y2, we can approximate the transition probability as a linear function of h:
The coefficient W is the probability of transition per unit time. It's obvious that the probability that the state didn't change is the remaining probability:
Therefore we can write the transition probability as
![p_{t}(y_2|y_1)=[1-a(y_1)h]\delta(y_2-y_1)+hW(y_2|y_1)](http://www.codecogs.com/png.latex?p_%7Bt%7D%28y_2%7Cy_1%29=%5B1-a%28y_1%29h%5D%5Cdelta%28y_2-y_1%29+hW%28y_2%7Cy_1%29)
Denotes the probability distribution at time t by 
. By total probability:
Now we can rewrite the Chapman-Kolomogorov equation
![\begin{align*} p_{t+h}(y_3|y_1)&=\int p_{t}(y_2|y_1) p_{h}(y_3|y_2)dy_2\\ &=\int p_{t}(y_2|y_1)\left[ (1-a(y_2)h)\delta(y_3-y_2)+hW(y_3|y_2) \right]dy_2\\ &=hp_t(y_3|y_1)(1-a(y_3))+h\int p_t(y_2|y_1)W(y_3|y_2)dy_2 \end{align*}](http://www.codecogs.com/png.latex?%5Cbegin%7Balign*%7D&space;p_%7Bt+h%7D%28y_3%7Cy_1%29&=%5Cint&space;p_%7Bt%7D%28y_2%7Cy_1%29&space;p_%7Bh%7D%28y_3%7Cy_2%29dy_2%5C%5C&space;&=%5Cint&space;p_%7Bt%7D%28y_2%7Cy_1%29%5Cleft%5B&space;%281-a%28y_2%29h%29%5Cdelta%28y_3-y_2%29+hW%28y_3%7Cy_2%29&space;%5Cright%5Ddy_2%5C%5C&space;&=hp_t%28y_3%7Cy_1%29%281-a%28y_3%29%29+h%5Cint&space;p_t%28y_2%7Cy_1%29W%28y_3%7Cy_2%29dy_2&space;%5Cend%7Balign*%7D)
differentiate it with respect to h, we get
![\partial_t p_t(y_3|y_1) = \int [W(y_3|y_2)p_t(y_2|y_1)-W(y_2|y_3)p_t(y_3|y_1)] dy_2](http://www.codecogs.com/png.latex?%5Cpartial_t&space;p_t%28y_3%7Cy_1%29&space;=&space;%5Cint&space;%5BW%28y_3%7Cy_2%29p_t%28y_2%7Cy_1%29-W%28y_2%7Cy_3%29p_t%28y_3%7Cy_1%29%5D&space;dy_2)
This is the master equation.
If we assume the initial distribution to be a delta function
, then the solution of the distribution at time t can be found firstly by total probability
and then apply master equation to get
![\partial_t\pi_t(y)=\int [W(y|y')\pi_t(y')-W(y'|y)\pi_t(y)]dy'](http://www.codecogs.com/png.latex?%5Cpartial_t%5Cpi_t%28y%29=%5Cint&space;%5BW%28y%7Cy%27%29%5Cpi_t%28y%27%29-W%28y%27%7Cy%29%5Cpi_t%28y%29%5Ddy%27)
This equation has prominent physical interpretation, it's a gain-loss equation in the sense that the change of probability Y(t)=y is sum of the probability that y' transfer to y substract the probability that Y(t) leaves y to y'.
If we assume the initial distribution to be a delta function
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