Kolomgorov's Equations

Hi.

Can any one tell me why the backwards equation has a minus sign in front of it for the inhomogeneous case but not for the homogeneous case?

Thanks

 

Isn't it just to do with the sigma(i,i) = -sum( sigma(i,j) )

 

No. It's because in the time-inhomogeneous case we are differentiating wrt s, ie the equations are of the form:

d/ds P(s,t) = ...

In the time-homogeneous case it's only the length of the time interval that is important and we have:

d/dt P(t) = ...