Kolmogorov backward equation-integral form

[indchaks]

In question 6.19 where we have been asked to derive the differential form of Kolomogorov backward equation from the integral form , the suggested solution makes a substituion of of s+w =v and gets working. However, if one does not make that substitution and goes about differentiating the integral directly, the differential form does not get derived (at least I hv not been able to).
Let me illustrate first, if we follow the notation given in the Tables, b(y) = t-s
a(y)=0..hence differentiating wrt s, b'(y)=(-)1, f(b(y),y) gets multiplied by
P[l to j](s+w,t) which being evaluated at w=t-s becomes P(t,t) =0 (for l not equal to j ) and hence the first terms is 0 a'(y)=0, heice the second term is 0 too, as regards differentiating within the integral we get an additional term lambda(s+w) multiplied by the integral in addition to lambda (s) multiplied by the integral which simplifies but not to backward form
What am I missing?

 

[BeckyBoo]

Do we need to be able to derive the integrated versions of the forward and backward equations for the exam?

 

[John Potter]

Yes, You need to be able to construct the integral equations in the exam:
1) they can ask you directly to do this (and have done)
2) being able to construct them is sometimes the easiest way to derive probs in a Markov jump model - it avoids deriving differential equations and then using integrating factor. it's good to be flexible with both methods though