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