KOLMOGOROV,S FORWARD DIFFERENTIAL EQUATION

Dear Team ,

Please provide solution for derivation of kolmogorov,s forward differential equation

i fail to under stand how Pij(t+h) = ∑ pik (t) pkj (h) = pij(t) + h ∑ pik (t) µkj + o (h)

as per my understanding:

Pij(t+h) = ∑ pik (t) pkj (h) --------------------------(1)

now pkj (h) is a transition from k to j state in small time interval h is given by formulae in chapter 5 page 16 equation 5.7 which is as follows :

pij(h) = {h µij + o (h) if i not equal to j and is 1+ h µij + o (h) if i = j

now we can substitute it in equation 1
Pij(t+h) = ∑ pik (t) pkj (h)

= ∑ pik (t) µkj + o (h)

please provide steps after this to derive the equation mentioned above.

regards
suresh sharma

 

You seem to have everything right here, except for where you substitute in for pkj(h).

First, you've missed an h multiplying the summation in your final formula (I think this is just a typo).

Next, note that you have (correctly) said that the formula for pij(h) is different when i is not equal to j, and when i = j. In your substitution, you appear only to have used the form where i is not equal to j.

When i = j, you get an extra 1 in the formula for pij(h), and this gives you the extra pij(t) term that you're missing.

 

Thank you sir

 

Dear sir
I am having problem in splitting of" I "not equal to j and "I "equal to j .

As I would like to move from state "i" to j in time t+h when I not equal to j . But when I equal to j . I am in state I during time t+h

It's confusion .please clarify the aplitting of time t+h for i=j and I not equal to j

Regards
Suresh sharma

 

I don't really understand your question here. But I've explained the difference between i = j and i not equal to j in my answer to your other post.

 

yes sir now its cleared.

 

Mark,
Thanks for the explanation and if I may request further explanation on term pij(t)*h*ujj which doesn’t appear in final equation.

Thanks,
Sunil

 

Thanks. Got it another thread after hours of search
Sunil