Can someone explain the intutive meaning of Kolmogorov forward and backward differential equations? What is the difference between them ? How does their application vary in different situations ? A simple theoretical example could really help me out. I know my questions are a bit awkward, but i really cant understand them! Thanking you in advance.
Kolmogorov forward and backward differential equations are just differential forms of Chapman-Kolmogorov equation.... you can use it to get probabilities.
Why do we represent a kolmogorov equation as a combination of probability of transition and the other part as transition rate ? I mean why transition from i to k as p_ik and the then transition from k to j as a transition rate; summing over all values of k in forward kolmogorov equation ? Similar in case of backward equations ? What is the difference in the equations ? Are they applicable in different cases ? I hope you know what I mean to say! Thank you. Hoping my question are specific this time !
Suppose, you want to find \( P_{ij}(s,t) \) for discrete time Markov Chain, then you can do it simply by \(P^{t-s} \) for same probabilities or by adding multiplying by all paths or by Chapman-Kolmogorov equations and setting a middle time. But, what about Continuous time Markov Process? how many times would you add the pathway(which you would get from multiplying Transit-to-transit)? suppose, unit time is years.. how many pathways will you get if you check daily/hourly/minutely/secondly transitions? We use differential equations to handle such pathways(so that you would get all pathways in time interval of size \(h,~where~h\to 0 \)... and we need transition rates for such, and we can get only 2 forms of differential equations(Backward and Forward), as long as we're looking for single differential equation. for time-homogeneous you'll get same Backward and Forward equations(as we get differential equations by time length)... But for time-in-homogeneous case you'll see different LHS(you just take care the differential parameter) for applicability, you can apply any form you feel easy for available data.
Thnx a lot Hemant bhai. Your explanation cleared many things. I just want to know only one thing more that, what do the components of the backward and forward equations mean? I mean that both have one part as transition rate and the other part a probability, but in a different order in both. So what do these actually mean, how does their different order in both forward and backward equation apply to situations? I want the concept behind the equations. I hope you know what I mean to ask. Thanking you once again. Anyone's reply shall be helpful.
Hi Sahil, You're Welcome... Yes! I got what you wanna know, I explained in my way... Now, let's wait for others reply
First of all read the latter half of page 26 chapter 5. That somewhat explains the difference b/w the two equations. then solve the example on page 22 using both forward and backward equations and compare which method was easier.
