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Is the integrated form of the Kolmogorov forward equation correct?

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asked Jul 2 in BUS 3018F - Models by 25b14c23 (400 points)

The intuitive derivation for the integrated form of the Kolmogorov backward equation makes sense and is given in the Act Ed notes on page 17 and 18. However, if I use the same logic when deriving the integrated form of the Kolmogorov forward equation I get a different equation from the notes. 

My attempt is uploaded here: http://imgur.com/hGzy1fu

I am trying to follow the logic used in the backward case, noting that I change the focus from the first jump to the last jump.

Please help

1 Answer

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answered Jul 2 by Yaseen (530 points)
selected Jul 2 by 25b14c23
Best answer

The probability statement given in your point one involves an integral over the time from s to t+w where the integrand involves the holding probability multiplied by the instantaneous transition rate. The time points considered in this integral would be the value to use for A because that is the time point at which you enter state k. However, this would be far more complicated as it would involve double integrals. Considering the result given on page 19, I believe it is easier to derive the forward equation using the method proposed in ActEd and you should rather focus on this.

commented Jul 2 by 25b14c23 (400 points)

Thanks for the response. I've looked over the full proof and it makes sense so my derivation was definitely wrong. Points 1 and 2 specifically were incorrect and I've figured out what I need to fix