Intuitively, why do we integrate from 0 to t-s in the formula for integrated Kolmogorov Backward equations? I understand the components of this formula but I don't understand why we have to integrate over 0 to t-s.

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For time-inhomogeneous Markov jump processes, the transition probabilities Pij(s,t) are defined as the probability that you are in state j at time t, given you were in state i at time s. You can think of times 's' and 't' as placeholders for any time period you want to consider the transition probability over. When we write the integrated CK equations, we want to take into account all possible 'starting time points' (s) and 'ending time points' (t). This is equivalent to considering all possible time intervals (t-s) over which the transition probability can occur.

Integrating from a lower bound of 0 to an upper bound of t-s is equivalent to integrating from a lower bound of s to an upper bound of t. The integral part of the Integrated CK equations accounts for all the different times at which transitions between states can occur. Since the Markov process is time inhomogeneous, in order to account for all possible values of 'w' in the integrand, we integrate from w=0 to w=t-s.

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