BACKWARDKOLMOGOROV EQUATIONS
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DOI: https://doi.org/10.1007/1-4020-0611-X_50
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In a continuous-time Markov chain with state X(t) at time t, define pij(t) as the probability that X(t + s) = j, given that X(s) = i, s, t ≥ 0, and rij as the transition rate out of state i to state j. Then Kolmogorov's backward equations say that, for all states i, j and times t ≥ 0, the derivatives dpij(t)/dt = Σk≠irikpkj(t) − vipij(t), where vi is the transition rate out of state i, vi = Σjrij. Markov chains; Markov processes.
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