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8.4.1 Kolmogorov's forward equations > 8.4.1 Kolmogorov's forward equations - Pg. 242

242 Time 0 Multiple state models t t+dt u Age Probability Event x ¢ 00 t p x x+t ¡ ¢ x+t+dt ¢ x+u ¡ µ 01 dt x+t transition to state 1 11 u-t p x+t in state 0 for t years in state 1 for u - t years Figure 8.6 The infinitesimal probability of this path is 00 01 t p x µ x+t 11 dt u-t p x+t 11 11 where we have written u-t p x+t instead of u-t-dt p x+t since the two are approx- imately equal if dt is small. Since the age at transfer, x + t, can be anywhere 01 between x and x + u, the total probability, u p x , is the `sum'(i.e. integral) of these probabilities from t = 0 to t = u. 2