why we can write down the following equation?
Please give me some advice.
dt is in every part, where is dX?So, the interpretation of P(S(t),I(t),t) is as the conditional expectation of I(T), as of time t (<T) given the current values S(t) and I(t). Since it is a conditional expectation process, it's a martingale and thus has expected increments equal to zero, which explains the right hand side, of the underlines equation. The left hand side follows from applying Ito's lemma to P and substituting in the drift and volatility terms of S(t) and I(t). I(t), in fact, has zero instantaneous volatility (quadratic variation) and thus no second order Ito term.
The left hand side is the expected value of dP.dt is in every part, where is dX?So, the interpretation of P(S(t),I(t),t) is as the conditional expectation of I(T), as of time t (<T) given the current values S(t) and I(t). Since it is a conditional expectation process, it's a martingale and thus has expected increments equal to zero, which explains the right hand side, of the underlines equation. The left hand side follows from applying Ito's lemma to P and substituting in the drift and volatility terms of S(t) and I(t). I(t), in fact, has zero instantaneous volatility (quadratic variation) and thus no second order Ito term.