In the book "Stochastic calculus for finance-II" http://home.ustc.edu.cn/~matheming/Steven%20E.%20Shreve-Stochastic%20Calculus%20for%20Finance%20II.pdf , the author mentions two definitions of Markov process as equivalent in equations 2.3.29 and 2.3.30 (page 74 and 75). The equations are -
- $$\mathbb{E}[f(X(t))|\mathcal{F}(s)] = g(X(s))$$
2)$$\mathbb{E}[f(t, X(t))|\mathcal{F}(s)] = f(s, X(s))$$
under the usual conditions(please see link for details). Can someone show me how to get 2) from 1) ?