The proof that shows independent increments imply Markov Property, I am struggling to understand how we get from 2nd to the 3rd step: Pr(Xt - Xs + x E A | Xs E x). This 3rd step is saying that the increment depends on the current value Xs but independent increments mean that the increment is not dependent on current value Xs right?
Start with Pr(X_t+1 E A | X0 = i_0 , X_1 = i_1 , ... , X_t = i_t) ** From this given you know X_t and i_t you can just add and subtract it as follows Pr(X_t+1 - X_t + i_t E A | X0 = i_0 , X_1 = i_1 , ... , X_t = i_t) Now by the independent increments assumption the process X_(t+1) - X_t is independent of the stochastic process {X_s : 0<=s<=t}, so this means that everything thats been conditioned on in ** can just be deleted (Except X_t because the LHS uses the fact we conditioned on it = i_t) So By recognising again that - X_t + i_t = 0 you arrive at the markov property using only the indep. increments assumption : Pr(X_t+1 E A | X0 = i_0 , X_1 = i_1 , ... , X_t = i_t)= Pr(X_t+1 E A | X_t = i_t) => Process with independent increments has the marvov property Q.E.D
