To prove it rigorously, for example, the Markov property for the Brownian Bridge, we have to do some calculus. Let (Wt)t≥0 be the standard Brownian motion issued from 0, and we have (Bt)s≤t≤T defined as
Bt=x+Wt−Ws−t−sT−s(x+WT−Ws),
a Brownian Bridge between s,T and at s it is x and at T its value is 0. One way to see this formula is that the term x+Wt−Ws is the Brownian Motion while we have to reduce the term at the endpoint T. Some simple calculus shows that it is equal to
Bt=T−tT−s(x+Wt−Ws)−t−sT−s(WT−Wt).
A Markov property is very simple but requires calculus: we would like to show that for s<r<t<T we have
Bt=Br+Wt−Wr−t−rT−r(Br+WT−Wr).(⋆)
Now we prove it. An intermediate step tells us
−t−rT−r(Br+WT−Wr)=−t−rT−s(x+WT−Ws) and we put it into the formula that
RHS(⋆)=x+Wr−Ws−r−sT−s(x+WT−Ws)+Wt−Wr−t−rT−s(x+WT−Ws)=x+Wt−Ws−r−sT−s(x+WT−Ws)−t−rT−s(x+WT−Ws)=x+Wt−Ws−t−sT−s(x+WT−Ws).
This proves the Markov property. Then the strong Markov property is just an approximation and the regularity of the trajectory.
