I am trying to solve exercise no. 1.13 page 86 of 'Continuous time Martingales & Brownian Motion', by Revouz-Yor. It says that a stochastic process $X_t$ is a Guassian process (i.e. the marginal distributions are all gaussian), and has Markov property, if and only if it is satisfied the relation: $\forall s < t< u, \gamma(s,u)\gamma(t,t)=\gamma(s,t)\gamma(t,u)$,where $\gamma(s,t)=Cov(X_s,X_t)$, the covariance function. A hint I was given is to consider the process as the integral of a Brownian motion, $X_s=\int_{0}^{s}B_rdr$.
Asked
Active
Viewed 56 times
1
-
That hint cannot be from Revuz & Yor because $X_s=\int_0^sB_r,dr$ being Gaussian is not Markov. – Kurt G. Jun 10 '23 at 06:03
-
In fact it is not. A friend told me, i tried to work it out but could not find anything – Tralfamadorian26 Jun 10 '23 at 11:52
-
This answer mentions a reference for that proof. – Kurt G. Jun 10 '23 at 13:06