Let $(\Omega_1, \mathcal{F}_1, P_1)$ and $(\Omega_2, \mathcal{F}_2, P_2)$ be probability spaces and suppose $(X_t)$ and $(Y_t)$ are real-valued stochastic processes defined on the respective spaces. Furthermore, assume $(Y_t)$ is a Markov process. My question is the following:

Suppose $P_1\circ X_t^{-1}=P_2\circ Y_t^{-1}$ as measures on $\mathbb{R}$. Must $(X_t)$ be a Markov process?


No, that both processes have the same one-dimensional marginals is not sufficient. In contrary, when $X$ is an arbitrary elliptic Itô-process, you can always find a Markov process with the same marginals. Cf. I. Gyöngy, Mimicking the One-Dimensional Marginal Distributions of Processes Having an Itô Differential. Probab.Theory Relat.Fields 71(4), 501–516 (1986)

  • 1
    $\begingroup$ @Stozek: I do not know if I understand your question right, but if $X$ and $Y$ have the same finite dimensional marginals, the define by Kolmogorov's extension theorem the same process... $\endgroup$ – Stephan Sturm May 22 '12 at 15:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.