Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question

1 Answer 1

up vote 3 down vote accepted

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)

share|improve this answer
1  
@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... –  Stephan Sturm May 22 '12 at 15:39

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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