Sufficient conditions for a Brownian motion

[economicsnerd]

I'm given a probability measure $\mathbb P$ on $\Omega = \{f\in C([0,1],\mathbb R): \enspace f(0)=0\}$ and told that $\mathbb P$ satisfies i.i.d. increments.

I'm interested in the weakest additional conditions that will ensure that $\mathbb P$ describes a Brownian motion, i.e. that there is some $\mu\in\mathbb R$ and $\sigma\in\mathbb R_+$ such that $\mathbb P$ is the law of $X$ as described by $dX_t = \mu dt + \sigma dZ_t$ for a standard Brownian motion $Z_t$.

Does it already follow from the assumptions of i.i.d. increments and continuity? It seems like I should be fine (by CLT) as long as increments have well-defined mean and finite variance. Do these properties come for free? If not, can I get away with assuming less? For instance, is it enough to only assume finite variance?

 

[mmwave]

Thank you for your question. In order to ensure that the probability measure $\mathbb P$ describes a Brownian motion, in addition to the assumptions of i.i.d. increments and continuity, you will need to assume that the increments have a well-defined mean and finite variance. This is because the defining characteristic of a Brownian motion is that its increments are normally distributed with mean $\mu$ and variance $\sigma^2$. Without these assumptions, the CLT may not hold and the resulting process may not be a Brownian motion.

However, if you are only interested in the existence of a Brownian motion with the given probability measure $\mathbb P$, then you may be able to get away with assuming only finite variance. This is because the CLT still holds for finite variance, and you can use it to construct a Brownian motion with the desired probability measure.

In summary, to ensure that $\mathbb P$ describes a Brownian motion, you will need to assume that the increments have a well-defined mean and finite variance. If you are only interested in the existence of a Brownian motion with this probability measure, then assuming finite variance may be enough. I hope this helps. Good luck with your research!