Dear all,

generally speaking, my question is about which properties of a stochastic process are preserved when I skip from the original to the augmented filtration.

Recall that if $(\mathcal{F}_t)_{t\geq0}$is a filtration on a probability space $(\Omega,\mathcal{A},P)$, the augmented filtration $(\mathcal{F}^0_t)_{t\geq0}$ is the smallest filtration such that

$\mathcal{F}_t\subseteq \mathcal{F}^0_t$ for all $t\geq0$

$N\subseteq A\in\mathcal{A}$ and $P(A)=0$ imply $N\in\mathcal{F}^0_0$ (complete)

- $\mathcal{F}_t^0=\mathcal{F}_{t+}^0$ for all $t\geq0$ (right-continuous)

It is well known that in many situations it is desirable to work with a complete and right-continuous filtration (for example, this ensures that martingales have a right-continuous modification, or that certain random times are stopping times).

Now here is my question in detail: Suppose we consider a stochastic process $(X(t))_{t\geq0}$and its natural filtration $(\mathcal{F}_t)_{t\geq0}$. And suppose we know that $X$ is a (strong) Markov process and a martingale. Do these properties still hold with respect to the augmented filtration? Or what about martingale problems: Existence results for martingale problems usually refer to the natural filtration of the potential solution process (that is the problem is "find a process which solves the martingale problem with respect to its own natural filtration"). So if I have established existence with respect to the natural filtration does the solution process solve the martingale problem with respect to the augmented filtration, as well?

In fact, my question reduces to the following: If working with the augmented filtration has many advantages why still bother with natural filtrations??

I am aware that these are a lot of questions packed into one post. Sorry, for not being more concise. I would greatly appreciate any comment which helps me to clarify these issues.

Best regards

lpdbw