Why Do we need a right-continuous filtration with $\mathcal{F}_0$ containing all P-null sets?

When we consider a Stochastic Differential Equation we let $(\Omega,\mathcal F, \{\mathcal F_t\}_{t\ge 0},P )$ be a complete probability space with a filtration $\{\mathcal F_t\}_{t\ge 0}$ satisfying the usual conditions, i.e. it is right continuous and $\mathcal F_0$ contains all $P$-null sets.

Why are these "usual" conditions important ? What if we omit them ?

In Karatzas-Shreve 2.7 "Brownian filtrations" they go over the motivation for augmenting by null-sets and the right continuity. See also Properties of Feller Processes.

Here are some reasons:

• For every strong-Markov/Feller process, its augmented filtration is right-continuous $$\mathcal{F}_{t}=\bigcap_{s>t}\mathcal{F}_{t}.$$

eg. see the above references or Proof that augmented filtration is right continuous.

• One implication of right-continuity is Blumenthal’s zero-one law. Consider, for example, a standard Brownian motion B. It can be shown $$B_t$$ hits zero infinitely often for t in any neighborhood of 0, with probability one. Events such as this are $$\mathcal{F}_t$$-measurable for each $$t>0$$ and, therefore, $$\mathcal{F}_{0+}$$ measurable.

• Also, Brownian motion needs to be modified at a measure zero set to even have continuity by the Kolmogorov continuity theorem.

• The hitting time of open sets is also sensitive to right-continuity eg. seeWhen is the hitting time of an open set a stopping time?.

• from Right-continuity of filtrations: Suppose that $$X \in L^1$$ is a random variable and $$(D_t)_{t \geq 0}$$ a right-continuous filtration. Then $$\lim_{s \downarrow t} \mathbb{E}(X \mid D_s) = \mathbb{E}(X \mid D_t).$$