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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 ?

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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 $$\lim_{s \downarrow t} \mathbb{E}(X \mid D_s) = \mathbb{E}(X \mid D_t).$$

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