# sigma algebra stochastic process

what is the sigma field created by X(t,w) where t belongs to [0,1] and X(t,w) =1 if t=w and zero otherwise.

-

## 1 Answer

$X_{t}$ is a stochastic process. Maybe you mean a natural filtration?

The sigma algebra generated by $X_{t}$ is $\sigma(X_{t})=\{\emptyset,[0;1],t,[0;t)\cup (t;1]\}$.

The natural filtration $\mathcal{F}_{t}$ is the minimal sigma algebra which contains all $\sigma(X_{u})$ for $u\in [0;t]$.

So $\mathcal{F}_{t}$ consists of all countable subsets of $[0;t]$ and their complements.

-
The question reads like this: omega=[o,1]. B =b([0,1]) and Lamda is the the lebesgue measure. Define the process Xt {0<=t<=1} where Xt(w)=1 if w=t and Xt(w) = 0 otherwise. What is the sigma field generated by Xt? so the question only says process not a stochastic process. And my professor does not like the answer {null, omega, t and [0,t)U(t,1] }. THANKS! –  itsow Nov 16 '10 at 13:18
@none: it does seem that the answer should be $\sigma(X_t) = \lbrace {\rm null}, {\rm omega}, \lbrace t \rbrace, [0,t) \cup (t,1] \rbrace$ (the only difference is in the curly brackets around $t$). –  Shai Covo Nov 16 '10 at 15:45
okay...thanks guys –  itsow Nov 17 '10 at 16:19