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Financial mathematician-beginner.


Nov
12
comment Why does $\sigma (X_t) \subset \sigma (X)$ hold?
I meant the definition of a (measurable) cylinder. I still don't see why $X_t^{-1}(B_t)=X^{-1}(B)$. Please comment a bit more on that.
Nov
12
accepted Why does $\sigma (X_t) \subset \sigma (X)$ hold?
Nov
11
awarded  Yearling
Nov
10
comment Why does $\sigma (X_t) \subset \sigma (X)$ hold?
Thanks, answers cleared my questions up, sadly, I still need a (hopefully) small nudge. Measurable cylinders are where most of my problems with probability theory mostly occur. a) do we agree on the definition that $B_t=S_t$ except for finitely many $t$'s? If so then is $B$ such a set? and finally b) Why does $X_t^{-1}(B) = X^{-1}(B)$? I am not even sure what $X_t^{-1}(B)$ means - since B seems to be a collection of elements of $\prod S_t$.
Nov
10
comment Why does $\sigma (X_t) \subset \sigma (X)$ hold?
Thanks, this answer is still quite cryptic to me though. A couple of questions: a) $X_t$ is measurable in what sense? $\forall B\in\mathcal{S}_t$ we have $X^{-1}_t\in\sigma(X)$? b) Could you please elaborate on "it suffices to show.."? (It mostly confuses me as in the definition or $\sigma(X)$ we consider $\mathcal{S}$ as opposed to $\mathcal{S}_t$). c) What does the notation $(s_r)_{r \in T} $ mean? Sorry for being kinda slow, I have poor background in measure theory (working on that..)
Nov
9
revised Why does $\sigma (X_t) \subset \sigma (X)$ hold?
typo
Nov
9
asked Why does $\sigma (X_t) \subset \sigma (X)$ hold?
Oct
12
accepted Smallest $\sigma$-algebra and $\sigma$-algebra generated by a function
Oct
11
comment Smallest $\sigma$-algebra and $\sigma$-algebra generated by a function
We've defined them as the smallest sigma-algebra containing something we called "measurable cylinders". I could write down the definition, but is that relevant? Say this is the definition and not a theorem - then my question is that I don't understand the definition properly.
Oct
11
comment Smallest $\sigma$-algebra and $\sigma$-algebra generated by a function
I added the definition in the original post.
Oct
11
revised Smallest $\sigma$-algebra and $\sigma$-algebra generated by a function
added 158 characters in body
Oct
11
revised Smallest $\sigma$-algebra and $\sigma$-algebra generated by a function
edited tags
Oct
11
asked Smallest $\sigma$-algebra and $\sigma$-algebra generated by a function
Oct
8
awarded  Popular Question
Oct
2
awarded  Tumbleweed
Sep
25
revised Extrema of a functional simple question
added 13 characters in body
Sep
25
asked Extrema of a functional simple question
Sep
24
awarded  Autobiographer
Sep
23
accepted Switching $\int$ and $\sum$ proof
Sep
23
comment Switching $\int$ and $\sum$ proof
Ah and as we know integral is linear so it could also be expressed as a (finite) sum of integrals right?