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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
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..)
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.
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?
Sep
23
comment Rational function inequality
Added that aswell
Sep
23
comment Rational function inequality
Ah right, sorry. Polynomials, I'l add that.
Sep
21
comment Limit Point of Zero Points Implying a Zero Function
Thanks. This has got to be it. Which brings me to the question why am I not aware of that. Is that perhaps a corollary of some more "famous" theorem?
Sep
19
comment Strong and weak extrema
I feel like I follow and agree with everything you wrote. But it is exactly this that confuses me. When we add the stronger condition $P'$, we, as you noted, we eliminate some functions. Take a function that we eliminate. This function would then be within the neighborhood of the strong extrema, but not the weak one. How does that then imply the implication strong $\implies$ weak? In other words, it seems as if the weak extremum is "more of an extremum" than the strong one.
Jun
5
comment A High School Exponential Decay Question
Ha, how neat! Of course, not going to explain this one to my student, but I like it.
Jun
3
comment probability that all players score
and one: The free throw awarded to a shooter who is fouled while scoring. Source: en.wikipedia.org/wiki/Glossary_of_basketball_terms
Jun
3
comment Calculating VaR, CVaR
@RRL Thank you. Are you sure the bounds in the CVaR should not then be reversed, since I am dealing with a profit, rather than a loss density function?
Jun
3
comment Discrete Mathematics - Is my answer correct?
So, in fact, my original answer was correct?
Jun
3
comment Fair Value Of a Call Option
Thank you very much. This question is a question from a past exam from a lecture and your answer seems to go beyond the scope of the lecture (don't get me wrong, it is perfectly understandable, I just cannot imagine a person that'd answer this, having only done the course, as we've never done anything like that and one would have to pull it out of thin air using a great deal of intuition). Thus I am wondering, are there perhaps any hidden assumptions or simplifications (even if they change the nature of the question a bit) that could've been made in a more rudimentary course?
Jun
2
comment A Quick Set Notation Question
Yes, you are correct, I was thinking of an imaginary second vertical bar, having two "such thats". Thanks.
Jun
2
comment A Quick Set Notation Question
Ah, right, because there's no vertical bar or colon between the two brackets denoting "such that", is that correct?
Jun
2
comment A Quick Set Notation Question
Is it always false? I think by now we can conclude that this is almost certainly a mistake, but I think that I am not understanding this wrong notation correctly. Could you elaborate on why is it an empty set?
Jun
2
comment Convexity of a rational function
@MichaelGrant you are correct, added (dis) in the first sentence.
May
26
comment Understanding this calculus simplification
In cases like this, it's best to take the end result and differentiate it, it usually gives you a clue of what happened.