1,700 reputation
516
bio website mathtm.blogspot.com
location University of California Los Angeles, CA
age
visits member for 2 years, 8 months
seen 19 hours ago

I am a recent graduate in applied mathematics at UCLA and currently trying to break into the quantitative investment/trading industry while also continuing to pursue advanced graduate-level mathematics as both a hobby and career necessity.


Jan
17
revised Deriving the definition of stochastic integrals with respect to Ito processes from first principles
deleted 176 characters in body
Jan
17
comment Deriving the definition of stochastic integrals with respect to Ito processes from first principles
@TheBridge - Regarding your second comment, the convergence is true for each $t\geq0$ in $L^{p}(\Omega)$ ($1\leq p\leq2$) no matter how you construct the sequence of partitions $\Pi_{n}$ as long as $||\Pi_{n}||\to0$. As is well known, you can construct a particular sequence $\Pi_{n}$ so that for a fixed $\omega$, $\text{QV}^{2}(W)(\omega)=\alpha$ for any $\alpha>0$ including $+\infty.$ However, you can avoid all of this by demanding that $||\Pi_{n}||\to0$ such that $\sum_{n=1}^{\infty}||\Pi||_{n}<\infty$; then by Borel-Cantelli the QV process does converge pathwise $\omega$-a.s. to $t$.
Jan
17
comment Deriving the definition of stochastic integrals with respect to Ito processes from first principles
@TheBridge - Regarding your first comment, please provide your definition if you disagree with mine; however, I am using the definition that is regularly encountered in mathematical finance texts. Also, there really isn't any need to be pedantic here with the conditions of the coefficient processes; they are more or less implied and in any case you can just assume the ones regularly imposed: adadpability, $L^{2}(\Omega\times[0,t])$ integrable, and $\omega$-a.s. continuous as a function of $t$.
Jan
15
revised Deriving the definition of stochastic integrals with respect to Ito processes from first principles
added 171 characters in body
Jan
15
revised Deriving the definition of stochastic integrals with respect to Ito processes from first principles
added 423 characters in body
Jan
15
revised Deriving the definition of stochastic integrals with respect to Ito processes from first principles
added 21 characters in body; edited title
Jan
15
revised Deriving the definition of stochastic integrals with respect to Ito processes from first principles
added 96 characters in body
Jan
15
revised Deriving the definition of stochastic integrals with respect to Ito processes from first principles
deleted 34 characters in body
Jan
15
revised Deriving the definition of stochastic integrals with respect to Ito processes from first principles
deleted 34 characters in body
Jan
15
revised Deriving the definition of stochastic integrals with respect to Ito processes from first principles
deleted 34 characters in body
Jan
15
revised Deriving the definition of stochastic integrals with respect to Ito processes from first principles
deleted 95 characters in body
Jan
15
revised Deriving the definition of stochastic integrals with respect to Ito processes from first principles
added 53 characters in body
Jan
15
asked Deriving the definition of stochastic integrals with respect to Ito processes from first principles
Jan
5
answered Inequality of fourier coefficients
Jan
4
answered Proving $P \bigg( \bigcup_n \bigcap_{k = n}^{\infty}A_k \bigg) = lim_{n \rightarrow \infty}P \bigg( \bigcap_{k = n}^{\infty}A_k \bigg) $?
Dec
23
comment Is Hoffman-Kunze a good book to read next?
"Covers quite a bit more" as in there are multiple topics and extensions of covered topics presented in H&K that are not available in Axler's presentation of the subject. The most noteworthy are systems of linear equations, determinants and multilinear forms.
Dec
23
revised Is Hoffman-Kunze a good book to read next?
added 322 characters in body
Dec
23
answered Is Hoffman-Kunze a good book to read next?
Dec
23
comment What is the explicit obstruction to almost sure convergence in stochastic integrals?
I think what's actually amazing is the fact that $\int f\;dg$ exists for all continuous $f$ if and only if $g$ is of bounded variation (uniform boundedness principle). The fact that we have convergence in $d\mathbb{P}$-mean isn't really surprising because of Ito's isometry (which is the only technical part of the proof; once that's established the result is basically an application of well known $L^{p}$ approximation theorems). The question I'm asking is along the lines of what is the nature of the "bad" set $E_{n}$ corresponding to $\Pi_{n}$ that causes pathwise convergence to fail?
Dec
21
revised What is the explicit obstruction to almost sure convergence in stochastic integrals?
added 76 characters in body