Taylor Martin
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 Jan 30 awarded Popular Question 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?