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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
Dec
21
comment What is the explicit obstruction to almost sure convergence in stochastic integrals?
Here's another observation that leads me to believe the usual uniform $t/n$ partition is not sufficient to force pointwise convergence. If $E_{n}$ is the "bad" set on which $X_{n}$ is not close to the limit $X$, and if it marches around the sample space, then it will obstruct pointwise convergence if $P(E_{n})\to0$ sufficiently slow such that there is enough mass to always cover a set of points of positive mass. But Borel Cantelli says that if $\sum P(E_{n})<\infty$, then $E_{n}$ loses mass sufficiently fast that it can't do this, resulting in pointwise convergence; but $\sum n^{-1}=\infty$
Dec
21
comment What is the explicit obstruction to almost sure convergence in stochastic integrals?
Since the question is somewhat imprecise, maybe answering this would be easier. Since convergence in probability implies a.s. convergence of a subsequence, if we take a rapidly decreasing sequence of partitions $\Pi_{n}$, we recover pathwise convergence. Is it possible to illustrate two explicit sequences of partitions whereby one results in convergence along both modes and the other just in probability? Furthermore, does the usual $t/n$ partition decrease rapidly enough to achieve a.s. convergence?
Dec
21
awarded  Custodian
Dec
21
revised What is the explicit obstruction to almost sure convergence in stochastic integrals?
added 225 characters in body
Dec
21
reviewed Reject What is the explicit obstruction to almost sure convergence in stochastic integrals?