Taylor Martin

University of California Los Angeles, CA

mathtm.blogspot.com

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.

7h
comment What is the explicit obstruction to the failure of pointwise convergence in the stochastic integral?
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?
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awarded Custodian
7h
revised What is the explicit obstruction to the failure of pointwise convergence in the stochastic integral?
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reviewed Reject suggested edit on What is the explicit obstruction to the failure of pointwise convergence in the stochastic integral?
8h
asked What is the explicit obstruction to the failure of pointwise convergence in the stochastic integral?
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awarded Tenacious
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answered Suppose that $f: \mathbb R \to \mathbb R$ is a continuous function and there is a number $p \in [a,b]$ so that $f(p) = q$.
Dec
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comment Proving uniform convergence of an integral-defined function on compact sets
Ah, then you will have to adjust the limits in the integral accordingly to take into account the domain. But note for instance that if $f$ is supported on $K\subset\mathbb{R}$, then $f_{\epsilon}$ is supported on a subset $K_{\epsilon}\subset K$. The specific range of $f$ is irrelevant, only that it allows for $\int_{\mathbb{R}} f=1$ and that $f\geq0$.
Dec
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revised Proving uniform convergence of an integral-defined function on compact sets
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Dec
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comment Proving uniform convergence of an integral-defined function on compact sets
Actually, since uniform convergence is usually harder to demonstrate than pointwise convergence, a good starting point would be to prove pointwise convergence first, even if restricted to compact sets. I elaborated on my answer a bit; hopefully it will get you started.
Dec
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revised If $\lim_{n \to \infty} f_n=f$ (Almost everywhere) then $\lim_{n \to \infty} f_n=f$ ( in measure on$E$)
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revised $y=ce^{y/x};\quad y'=y^2/(xy-x^2)$
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answered $y=ce^{y/x};\quad y'=y^2/(xy-x^2)$
Dec
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revised If $\lim_{n \to \infty} f_n=f$ (Almost everywhere) then $\lim_{n \to \infty} f_n=f$ ( in measure on$E$)
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Dec
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answered Proving uniform convergence of an integral-defined function on compact sets
Dec
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answered If $\lim_{n \to \infty} f_n=f$ (Almost everywhere) then $\lim_{n \to \infty} f_n=f$ ( in measure on$E$)
Dec
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comment Proving uniform convergence of an integral-defined function on compact sets
In the statement of your problem, does $f$ have compact support?
Dec
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comment Orders of growth of typical sequences
Are you using the $\ll$ symbol to mean $P(n)\ll Q(n)$ if there is an absolute constant $C$ such that $P(n)\leq CQ(n)$ or that $P(n)/Q(n)\leq C$ for $n\geq N$ for some $N$ sufficiently large; or is it notation for one quantity being "much smaller" than the other? Also, are the numbers $p_{j},q_{j}$ and $a_{j},b_{j}$ arbitrary pairs with no relation between the indices?
Nov
20
comment Sum to infinity of the sum 1/n^2
Look to the right as you type this question and you'll see a ton of similar questions w/ answers. There's even one with literally dozens of solutions.
Nov
17
answered Equivalency of Norms and the Open Mapping Theorem
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