bio | website | users.wpi.edu/~ssturm |
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location | Worcester, MA | |
age | 37 | |
visits | member for | 3 years, 4 months |
seen | 3 hours ago | |
stats | profile views | 408 |
Assistant Professor at Worcester Polytechnic Institute
Research interests: Mathematical Finance, Stochastic Analysis
Mar 6 |
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$\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ for continuous semimartingales?
Yes. Note also that $L^2$ convergence implies a.s. convergence only along a subsequence, but this is sufficient for your argument. |
Mar 6 |
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$\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ for continuous semimartingales?
While I agree that adding some $L^2$ integrability will make the lemma and in essence the proof correct, I fail to see why the stochastic integral may not exist. In general it is not in $L^2$ and might even not define a martingale, bu exists as local martingale... |
Mar 6 |
answered | $\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ for continuous semimartingales? |
Feb 3 |
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Question about the stochastic integral of martingales
Why should the expectation of the quadratic variation be finite? I thought this is true if and only if the martingale is square integrable? |
Jan 23 |
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Book on Convergence Concepts in Probability without Measure Theory
While I agree that concepts of convergence can be understood only via measure theory, one can get to a certain degree a good understanding of limit theorems without it. E.g., Emanuel Lesigne's "Heads or Tails" (ams.org/bookstore-getitem/item=STML-28) is a very readable introduction. |
Dec 18 |
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Do we need Feller condition if the process jumps?
I agree with this, only that the process can become numerically negative even in the case that Feller's condition is satisfied. An alternative way to circumvent this (due to Alfonsi, cermics.enpc.fr/~alfonsi/SC_preprint.pdf) is to rewrite the Ito Integrals into Stratonovich integrals and discretize only then. This leads to an implicit FD scheme that can even be turned into an explicit one if Feller's condition is satisfied. |
Dec 18 |
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Do we need Feller condition if the process jumps?
I can confirm that when calibrating the Heston model to equity data, the parameter are usually close to the boundary, sometimes they satisfy Feller, sometimes they don't. But this is a problem only insofar, as some theoretical results on the model require the condition to hold and you have to be careful to check if a given paper requires the Feller condition or it doesn't. |
Dec 17 |
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Do we need Feller condition if the process jumps?
And if you make this assumption, neither the diffusion nor the jump diffusion you specified reach zero a.s. |
Dec 17 |
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Do we need Feller condition if the process jumps?
I still do not understand what you mean by "The Feller condition holds". It is just an assumption on parameters. |
Dec 16 |
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Do we need Feller condition if the process jumps?
What do you mean by "need" the Feller condition. Do you have some specific proof in mind where as it is needed? the only thing it does is to assure you that you do not hit 0 a.s. As you have in your example only an upward jump part with finite jump actitvity, its presence does not change anything in this consideration. |
Nov 28 |
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Teaching stochastic calculus to students who know no measure theory (or PDE, or…)
I agree that he sweeps some things under the rug, one could add his treatment of quadratic variation or the non-treatment of local martingales (a quite important topic even for option pricing, e.g. in local and stochastic volatility models). But the balance between ease of exposition and rigor is hard to strike (do you really want to discuss tightness of probability measures in such a class?) and I think he is doing a very good job. I feel amending a textbook by other sources is now easier than ever, as many textbooks are in electronic form available for students for free... |
Nov 28 |
answered | Teaching stochastic calculus to students who know no measure theory (or PDE, or…) |
Nov 28 |
awarded | Yearling |
Nov 28 |
answered | Does $L^2$ progressive measurable processes form a Hilbert space? |
Nov 20 |
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First passage time of a pure drift process
I assume by moments you mean the moments of the marginals for some fixed t? I doubt that this will help much without knowing the transition probabilities of $X$. Could you be more precise how your process $X$ is given? |
Nov 20 |
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First passage time of a pure drift process
Your process $Z_t$ admits the explicit representation $e^{-\kappa t}\Bigl(Z_0 + \kappa \int_0^t e^{\kappa s} X_s \, ds\Bigr)$. So without knowing the distribution of $X_t$ or some particular properties (as bounded away from zero), I do not expect a general result. |
Jul 20 |
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Progressively measurable vs adapted
I think this is most easily seen by considering a counterexample. Let $(X(t))_{t\geq 0}$ be a deterministic, non-Borel-masurable function. Then this is clearly a stochastic process that is not progressively measurable. However, as for every $t$, $X(t)$ is just a constant, the process is trivially adapted to any filtration you want (as constants are measureable w.r.t. the trivial sigma field). |
Jul 19 |
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Rediscovery of lost mathematics
A good overview of Doblin's work in the "pli chachete" is given in this article by Bernard Bru and Marc Yor: ams.org/mathscinet/search/… |
Jul 19 |
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Rediscovery of lost mathematics
I am not sure how much this can count as rediscovery. There is no stochastic integral in Doblin's paper, and hence no actual Ito formula. What Doblin does, is to give a probabilistic characterization of a diffusion in terms of a time-changed Brownian motion. On the other hand this can be considered as even more anticipating, as the connection of random time changes and stochastic integrals was only discovered in 1965, by Dubins-Schwarz and Dambis... |
Apr 30 |
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Applied Math author order
@IgorRivin: SSRN has annoying feature to list first the submitting author, if you take not great precaution to avoid it. And after fact it is hard to change this. Thus the ordering in the html mask of SSRN differs often from the ordering in the actual paper. One should check really pdf. E.g, the first paper following your link is given as Goetzmann, Ibbotson and Brown, however if you do the actual work and load down the pdf, you will see that the actual order is Brown, Goetzmann and Ibbotson. |