532 reputation
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bio website users.wpi.edu/~ssturm
location Worcester, MA
age 37
visits member for 3 years, 5 months
seen 3 hours ago
Assistant Professor at Worcester Polytechnic Institute Research interests: Mathematical Finance, Stochastic Analysis

May
12
revised Is $B(t-1)$ an Ito process?
Tried to improve explantaion.
May
10
revised Is $B(t-1)$ an Ito process?
added 13 characters in body
May
10
revised Is $B(t-1)$ an Ito process?
added 126 characters in body
May
9
answered Is $B(t-1)$ an Ito process?
Mar
6
comment $\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ 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
comment $\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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.