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1h
comment Martingale with respect to a decreasing filtration
@ Iliana : Using time inversion $s=1/T$ you get $B_s$ is a brownian motion in the filtration $\mathcal{G}_s=\mathcal{F}_{1/t}$ which puts you back in the usual situation. Best regards
5h
comment Girsanov theorem for Ito diffusion process
@ zebullon : You should notice that $Z$ is the radon nikodym process so that $E_P[\bar{W}_t.Z_t]=E_Q[\bar{W}_t]$. Your second point is still unclear to me. Best regards
2d
comment Modified Doob's $L^1$ inequality
@ userXXX : Write the right hand side as an expectation then condition over the closest sigma algebra F_{n_1}, use positivity and submartingale property and repeat the operation. I dont know if it that works though. Best regards.
2d
revised Modified Doob's $L^1$ inequality
edited body
2d
comment Modified Doob's $L^1$ inequality
@ use207952 : I would go for tower property using it iteratively in order to try to show successive inequalities by conditioning. No warranty this work but might worth a try. Best regards.
Jan
26
comment Modified Doob's $L^1$ inequality
@ user207952: In your comment about point (1), do you mean $ \int_{sup_{k \leq n} X_k \geq 2\lambda} X_n dP$ ? Best regards
Jan
24
reviewed Approve How do I solve these four simultaneous equations?
Jan
22
revised Meaning of $\mathcal A_{\tau}$ for stopping time $\tau$.
added 18 characters in body
Jan
22
revised Meaning of $\mathcal A_{\tau}$ for stopping time $\tau$.
added 942 characters in body
Jan
21
reviewed Approve Is it always possible to extend a ring to a unital ring?
Jan
21
answered Meaning of $\mathcal A_{\tau}$ for stopping time $\tau$.
Jan
21
revised Meaning of $\mathcal A_{\tau}$ for stopping time $\tau$.
corrected a few typos that had made the question meaningless
Jan
20
comment Deterministic integrals involving a Brownian motion
@ Did : No need to be sorry, if it has already been solved in an old post then everything is fine. Best regards
Jan
19
comment Deterministic integrals involving a Brownian motion
@ Adam : Think about Integration by Parts formula, then you should arrive to a sum of Gaussian unless mistaken (I haven't the calculations though too lazy for that). Best regards
Jan
17
reviewed Reject Prove that $\lim_{b\to\infty}_{a\to0+}\int_a^b\frac{\hat{f}(\xi)}\xi d\xi=-\pi i\int_0^\infty f(x)dx$
Jan
16
comment Deriving the definition of stochastic integrals with respect to Ito processes from first principles
Last remark I have read the "Ito's proof" in your blog but I think you have missed a detail, the convergence of the quadratic variation of Brownian motion to $T$ is true in probability over the set of all partition of time interval [0,T]. Otherwise the convergence is a.s. to $\infty$. To keep an almost sure CV you have to add the constraint of refining sequence of partitions (this is why stochastic integration need a probability measure and cannot be performed in an probabilistically independent way).You have to be cautious with the way you use limits. Best regards.
Jan
16
comment Deriving the definition of stochastic integrals with respect to Ito processes from first principles
@ Taylor Martin : I am not sure to agree with the way you seem to define your stochastic integral with respect to $X$ (at least I don't consider this as a classical construction, maybe a result ca be build in this direction though). Moreover you shoud add conditions on $\Theta, \Delta, \Gamma$ processes for all this to make sense in the first place (unless you want to reason heuristically first). My advice is to start with elementary processes integrable with respect to $X$ then extend the result to more general process $\Gamma$. Best regards
Jan
13
comment Natural Filtration and Sigma-Field Generated by path function
If you have a borel sigma field you should precise the topology associated with it. I know that in Karatzas and Shreve's book on Brownian motion and stochastic calculus there is an exercise that allow to show if I remember well that both are equivalent for the sup over compact topology, I don't know the answer for the general problem but it might be possible to elaborate further from this example. Best regards
Jan
6
reviewed Reject How do I find this limit?
Jan
5
reviewed Approve Does the eigenvectors of a sub-block matrix are contained in the eigenvectors of the original matrix?