User hicham - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T07:15:13Z http://mathoverflow.net/feeds/user/30889 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/130609/generalisations-of-the-gronwalls-lemma/130617#130617 Answer by Hicham for Generalisations of the Gronwall's lemma Hicham 2013-05-14T18:18:58Z 2013-05-14T18:18:58Z <p>You can calculate the seconde derivative of $F^{1/p}$. you'll find that it is negative, and the seconde derivative of $G^{1/p}$ is 0, so we have $(F^{1/p})^{''}\leq (G^{1/p})^{''}$ Then by integrating twice you'll have $F^{1/p}\leq G^{1/p}$ and then $F\leq G$</p> http://mathoverflow.net/questions/48477/never-appeared-forthcoming-papers/124001#124001 Answer by Hicham for Never appeared forthcoming papers Hicham 2013-03-08T17:37:28Z 2013-03-08T17:37:28Z <p>There is a result by Oesterle, that proves that you can find the first non residue quadratic modulo a prime in no more than $70\log(p)^2$ step assuming the GRH, this result was then improved by Bach who replaced the constant $70$ by $2$. The result of Oesterle was never published and when I asked him why, he told me because the laptop containing the proof was stolen from his car. However I think he exposed his proof to the mathematical community, so it is widely recognized. </p> http://mathoverflow.net/questions/123735/relationship-between-the-derivative-of-a-matrix-and-its-eigenvalues/123770#123770 Answer by Hicham for Relationship between the derivative of a matrix and its eigenvalues Hicham 2013-03-06T15:10:37Z 2013-03-06T16:06:24Z <p>Sorry for posting this as an answer, I cannot leave comment.</p> <p>Are you sure your matrix is $$[A(k)]_{j\,l}= -\frac{e^{ik|y_j-y_l|}}{4\pi|y_j-y_l|}\, j\neq l$$ and not $$[A(k)]_{j\,l}= -\frac{e^{ik|y_j-y_l|}-1}{4\pi|y_j-y_l|}\, j\neq l$$</p> <p>If you want to extend some properties of the derivative of a matrix to its eigenvalue, the eigenvectors have to be independent of the derivation variable. </p> http://mathoverflow.net/questions/120124/compute-the-expected-value-of-the-product-between-a-lebesguestieltjes-type-integ/120125#120125 Answer by Hicham for Compute the expected value of the product between a Lebesgue–Stieltjes type integral and an Ito integral Hicham 2013-01-28T16:38:19Z 2013-01-28T17:44:39Z <p>You can write $h(s)=\int_0^Tf(t)dtH(s)$ then your expectation could be written as $E[ \int_{o}^{T} h(s) dW(s)]$. This integral is $0$ if you can prove $(\int_{o}^{t} h(s) dW(s))$ to be a martingale, for example $E(\int_{o}^{T} h^2(s) ds)&lt;+\infty$. </p> <p>As you noticed in your comment, this procedure is correct if we can suppose that the process $h(s)$ is adapted, which is the case if $f$ is deterministic. Otherwise we can the decompose the integral as the in the following</p> <p>$$\int_0^T(\int_0^Tf(t)dt)H(s)dW_s =\int_0^T(\int_0^sf(t)dt)H(s)dW_s+\int_0^T(\int_s^Tf(t)dt)H(s)dW_s$$</p> <p>then with interverting of the order of integration in the second integral we can write $$\int_0^T(\int_s^Tf(t)dt)H(s)dW_s=\int_0^T(\int_0^tH(s)dW_s)f(t)dt$$</p> <p>The process apperaing in the integrals are now adapted, and you can add the condition so that Fubini works and to justify the martangality of the integrals. Hope thsi help.</p> http://mathoverflow.net/questions/119713/financial-mathematics-books/119739#119739 Answer by Hicham for Financial Mathematics Books Hicham 2013-01-24T10:09:32Z 2013-01-24T10:17:36Z <p>I definitely recommand "Introduction to Stochastic Calculus Applied to Finance" by Lamberton and Lapeyre. It is concise, precise, introduce all the mathematics you want by constructing the Ito calculus, the Black-Scholes model, formulation of the pricing via martingales and PDE, some interest rate theory and then introducing jumps and finishing by the algorithmic side. Then you can read "The Concepts and Practice of Mathematical Finance " by Mark Joshi where you can gain more insight on the financial side and the technics used by the practionners. I think this could be a good start.</p> <p>There is also the excellent lectures by Emmanuel Derman that you can find here <a href="http://www.ederman.com/new/docs/laughter.html" rel="nofollow">http://www.ederman.com/new/docs/laughter.html</a>. You can read theme even before the Joshi book. It goes from the basics to local-stochastic volatility in a physicist spirit with lot of intuition</p> http://mathoverflow.net/questions/126905/proof-that-the-browian-motion-is-a-martingale/126908#126908 Comment by Hicham Hicham 2013-04-08T22:06:07Z 2013-04-08T22:06:07Z I replaced $(B_t)_t$ by $(B_t)_{t\leq s}$, hope it's clearer. http://mathoverflow.net/questions/126905/proof-that-the-browian-motion-is-a-martingale/126908#126908 Comment by Hicham Hicham 2013-04-08T21:59:41Z 2013-04-08T21:59:41Z By $(B_t)_t$ I mean the whole process not just a particular value. http://mathoverflow.net/questions/125316/is-there-a-method-to-solve-this-number-theoretic-problem-concerning-primes/125317#125317 Comment by Hicham Hicham 2013-03-22T20:14:06Z 2013-03-22T20:14:06Z Yes, see the general form of the Dirichlet theorem. http://mathoverflow.net/questions/123735/relationship-between-the-derivative-of-a-matrix-and-its-eigenvalues/123770#123770 Comment by Hicham Hicham 2013-03-06T16:20:58Z 2013-03-06T16:20:58Z which page are you looking at? http://mathoverflow.net/questions/123735/relationship-between-the-derivative-of-a-matrix-and-its-eigenvalues/123770#123770 Comment by Hicham Hicham 2013-03-06T16:08:34Z 2013-03-06T16:08:34Z I don't see how to make further progress, could you give the reference of the book you've mentioned? http://mathoverflow.net/questions/123735/relationship-between-the-derivative-of-a-matrix-and-its-eigenvalues/123770#123770 Comment by Hicham Hicham 2013-03-06T15:29:35Z 2013-03-06T15:29:35Z Do you have additional information on the $\alpha_i$ or they are of general form? http://mathoverflow.net/questions/120124/compute-the-expected-value-of-the-product-between-a-lebesguestieltjes-type-integ/120125#120125 Comment by Hicham Hicham 2013-01-28T17:45:27Z 2013-01-28T17:45:27Z I edited the first answer to take into account your comment