I am trying to prove the following via the direct definition of Ito integrals (i.e. without the use of the Ito formula).     $$ \int^{t}_{0}B^{2}_{s}dB_{s}=\frac{1}{3}B^{3}_{t} - \int^{t}_{0}B_{s}ds$$   Progress so far     So Using the Definition of the Ito integral we can write    $$\int^{t}_{0}B^{2}_{S}dB_{s}=\lim_\limits{\Delta t_{j} \ \to 0}\sum_\limits{j}B^{2}_{j}\Delta B_{j}$$    Where $\Delta B_{j}=[B_{j+1}-B_{j}]. $      From here i then started by trying to calculate $\Delta(B^{3}_{j})$ which i got       $$ \Delta(B^{3}_{j}) = B^{3}_{j+1}-B^{3}_{j} = (B_{j+1}-B_{j})^{3}+3B_{j}B_{j+1}\Delta B_{j}$$   However this doesn't really get the desired result i'm not sure how to proceed i assume i have calculated $\Delta (B^{3}_{j})$ incorrectly?
I am trying to solve the exercises from the book Stochastic differential equations -An Introduction with applications by Bernt Oksendal and I am stuck on 1 question.  Prove directly from the definition of Ito's integral that   $$\int_0^t B_s^2 dB_s=\frac{1}{3}B_t^3-\int_0^t B_s ds$$  I have tried for a couple of hours but I simply cannot prove it . I would be grateful if somebody could help me out , even a hint would be very helpful
Displaying attachments on visual force page Is it possible to take screenshots on the 3DS? In Pokemon X and Y there are specific locations in the game that allows you to save what are essentially 3D screenshots from the upper screen using a virtual camera. This made me wonder - is it possible to take screenshots, 3D or otherwise, from within the 3DS? Why can't a single photon produce an electron-positron pair? In reading through old course material, I found the assignment (my translation):     Show that a single photon cannot produce an electron-positron pair, but needs additional matter or light quanta.   My idea was to calculate the wavelength required to contain the required energy ($1.02$ MeV), which turned out to be $1.2\times 10^{-3}$ nm, but I don't know about any minimum wavelength of electromagnetic waves. I can't motivate it with the conservation laws for momentum or energy either.  How to solve this task? Because of past conflicts, I don't believe my advisor will write me a good LOR for postdoc positions. What should I do? How essential is having my advisor's letter for post doc application?   I had a pretty bad relationship with my advisor for more than 4 years since I joined his lab and sometimes the relationship feels like slow inflammation that can easily flare up. We had very different opinion on how I want to approach my thesis and we argue very often on the experimental approach and the angle of attack. After that, he turned passive aggressive and completely hands-off. I noticed that he started slow to response to my email and given me very small amount time for meeting in comparison to other people in the lab. However, I proved to him that my approach was right and wrote the full paper myself and got published.   Recently, I got contact by a post doc employer who interested in my own work and request letter of recommendation. I asked my advisor for letter and he told me that I should write my own from which he can modify. I feel this is quite strange as he had wrote for all the other people in the lab, including a departing post doc who is currently interviewing for job. He told me he had done the same for other people but obviously he lied to me.   I further asked if he can provide me a copy of a completed version of the recommendation but he refused.   My question is should I still trust him to write my letter and is it essential? This post doc position is a great opportunity for me and either way I do is a big gamble.  Update: I decided to let my advisor wrote a letter for me and I think I may have made a big mistake. Since he wrote my letter, I asked the post doc lab head if my letter is completed and I no longer be able to hear back from him. Before that, he was quick to response to my email and showed great enthusiasm. As an insurance, I use one opportunity I have less desire to as bait to test the water and I still have one more shot at another position I really want.  Now, I'm thinking requesting open letter from advisor from now on for any future position or use letter service to check the letter.   Any suggestion? Thanks.  update: Great news! Got interviewed and offered at top ones! Thanks everyone! Movie about gene therapy and genetics-based jobs I saw a movie several years ago and I have not been able to find it since.  It was about a dystopian society where parents could modify the genes of their children before birth.  Some individuals, including the protagonist, had not had this done, and so had a somewhat inferior genetic makeup.  Jobs are based on the results of genetic tests, and the protagonist, a male, was slated for a less than ideal job. In the movie, he fakes his identity and has major surgery to make him a little taller to match his ID. 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Lemma vs. Theorem I've been using Spivak's book for a while now and I'd like to know what is the formal difference between a Theorem and a Lemma in mathematics, since he uses the names in his book. I'd like to know a little about the ethymology but mainly about why we choose Lemma for some findings, and Theorem for others (not personally, but mathematically, i.e. why should one classify a finding as lemma and not as theorem). It seems that Lemmas are rather minor findings that serve as a keystone to proving a Theorem, by that is as far as I can go.  NOTE: This  doesn't address my concern, so please avoid citing it as a duplicate. what is the relationship between the dynamical casimir effect and virtual particles? I am Haitian, and traveling with a Haitian passport. I have to go to London with a work permit (tier 2 visa) on the 29th of July but I have 2 hour transit in Dublin.  Do I need a transit visa? Can I get it at the airport?  I also have a valid US visa. To hyphenate or not? As a non-native speaker of English and an engineer by training, I always get confused about hyphenation and almost always end up referring to Google every time I need to make that decision.  Does anybody know of a concise, comprehensive style guide to hyphenation that explains this rather complex issue, once and for all? Bash variable scope Please explain to me why the very last echo statement is blank? I expect that XCODE is incremented in the while loop to a value of 1:  #!/bin/bash OUTPUT="name1 ip ip status" # normally output of another command with multi line output  if [ -z "$OUTPUT" ] then         echo "Status WARN: No messages from SMcli"         exit $STATE_WARNING else         echo "$OUTPUT"|while read NAME IP1 IP2 STATUS         do                 if [ "$STATUS" != "Optimal" ]                 then                         echo "CRIT: $NAME - $STATUS"                         echo $((++XCODE))                 else                         echo "OK: $NAME - $STATUS"                 fi         done fi  echo $XCODE   I've tried using the following statement instead of the ++XCODE method  XCODE=`expr $XCODE + 1`   and it too won't print outside of the while statement. I think I'm missing something about variable scope here, but the ol' man page isn't showing it to me. More options when flagging for migration What does the number next to the review link in the top bar actually mean? Determine if the convergence of $f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$ is uniform. $f_n(x) = \sqrt{x^2 + \frac{1}{n^2}}$ converges pointwise in a set $E = [0, \infty)$ to $f(x) = x$.   This problem reminds me a lot of how $\frac{x}{n}$ fails to converge uniformly to $f=0$ on $[0,\infty]$, but does converge uniformly on $[0,a]$. However, when we proved that we used the definition of uniform convergence. I do not wish to do that here.  I wish to use the Weierstrass test here. And that is where I'm stuck. I cannot find $M_n = \sup\limits_{x \in E} |f_n(x)-f(x)|$. Can someone help me out here? Keyboard shortcut to focus on (idle) terminal I would like to know, if there is possible to use keyboard shortcut to focus on first idle (non working) terminal window in unity (13.10)? Something like Ctr+Alt+T, but before opening the new instance of terminal, searching for any idle instance? How to integrate PHP forum system with new Django website Should import statements always be at the top of a module? Notify close voters when a closed question gets edited As mentioned in , a question closed because it was not well-phrased is currently very unlikely to be reopened after an edit clarifying the issues. Those who voted to close should at least be asked to reconsider and reopen the question after such an edit. How do you Programmatically Download a Webpage in Java I would like to be able to fetch a web page's html and save it to a String, so I can do some processing on it.  Also, how could I handle various types of compression.  How would I go about doing that using Java?