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Context : This post is the first of a post taken from exercises in Revuz and Yor's Book "Continuous Martingales ans Brownian Motion". The reason for doing so is that the exercises of this book are hard sometime very hard but still very interesting and that there is no definitive source for the solutions. I am not alone on this project but I am testing and starting the idea with the first exercise of the book (which is easy and for which I will post shortly an answer as soon as the moderators will allow me to do so). Moreover I want those exercises to be posted in "community wiki" way but do not have clearance for doing so.

Let $B$ be the standard linear Brownian Motion on $[0,1]$, i.e. we consider only $t\in [0,1)$. Prove that the process $\tilde B$ defined by : $$\tilde B_t =B_{1-t}-B_1$$ is another version of $B$, in other words, a standard BM on $[0,1)$.

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closed as off-topic by Rhys Steele, saz, Math1000, NCh, Tianlalu Apr 2 at 4:33

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    $\begingroup$ This is copied word for word from the textbook (and hence definitely does not show your own attempts at the problem). Have you tried anything? $\endgroup$ – Rhys Steele Mar 28 at 14:33
  • $\begingroup$ Hi I understand your position but please be patient, I want this question (and more to come) to be "community wiki" flagged but I can't do this by myself. The aim is to get for this exercise (and more to come) a place to post solutions and elect the best one. For this particular I have got one solution that I will post later. $\endgroup$ – TheBridge Mar 28 at 15:07
  • $\begingroup$ Moreoover I am not alone o this subject and other people that I know want to post some exercises form this book here to have a common and public place to discuss solutions regards. PS / I am not a student and the other people I mentioned neither so this is not homework but pleasure to share knowledge. $\endgroup$ – TheBridge Mar 28 at 15:09
  • $\begingroup$ I have some sympathy with your reasons for wanting such a reference, since Revuz and Yor is a book with many exercises that are difficult for beginners to the subject. It might help in future to post your solution more quickly to avoid accumulating the response I gave here (because otherwise this question bears all the hallmarks of someone taking a first course on continuous time martingale theory based on Revuz and Yor cheating on their homework, which is common given the relative difficulty of the subject). $\endgroup$ – Rhys Steele Mar 28 at 15:29
  • $\begingroup$ Well that is our point the book is wonderfull but too hard for guys with intermediate level (like me) and it's so very frustrating. As "book tags" are forbidden here (which is a pity I think but democracy here has the final say) so we will resort to use the following title naming convention : Revuz and Yor's Book “Continuous Martingales ans Browian Motion” - Chapter CC - Exercise N.nn" Last about the homework issue I would say ... well who's to blame really the students or the teacher for not being able to get original exercises ;-) needless to say where my opinion leans $\endgroup$ – TheBridge Mar 28 at 15:43