Category Archives: Uncategorized

Solve Problems 1,2,8,9,10,11 in Milnor’s book. (Extra credit for problem 6)

Exercise 1. Let . Let be a continuous Gaussian process such that for , Show that for every , there is a positive random variable such that , for every and such that for every , \textbf{Hint:} You may use … Continue reading →

Here are some videos to visualize non orientability.  

Exercise. Let be a filtered probability space that satisfies the usual conditions. We denote and for , is the restriction of to . Let be a probability measure on such that for every , Show that there exists a right continuous … Continue reading →

Exercise 1. Let be a smooth manifold and be a linear operator such that for every smooth functions , . Show that there exists a vector field on such that for every smooth function , . Exercise 2. Let be the open … Continue reading →

Exercise. Let be a subset homeomorphic to the closed ball .  Show that if is continuous, then there exists such that . Exercise. Let be a one-dimensional compact manifold with boundary. Show that is diffeomorphic to a finite union of segments and … Continue reading →

Exercise. (First hitting time of a closed set by a continuous stochastic process) Let be a continuous process adapted to a filtration . Let where is a closed subset of . Show that is a stopping time of the filtration . … Continue reading →

Exercise 1. Let and let be a symmetric and positive function. Show that there exists a probability space and a Gaussian process defined on it, whose mean function is and whose covariance function is . Exercise 2. Let be a continuous process … Continue reading →

Exercise 1. Let , . Show that is a 2-dimensional smooth manifold homeomorphic to the torus . Exercise 2. Let be the stereographic projection from the north pole , and be the stereographic projection from the south pole . Show that for … Continue reading →

Exercise 1. Show that the  -algebra is also generated by the following families: where and where are Borel sets in . where and where is a Borel set in .   Exercise 2.  Show that the following sets are in :