Let B(t), t 0 be a standard Brownian Motion and define the stochastic process N (t) = min 0st B(s). What is the probability density function (pdf) of B(t)? Calculate the pdf of N (t), and also calculate its expected value, E[N (t)]. Problem 2. Properties of the Brownian Motion. Let B(t), t 0, be a standard Brownian Motion on the probability space (, F, P). (i) Let t i , t j 2 R be fixed such that 0 < t i < t j . Prove that CovB(t i ), B(t j ) = t i . Find CovB(t i ), B(t j ) in the general case, i.e., for any t i , t j nonnegative. (ii) Define the stochastic process X(t) = 1 aB(a 2 t), t 0. Show that X(t) is also a standard Brownian Motion, for any a > 0. Hint: verify that the properties in the definition of a BM hold. Also check that the covariance structure is the same, i.e. compute CovX(t i ), X(t j ) for any t i , t j in some partitionchosen a priori, i.e.= {0 = t 0 , t 1