MATH 545, Stochastic Calculus Problem set 2

MATH 545, Stochastic CalculusProblem set 2 January 24, 2019 These problems are due on TUE Feb 5th. You can give them to me in class, drop them in my box. In allof the problemsEdenotes the expected value with respect to the specified probability [Klebaner], Chapter4 and Brownian Motion Notes (by FEB 7th) Problem 1(Klebaner, Exercise ).Let{Bt}t 0be a standard Brownian Motion. Show that,{Xt}t [0,T],defined as below is a Brownian )Xt= Bt,We check that the defining properties of Brownian motion hold. It is clear thatB0= , and thatthe increments of the process are independent. Fort > s, the increments can be written as( Bt) ( Bs) = (Bt Bs).BecauseBt Bsis a gaussian RV with mean0and variancet s, (Bt Bs)must have the )Xt=BT t BTforT < ,It is clear thatB0= Fort > s, the increments of the process are given byXt Xs= (BT t BT) (BT s BT) =BT t BT increments are independent ofXs=BT s BTby the inependent increments propery ofBrownian motion.

A standard bivariate normal distribution is a bivariate normal distribution where the means of both coordinate variables are zero and the covariance matrix is the identity matrix. You can use the fact that any linear combination of random variables following a multi-variate normal distribution has a normal distribution. Let [Z 1 Z

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  Normal, Bivariate, Bivariate normal

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