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.

d) X t = ˆ tB 1=t; if t>0 0; if t= 0;1 By Theorem 3.3 in [Klebaner]: X t is a mean zero gaussian process with covariance structure Cov(X s;X t) = min(s;t).Because rescaling time and brownian motion paths does not affect the mean of the process not its Gaussian structure, the first two points above are trivial.

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  Structure, Covariance, Covariance structure

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