This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 236, Stochastic Differential Equations Problem set 2 January 22, 2009 These problems are due on Friday January 30. 5pm . You can give them to me in class, drop them in my office, or put them in my mailbox outside the mathematics department. Problem 1 Let 0 = t &lt; t 1 &lt; t 2 &lt; &lt; t N = T be a partition of the interval [0 ,T ] and let B t , t , be the standard Brownian motion process. Show that N 1 summationdisplay k =0 ( B t k +1 B t k ) 2 converges with probability one to T as N and max k N 1 ( t k +1 t k ) 0. That is P { lim N N 1 summationdisplay k =0 ( B t k +1 B t k ) = T } = 1 . Use the BorellCantelli lemma and the Chebychev inequality P { X  } E { X  p } p with p = 3. Problem 2 Let u ( t,x ) be a smooth (classical) solution of the terminal value PDE u t ( t,x ) + 1 2 2 ( x ) u xx ( t,x ) + b ( x ) u x ( t,x ) + c ( x ) u ( t,x ) = 0 , t &lt; T , x R with terminal condition u ( T,x ) = f ( x ). Let X t be the Ito process defined by...
View
Full
Document
This note was uploaded on 11/08/2009 for the course MATH 236 taught by Professor Papanicolaou during the Winter '08 term at Stanford.
 Winter '08
 Papanicolaou
 Differential Equations, Equations

Click to edit the document details