Stats 219 - Stochastic Processes Homework Set 7, Fall 2009 1. Exercise 4.2.4 (a) This can be easily proved by replacing t + h with t and t with s in the identity as given in hints. Note that At is a deterministic process and hence comes out of the conditi
Math 136 - Stochastic Processes Homework Set 6, Autumn 2009, Due: Nov 4 1. Exercise 4.1.5. Suppose (Xn , Fn ) is a martingale. Show that then cfw_Xn is also a martingale with respect to its canonical ltration and that a.s. E [Xl |Fn ] = Xn for all l > n.
Stochastic Processes Amir Dembo (Revised by Kevin Ross) September 25, 2008
E-mail address: amir@stat.stanford.edu, kjross@stat.stanford.edu Department of Statistics, Stanford University, Stanford, CA 94305.
Contents
Preface Chapter 1. Probability, measure
Stat219 / Math 136 - Stochastic Processes Homework Set 1, Fall 2009. Due: Wednesday, September 1 1. Exercise 1.1.3. Let (, F , IP) be a probability space and A, B, Ai events in F . Prove the following properties of IP. (a) Monotonicity. If A B then IP(A)
Math 136 - Stochastic Processes
Homework Set 7, Autumn 2013, Due: November 13
1. Exercise 4.3.4. Show that the rst hitting time () = mincfw_k 0 : Xk () B of a Borel set B R by
a sequence cfw_Xk , is a stopping time for the canonical ltration Fn = (Xk , k
Math 136 - Stochastic Processes
Homework Set 4, Autumn 2013, Due: October 23
1. Exercise 2.3.19. Suppose that X and Y are square integrable random variables.
(a) Show that if E(X|Y ) = E(X) then X and Y are uncorrelated.
ANS: By the tower property and tak
Math 136 - Stochastic Processes
Homework Set 6, Autumn 2013, Due: November 6
1. Exercise 4.1.6 Provide an example of a probability space (, F, P), a ltration cfw_Fn and a stochastic
process cfw_Xn adapted to cfw_Fn such that:
(a) cfw_Xn is a martingal
Math136/Stat219 Fall 2009 Sample Final Examination Write your name and sign the Honor code in the blue books provided. You have 3 hours to solve all questions, each worth points as marked (maximum of 100). Complete reasoning is required for full credit. Y
Math136/Stat219 Fall 2009 Sample Final Examination Write your name and sign the Honor code in the blue books provided. You have 3 hours to solve all questions, each worth points as marked (maximum of 100). Complete reasoning is required for full credit. Y
Conditional expectation
Jason Swanson April 17, 2009
1
Conditioning on -algebras
P (B A) , P (A)
Let (, F , P ) be a probability space and let A F with P (A) > 0. Dene Q(B ) = P (B | A) = for all B F .
It is easy to to check that Q is a probability measur
Stochastic Processes Amir Dembo (revised by Kevin Ross) April 8, 2008
E-mail address : amir@stat.stanford.edu Department of Statistics, Stanford University, Stanford, CA 94305.
Contents
Preface Chapter 1. Probability, measure and integration 1.1. Probabil
STAT 219/ MATH 136 Fall 2009 Midterm Solutions
Friday, October 23
Write your name and sign the Honor code in the blue books provided.
This is a closed note and closed book exam. You have 50 minutes to solve all THREE questions, each worth points as marked
STATS 219 - Stochastic Processes HW 9, Autumn 2009 1. Exercise 4.6.8. Suppose cfw_Zn is a branching process with P(N = 1) < 1 and Z 0 = 1. Show that P( lim Zn = ) = 1 pex ,
n
rst in case m 1, then in case P(N = 0) = 0 and nally using the preceding exerci
Stat 219 - Stochastic Processes Homework Set 8, Fall 2009, Due: November 18th 1. Exercise 5.2.5. Show that E(, ) = by applying Doobs optional stopping theorem for the uniformly integrable stopped martingale Wt2 , t , . ANS: We have seen en-route to (5.2.2
Math 136 - Stochastic Processes Homework Set 5, Fall 2009, Due: October 28 1. Exercise 3.2.21. Consider the random variables Sk of Example 1.4.13. (a) Applying Proposition 3.2.6 verify that the corresponding characteristic functions are Sk () = [cos(/ k )
Math 136 - Stochastic Processes Homework Set 4, Fall 2009 1. Exercise 2.3.8. The left hand side is the smallest distance from G2 , while by embeddedness, right hand side is a distance from G2 , and hence the inequality holds.(no need for algebra) 2. Exerc
Stat 219 - Stochastic Processes Homework Set 3, Autumn 2009 1. Exercise 1.4.31. Prove Proposition 1.4.3 using the following steps. (a) Verify that the identity (1.4.1) holds for indicator functions g (x) = I B (x) for B B . ANS: Let B B be an arbitrary Bo
Stat 219 - Stochastic Processes Homework Set 2, Fall 2009, Due: October 7 1. Exercise 1.2.22. Write (, F , P ) for a random experiment whose outcome is a recording of the results of n independent rolls of a balanced six-sided dice(including their order).
Math 136 - Stochastic Processes
Homework Set 5, Autumn 2013, Due: October 30
1. Exercise 3.2.21. Consider the random variables Sk of Example 1.4.13.
(a) Applying Proposition 3.2.6 verify that the corresponding characteristic functions are
S () = [cos(/ k)