Solutions to Homework 5
Exercise 9.4. It suces to prove that Mt Mt
generated by W (1) and W (2) . For any 0 s < t,
is a martingale. Let F is the ltration
E Wtt0 Wtt0 |Fst0
= E (Wtt0 Wst0 )(Wtt0 Wst0 )|Fst0
10.6.6 Suppose that H is a nonempty compact subset of X and that Y is a Euclidean space.
a) If f : H Y is continuous, prove that
| f |H := sup | f (x)|Y
is nite and there exists x0 H such that | f (x0 )|Y = | f |H .
10.4.2 Let A and B be compact subsets of X . Prove that A B and A B are compact.
Proof: Let U = cfw_u be an open cover of A B. Then U is also an open cover of A and B since
Since A and B are compact, ther
10.3.8 Let Y be a subspace of X .
a) Show that V is open in Y if and only if there is an open set U in X such that V = U Y .
Note that Y is given a topology in two different ways here: One topology is the metric space
8.2.11 Fix T L (Rn ; Rm ). Set
M1 := sup |T (x)|
cfw_C > 0 : |T (x)| C|x| for all x Rn .
a) Prove that M1 |T |.
Proof: Let x Rn and note that x/|x| = 1. Then
= |T |.
7.3.2 Find the interval of convergence of the following power series.
(1)k + 3)k (x 1)k .
Solution: First compute the radius of convergence for this power series
lim sup(1) + 3 4
So the power series
7.2.3 Let E (x) = =0 x !
a) Prove that the series dening E (x) converges uniformly on any closed interval [a, b].
Proof: Let [a, b] R be a closed interval and dene M = max(|a|, |b|). Then
7.1.3 Suppose that for each n N, fn : E R is bounded. If fn f uniformly on E as n , then cfw_ fn
is uniformly bounded on E and f is a bounded function on E .
Proof: For each n N, there exists Mn > 0 such that | fn (x)| Mn
Solutions to Homework 6
Exercise 11.1. Note that 1cfw_a (Ws ) 0. Since
1cfw_a (Ws )ds =
we conclude that
1cfw_a (Ws )ds = 0 a.s.
Exercise 11.2. Let
Dene fa,b, (x) =
E1cfw_a (Ws )ds = 0 ,
ga,b, (x) = 1
+ + 1
Solutions to Homework 3
Exercise 4.1. If W is a Brownian motion, let St = supst Ws . Find the density for St .
Solution: Let Tb = inf cfw_t : Wt b, then
P cfw_St b = P cfw_Tb t = P cfw_Tb t, Wt < b + P cfw_Tb t, Wt b = 2P cfw_Tb t, Wt b
= 2P cfw_Wt b
Solutions to Homework 2
Exercise 3.1. Let Ft be the ltration for the Brownian motion W . It is easy to see that
Ws ds| <
for any t 0. Let u < t. We have
E[Wt3 |Fu ] = E[(Wt Wu )3 + 3(Wt Wu )2 Wu + 3(Wt Wu )Wu + Wu |Fu ]
= E(Wt Wu )3 )
Solutions to Homework 1
Exercise 1.3. (1) Let
Xsn) ( )
Xk/2n ( )1[k/2n ,(k+1)/2n ) (s).
Fix t 0 and consider X as a map from [0, t] to R. Each term in the above sum
Xk/2n ( )1[k/2n ,(k+1)/2n ) (s) ,
is a product of two factors, one Xk/2n ( ) is me
Problem 1 Let cfw_W (t), t [0, 1] be a Brownian motion. The process
X (t) = W (t) tW (1),
is called the Brownian bridge. Show that X is a Gaussian process and compute its mean
and covariance function.
Answer Fix t1 < <
Solutions to Homework 4
Exercise 6.4. (1) Fix i 1. By the denition of ij (x), we have
0 ij (t)
dx = 2
The set cfw_t : ij (t) > 0 coincides with ( 2j21 , 2j ), and these sets are clearly disjoint for xed
i and dierent j
1) For E1 , E2 Rn , dene
E1 + E2 = cfw_x + y : x E1 , y E2 .
(a) Prove that if E1 and E2 are compact, then E1 + E2 is compact.
Proof: Since E1 + E2 Rn , it is sufcient to prove that E1 + E2 is closed and bounded.
E1 + E