Some solutions to HW2
Phillip Monin
April 30, 2012
0.1
Exercise Sheet 1
Let cfw_Xi be a sequence of iid random variables with probability distribution P(Xi = 1) = p and
P(Xi = 1) = 1 p. Let S0 = 0 and Sn = n Xi be the random walk started from zero. The
i
Solutions to Problem Set 5
April 16, 2012
Problem 1: Applying Jensens inequality to the concave function u (or the convex function u ) yields:
i) EP (u (x + Wt ) u (EP (x + Wt ) = u (x).
ii) EP (u (x + Wt ) u (EP (x + Wt ) = u (x).
iii) EP u x + e
2
2
iv)
April 30, 2012
Problem 1 1. Let Ft , t 0, be the ltration generated by the Brownian motion. Then an
admissible control is a process Cs , s [t, T ], such that
Cs is Fs -measurable.
E
T
t
C
a2 Xs
2
ds < .
C
Cs 0 and Xs 0 for all s [t, T ].
2. Here DPP is