Math 219 Final exam (due 4PM April 24)
Please do not collaborate with other students in solving these problems
1. Consider Brownian motion with drift given by dYt = dt + dBt . Let = inf cfw_t : Yt
[0, N ]. Compute v (x) = Ex . It is enough to write down
Practice Problems for Exam 1
1. Find P0 (B2 > B1 > B3 ).
Ans. This is equal to P0 (B1 > 0 > B2 ) = P (B1 > 0, B2 B1 < B1 ). P (B1 > 0, B2 B1 <
0) = 1/4. Symmetry implies that
P (B2 B1 < B1 |B1 > 0, B2 B1 < 0) = 1/2
so the answer is 1/8. For another approa
Math 219, Stochastic Calculus: Introductory Lecture
As in ordinary calculus much of our time will be spent on stochastic integration and stochastic
dierential equations. We begin with some nonrandom examples written in slightly dierent
notation.
Exponenti
Homework of SDE 2
Yuan Zhang
Feb/4/2012
Problem I. Show that with probability 1, lim supt Bt /t1/2 = and that
lim supn Bn /(n log n)1/2 2 where the second limit is through the integer
Proof:
For any m > 0, it is easy to see that, for any t, cfw_sups>t Bs
Topics for Exam 1 in Math 219
Brownian motion
Denition 1. A one-dimensional Brownian motion is a real-valued process Bt , t 0 that
has the following properties:
(a) If t0 < t1 < . . . < tn then B (t0 ), B (t1 ) B (t0 ), . . . , B (tn ) B (tn1 ) are indepe
Math 219 Exam 1 Grade Improvement Exercises
Due Thursday March 1
1. (i) Use the Markov property to write a formula for Px (Bs > 0, Bs+t > 0). (ii) Use the
fact that y Py (Bt > 0) is increasing to conclude that
Px (Bs+t > 0|Bs > 0) > Px (Bs+t > 0|Bs < 0).
Chapter 5
Convergence to SDE
5.1
Weak convergence
We begin with a treatment of weak convergence on a general space S with a metric
, i.e., a function with (i) (x, x) = 0, (ii) (x, y ) = (y, x), and (iii) (x, y ) + (y, z )
(x, z ). Open balls are dened by
Chapter 4
SDE as Markov processes
4.1
Discrete state space
To prepare for our discussion of diusion processes, we recall the analogous results
for a countable state space S . A Markov chain on S is described by giving the rate
q (i, j ) at which jumps for
CHAPTER 3. SEMIMARTINGALE INTEGRATION
62
3.5
OUSDE
Explict solutions of SDE
Example 3.3. Ornstein Uhlenbeck Process. Let Bt be a one dimensional Brownian motion and consider
(3.10)
dXt = Xt dt + dBt
OUsde
which describes one component of the velocity of a
Chapter 3
Semimartingale Integration
3.1
Basic denitions
The solutions to our SDEs have the form
t
Xt X0 =
t
(Xs ) dBs +
0
b(Xs ) ds
0
where for the moment we suppose Xt , b(Xt ), and (Xt ) are real numbers. and b
and are continuous. To have a useful int
Chapter 1
Brownian motion
Brownian motion is a process of tremendous practical and theoretical signicance. It
originated (a) as a model of the phenomenon observed by Robert Brown in 1828 that
pollen grains suspended in water perform a continual swarming m
Math 219 HWK 4 (due April 10)
To match the other problem sets, each problem is worth 12 points.
1. Consider exponential Brownian motion. St = S0 exp(t + Bt ). Find the transition
probability pt (x, y ) and show that it satises
t pt =
+
2
2
xx pt +
2 x2 2
Math 219 HWK 3 (due March 20) - Meet the Martingales
Recall the following fact from the notes: let Xm , 0 m n be a submartingale with respect to
Fm . Let 0 Hm Fm1 . Then (H X )n = n =1 Hm (Xm Xm1 ) is a submartingale and hence
m
E (H X )n 0.
1. (a) Let N
Homework of SDE 1
Yuan Zhang
Jan/18/2012
Problem I. Borel Cantelli lemma. Given events Fn dene
lim sup Fn = =1 n=N Fn
N
(a) Show that lim sup Fn = cfw_Fn i.o., where the right-hand side is dened to be
the set of outcomes that appear in innitely many of th