ma414soln5.tex
MA414 SOLUTIONS 5. 13.2.2012
Q1. Proof (Doobs Submartingale Inequality). Let
F := cfw_max Xk c,
kn
Fk := cfw_X0 < ccfw_X1 < c. . . cfw_Xk1 < ccfw_Xk c.
Then F is the disjoint union F = F0 . . . Fn . Also Fk Fk , and Xk c
on Fk . So
E [Xn I
ma414l3.tex
Lecture 3. 26.1.2012
9. The Borel-Cantelli lemmas and the zero-one law.
First, recall from Real Analysis the denition of the upper and lower
limit, lim sup and lim inf, of a real sequence xn :
lim sup xn := inf sup xk , = lim sup xk
n
k n
n
k
ma414l4.tex
Lecture 4. 2.2.2012
II. STOCHASTIC PROCESSES
1. Conditional expectations.
Suppose that X is a random variable, whose expectation exists (i.e.
E |X | < , or X L1 ). Then EX , the expectation of X , is a scalar (a
number) non-random. The expecta
ma414l5.tex
Lecture 5. 9.2.2012
5. Martingales: discrete time. We refer for a fuller account to [W]. The
classic exposition is Ch. VII in Doobs book [D] of 1953.
Denition. A process X = (Xn ) in discrete time is called a martingale
(mg) relative to (cfw_F
ma414l6.tex
Lecture 6. 16.2.2012
Corollary (Doob). A non-negative supermg Xn is a.s. convergent.
Proof. As Xn is a supermg, EXn decreases. As X 0, E [Xn ] 0. So
E [|Xn |] = E [Xn ] is decreasing and bounded below, so (convergent and)
bounded: Xn is L1 -bo
ma414l7.tex
Lecture 7. 23.2.2012
Quadratic Variation.
A non-negative right-continuous submartingale is of class (D). So it has
a Doob-Meyer decomposition. We specialize this to X 2 , with X cM2 :
2
X 2 = X0 + M + A,
with M a continuous martingale and A a
ma414l8.tex
Lecture 8. 1.3.2012
Proof.
n W t =
ti n
cfw_(W (ti+i )W (ti )2 (ti+1 ti ) =
cfw_(i W )2 (i t) =
i
Yi ,
i
where since i W N (0, i t), E [(i W )2 ] = ti , so the Yi have zero mean,
and are independent by independent increments of W . So
E [(n W
ma414l9.tex
Lecture 9. 8.3.2012
Example. We calculate W (u)dW (u). We start by approximating the integrand by a sequence of simple functions.
W (0) = 0
W (t/n)
if
if
.
.
.
if
Xn (u) = .
.
.
W (n 1)t/n)
0 u t/n,
t/n < u 2t/n,
(n 1)t/n < u t.
By denition,
ma414l10.tex
Lecture 10. 15.3.2012
Theorem (Brownian Martingale Representation Theorem). Let M =
(M (t)t0 be a RCLL local martingale with respect to the Brownian ltration
(Ft ). Then
t
M (t) = M (0) +
0
H (s)dW (s), t 0
with H = (H (t)t0 a progressively m
ma414prob2.tex
MA414 SOLUTIONS 2. 26.1.2012
Q1.
k := X k E [X k ] = k
a.s. (n ).
The k th central moment is 0 := (X X )k . Then
k
0
k
:= (X
X )k
()
k k
X i ()ki (X )ki
i
()
k k
=
(X i )()ki (X )ki .
0i
=
0
By SLLN, as n this tends a.s. to
()
k k
0
i
E [X
ma414soln3.tex
MA414 SOLUTIONS 3. 7.2.2011
Q1. We are using almost all for numbers on the line; this means under
Lebesgue measure (unless otherwise stated). We are talking about decimal
expansions; this relates to the fractional part, which is in [0, 1];
ma414l2.tex
Lecture 2. 19.1.2012.
5. Modes of convergence.
We need (at least) four modes of convergence two strong, one intermediate, one weak. We begin with the strong modes.
We say that Xn X almost surely, or a.s., if Xn X with probability
1: P (Xn X )