CHAPTER 1
Martingale Theory
We review basic facts from martingale theory. We start with discretetime parameter martingales and proceed to explain what modifications are
needed in order to extend the results from discrete-time to continuous-time.
The Doob-

CHAPTER 3
Stochastic Integration and Itos Formula
theory of stochastic integration. This is a
In this chapter we discuss Itos
vast subject. However, our goal is rather modest: we will develop this theory only generally enough for later applications. We w

CHAPTER 2
Brownian Motion
In stochastic analysis, we deal with two important classes of stochastic processes: Markov processes and martingales. Brownian motion is the
most important example for both classes, and is also the most thoroughly studied stochas

CHAPTER 4
Applications of Itos Formula
In this chapter, we discuss several basic theorems in stochastic analysis.
formula.
Their proofs are good examples of applications of Itos
1. Levys martingale characterization of Brownian motion
Recall that B is a B

CHAPTER 5
Stochastic Differential Equations
1. Simplest stochastic differential equations
In this section we discuss a stochastic differential equation of a very
simple type.
Let M be a martingale in and A a process of bounded variation. Let a
and b be tw

CHAPTER 6
Financial Mathematics
1. Portfolio management
Let F be the filtration generated by the Brownian motion. It represents
the information of the market up to time t. We consider the simplest case
where the market consists of a risk-free investment c

Stat/219 Math 136 - Stochastic Processes
Notes on Markov Processes
1
Notes on Markov processes
The following notes expand on Proposition 6.1.17.
1.1
Stochastic processes with independent increments
Lemma 1.1 If cfw_Xt , t 0 is a stochastic process with in

Stat/219 Math 136 - Stochastic Processes
Notes on Section 4.1.2
1
Independence, Uncorrelated-ness, and Something in Between
Suppose (X, Y ) is a random vector defined on some probability space (, F, IP) with IE(X 2 ) < , IE(Y 2 ) < .
The following implica

21
(2/23/2015)
Example 21.1 (Example 19.2 revisited). [Skipped this example]. Draw a closed
contour, C, which winds around 0 counterclockwise one time which starts and
ends at a say. For example, C : z : [0, 2] C\ cfw_0 crosses (, 0) only at
z(0) = z(2),