Chapter 3 Stochastic Integration and Ito Calculus
3.1
Wiener integral
The most basic integration theory is based on the Riemann integral of a realvalued function on an interval [0, T ]. It is easy to extend the integrand to be a vector-valued or a Banach
2.3. SOME BASIC PROPERTIES
65
2.3
Some basic properties
Let f : [0, t] R be a real-valued function, we say that f is of bounded variation if V (f ) = sup
P i=1 n
|f (ti ) f (ti1 )| < .
where the supremum is taken over all partition P = cfw_0 = t1 < t2 < .
2.2. BROWNIAN MOTION AND SAMPLE PATHS
53
2.2
Brownian motion and sample paths
For a normal r.v. X N (0, 2 ), the symmetry implies E (X 2k+1 ) = 0, and the integration by parts yields E (X 2k ) = 1 3 5 (2k 1) 2k . We also need the following elementary prop
Chapter 2 Brownian Motion
2.1 Continuous time stochastic processes
We call a family of random variables cfw_Xt t0 on (, F , P ) a continuous time stochastic process. For each , X (, ) = X() ( ) is called a sample path. Usually we treat X (, ) = (t) (this
1.4. MARTINGALES
31
1.4
Martingales
We rst consider a simple example in analysis. Let f be an integrable function on [0, 1], let Pn = cfw_0 =
1 2n
k 2n
1 be a partition of [0, 1] and let
+1 In,k = [ 2k , k2n ). We dene the average function fn of f on t
20
CHAPTER 1. BASIC PROBABILITY THEORY
1.3
Markov Property
Let A be an index set and let cfw_F : A be family of sub- -elds of F . We say that the family of F s are conditionally independent relative to G if for any i Fi i = 1, , n,
n n
P(
j =1
j |G ) =
j
1.2. CONDITIONAL EXPECTATION
13
1.2
Conditional Expectation
Let F with P () > 0, we dene P (E |) = P ( E ) P () where P () > 0.
It follow that for a random vector (X, Y ), P (Y y, X = x) , P (X = x) P (Y y |X = x) = 0 ,
if P (X = x) > 0 , otherwise.
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1 MAT6082 Topics in Analysis II 1st term, 2009-10
Teacher: Schedule: Venue: Topics:
Professor Ka-Sing Lau Every Tuesday, 7:00pm to 9:30pm Room 222, Lady Shaw Building, CUHK Introduction to Stochastic Calculus
In the past thirty years, there has been an in