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STOCHASTIC CALCULUS ON BROWNIAN MOTION AND
STOCHASTIC INTEGRATION
LINGYUEYU
Abstract. In this paper, I will first introduce the basics of measure theo-
retic probability and give a proof of Central Limit Theorem using moment
generating functions. This section will allow us to explore stochastic processes
and Brownian motion in a more rigorous way. Finally, building upon Brow-
nian motion, we can formally explain Ito’s Integral and Ito’s Formula on the
stochastic calculus.
Contents
1. Introduction 1
2. Probability Measure, Random Variable, and Expectation 1
3. Stochastic Processes 5
4. Brownian Motion 6
5. Itˆo’s Formula 8
Acknowledgments 11
References 11
1. Introduction
Calculus is the study of continuous change and typically characterized by differ-
entiable functions. However, a stochastic process is a collection of random variables
and its derivative is not easily defined, which renders ordinary calculus ineffective.
To define a stochastic integral, we need to deal with randomness in stochastic pro-
cesses. Essential definitions and theorems in probability introduced in the first
section allow our further discussions. In the remaining sections, we will delve into
the stochastic integration on Brownian motion, a stochastic process modeling con-
tinuous random motion, and examine the construction of stochastic integrals under
the framework of Itˆo’s Integral and Itˆo’s Formula.
2. Probability Measure, Random Variable, and Expectation
Definition 2.1 (Sample Space). A sample space Ω is a non-empty set of outcomes.
Definition 2.2 (Algebra of Sets). Let X be a set (We usually work within the
context of sample space Ω in probability). An algebra is a collection A of subsets
of X such that
(1) ∅ ∈ A and X ∈ A.
(2) If A ∈ A, then Ac := X \A ∈ A.
Date: DEADLINES: Draft AUGUST 15 and Final version AUGUST 29, 2020.
1
2 LINGYUE YU
S T
(3) If A ,...,A ∈ A, then n A ∈Aand n A ∈A.
1 n k=1 k k=1 k
Definition 2.3 (σ-Algebra of Sets). For an algebra of sets, we have in addition
that S T
(4) If A1,A2,··· ∈ A, then ∞ Ak∈Aand ∞ Ak∈A.
k=1 k=1
then we call A an σ -Algebra of Sets. T
In (4), we only allow countable unions and intersections. Since ∞ Ai =
S T i=1
∞ c c ∞
( i=1 Ai) , the requirement that k=1Ak ∈ A would be redundant. The pair
(X,A) is called a measurable space. A set A is measurable or A − measurable if
A∈A.
Definition 2.4 (Probability Measure). Let Ω be a sample space and let F be a
σ-Algebra on Ω. A probability measure is a function P : F → [0,1] such that
(1) P(∅) = 0 and P(Ω) = 1.
(2) If events E ,E ,... are pairwise disjoint, then
1 2
∞ ∞
XP(Ei)=P [Ei .
i=1 i=1
Definition 2.5 (Probability Space). Given a set Ω and a σ-Algebra F on Ω, a
probability space is the triple (Ω,F,P), where P is the probability measure.
Definition 2.6 (Borel σ-Algebra). If we have an arbitrary collection C of subsets
of X, define \
σ(C) := A.
A is a σ−Algebra
C⊂A
We call σ(C) the σ-Algebra generated by C. If G is the collection of all open sets
on X, then we define B = σ(G) to be the Borel σ-Algebra on X.
Definition 2.7 (Borel σ-Algebra on R). R is the Borel σ-Algebra generated by R.
Proposition 2.8 (LebesgueMeasure). Let X = [0,1] and B be the Borel σ-Algebra.
There is a unique measure λ on (X,B) such that for any interval J ⊂ X, we have
that λ(J) = length(J). We call the measure as Lebesgue Measure.
Proof. The construction of Lebesgue Measure involves Carth´eodory Extension The-
orem using the algebra of finite unions and intersections of intervals. This proposi-
tion also works with X = R because the space is σ-finite. The whole construction
of Lebesgue measure please see Chapter 4 of [7] from Page 24 to 41.
Definition 2.9 (Random Variable). Let (Ω,F,P) be a probability space and let
(R,B,λ) be the real line endowed with the Borel σ-Algebra and the Lebesgue mea-
sure. A real random variable is a measurable function X : Ω → R.
Definition 2.10 (Expectation). The expectation for a discrete random variable is
E[X] = XiP(X =i).
i
For a continuous random variable, the expectation is defined as the following inte-
gral: Z
E[X] = XdP.
Ω
STOCHASTIC CALCULUS ON BROWNIAN MOTION AND STOCHASTIC INTEGRATION 3
Definition 2.11 (Variance). The variance of random variable X is defined as
2 2 2
Var[X] := E[(X −E[X]) ] = E[X ]−E[X] .
Definition 2.12 (Conditional Expectation). Let X be a random variable with
E[X] < ∞. Then there exists a unique G-measurable random variable E(X|G) such
that, for very bounded G-measurable random variable Y, we have
E(XY)=E(E(X|G)Y).
The unique random variable E(X|G) is called the conditional expectation.
Definition 2.13 (Distribution Function). The distribution function F : R → [0,1]
of a random variable X on Ω is defined by
F (x) := P ((−∞,x]) = P(X ≤ x).
X X
F is an increasing function whose corresponding Lebesgue measure is P .
X X
Definition 2.14 (Density and Distribution). Let X be a continuous random vari-
able. A function F is the distribution function of X if
F(x) = Z x f(y)dy.
−∞
Function f is the density of X if
PX(A)=Z f(x)dx.
A
Definition 2.15 (Standard Normal Distribution). A standard distribution is de-
fined to be Z
b 2
1 −x
Φ(b) = √ e 2 dx.
−∞ 2π
Definition 2.16 (Independence). Two events E ,E are independent if
1 2
P(E ∩E )=P(E )P(E ).
1 2 1 2
Lemma2.17 (Borel-Cantelli). Let {En} be a sequence of events in the probability
space (Ω,F,P). If the sum of the probabilities of the E is finite
n
∞
XP(En)<∞,
n=1
then the probability of infinitely many of them to occur is 0, which is
∞ ∞
P(\ [ EmEn)=0.
n=1m=n
Proof. By definition, for each n,
∞ ∞ ∞ ∞
P(\ [ E E )≤P([ E )≤ XP(E )<∞.
m n m m
n=1m=n m=n m=n
P
When n→∞, ∞ P(E )→0. Therefore, we have proved the claim.
m=n m
4 LINGYUE YU
Definition2.18(MomentGeneratingFunction). Themoment-generatingfunction
of a random variable X is given by
M(t) := E[etX], t ∈ R.
If two random variables have the same moment-generating function, they are said
to be identically distributed.
Lemma2.19. LetZ1,Z2,... be a sequence of random variables having distribution
functions F and moment generating function M , n ≥ 1. Furthermore, let Z be
Z Z
n n
a random variable having distribution function F andmomentgeneratingfunctions
Z
M . If M →M forallt, then we have F (t) → F (t) for all t at which
Z Zn(t) Z(t) Zn Z
F is continuous.
Zt
This lemma is integral to the proof of the central limit theorem. As it is an
advanced and technical proof, we will not prove this lemma in the paper. However,
the whole proof can be seen in Probability and Random Processes [5].
Theorem 2.20 (Central Limit Theorem). Let X ,X ,...,X be independent,
1 2 n
identically distributed random variables with E[X ] = µ and Var[X ] = σ2 < ∞.
i i
Let
(X +X +···+X )−nµ
Z = 1 2 √ n .
n σ n
Then as n → ∞, the distribution of Zn approaches a standard normal distribution.
Precisely, this means: if a < b
lim P(a ≤ Zn ≤ b) = Φ(b)−Φ(a).
n→∞
Proof. We begin the proof with the assumption that µ = 0,σ2 = 1 ,and the mo-
ment generating function of X ,M(t) exists and is finite. If not, we consider its
i
standardized random variable X∗ = (Xi − µ)/σ2 for the same. By definition, the
i
Xi
moment generating function of √ is given as:
n
tX
t √i
M(√ )=E[e n].
n
Pn Xi t n
Thus, the moment generating function of i=1 √ is [M(√ )] since:
n n
" n !# " n !#
XX XtX
E exp t √i =E exp √i
i=1 n i=1 n
n
Y tXi
= E exp √
i=1 n
n
Y tX
√i
= E[e n]
i=1
t n
=[M(√n)] .
Then, we define
L(t) = logM(t),
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