Brownian Motion

Chuan-Hsiang HanNovember 24, 2010

Symmetric Random Walk

Given ; let and , and denotes the outcome of th toss. Define the r.v.'s that for each

A S.R.W. is a process such that and

Independent Increments of S.R.W.

Choose , the r.v.s are independent, where the increment is defined by Note:(1) Increments are independent.(2) The increment has mean 0 and variance .(Stationarity)

Martingale Property of S.R.W.

For any nonnegative integers ,

contains all the information of the first coin tosses.If R.W. is not symmetric, it is not a martingale.

Markov Property of S.R.W.

For any nonnegative integers and any integrable function

If R.W. is not symmetric, it is still a Markov process.

Quadratic Variation of S.R.W.

The quadratic variation up to time is defined to be

Note the difference between (an average over all paths), and (pathwise property)

Scaled S.R.W.

Goal: to approximate Brownian Motion

1. new time interval is "very small" of instead of 12. magnitude is "small" of instead of 1.For any can be defined as a linear interpolation between the nearest such that .

Properties of Scaled S.R.W.

(i) independent increments: for any are independent. Each is an integer.

(ii) Stationarity: for any ,

(iii) Martingale property

(iv) Markov Property: for any function , these exists a function so that

(v) Quadratic variation: for any ,

Limiting (Marginal) Distribution of S.R.W.

Theorem 3.2.1. (Central Limit Theorem)For any fixed ,

in dist.or

Proof: shown in class.

A Numerical Example

Log-Normality as the Limit of the Binomial Model

Theorem 3.2.2. (Central Limit Theorem)For any fixed ,

in the distribution sense, where , ,and

What is Brownian Motion?

"If is a continuous process with independent increments that are normally distributed, then is a Brownian motion."

Standard Brownian Motions

check Definition 3.3.1 in the text.Definition of SBM: Let the stochastic process under a probability space be continuous and satisfy:1. 2. 3. is independent of for .

Covariance Matrix

Check for any nonnegative and For any vector with

In fact,

Joint Moment-Generating Function of BM

𝜑 (𝑢1 ,𝑢2 ,⋯ ,𝑢𝑚 )=𝐸 [𝑒𝑥𝑝 (�⃗� ∙𝑉 ) ]=𝐸 [𝑒𝑥𝑝 (𝑢1𝑊 𝑡1+𝑢2𝑊 𝑡 2

+⋯+𝑢𝑚𝑊 𝑡𝑚 )]=𝑒𝑥𝑝 {12 (𝑢1+𝑢2+⋯+𝑢𝑚 )2𝑡1+12

(𝑢2+⋯+𝑢𝑚 )2 (𝑡 2−𝑡 1 )+⋯+ 12𝑢𝑚2 (𝑡𝑚−𝑡1 )}

Alternative Characteristics of BrownianMotion (Theorem 3.3.2)

For any continuous process with , the following three properties are equivalent.(i) increments are independent and normally distributed.(ii) For any , are jointly normally distributed.(ii) has the joint moment-generating function as before.If any of the three holds, then , is a SBM.

Filtration for B.M.

Definition 3.3.3 Let be a probability space on which the B.M. is defined. A filtration for the B.M. is a collection of -algebras , satisfying(i) (Information accumulates) For , .(ii) (Adaptivity) each is -measurable.(iii) (Independence of future increments) , the increment is independent of . [Note, this property leads to Efficient Market Hypothesis.]

Martingale property

Theorem 3.3.4 B.M. is a martingale.Proof:

Levy's Characteristics of Brownian Motion

The process is SBM iff the conditional characterization function is

Variations: First-Order (Total) Variation

Given a function defined on , the total variation is defined by

where the partition and

If is differentiable,

for some . Then .

Quadratic Variation

Def. 3.4.1 The quadratic variation of up to time is defined by

If is continuous differentiable,

for some . Then

Quadratic Variation of B.M.

Thm. 3.4.3 Let be a Brownian Motion. Then for all a.s..B.M. accumulates quadratic variation at rate one per unit time.Informal notion:, ,

Geometric Brownian Motion

The geometric Brownian motion is a process of the following form

where is the current value, is a B.M., is the drift and is the volatility.For each partition , define the log returns

Volatility Estimation of GBM

The realized variance is defined by

which converges to as

BM is a Markov process

Thm. 3.5.1 Let be a B.M. and be a filtration for this B.M.. Then(1)Wt0 is a Markov process.

Thm. 3.6.1.(2) is martingale.(We call exponential martingale.)Note: