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Brownian Motion

A stochastic process is collection of random variables indexed by time t, or a function from the reals to a set of random variables. A Brownian motion is a stochastic process that is a random walk occurring in continuous time, with continuous movement. More precise definition:

Let Z(t) be the value of a Brownian motion at time t . It has the following properties:

Z(0)=0 .
Z(t+s)-Z(t) is normally distributed with mean 0 and variance s ; in particular,

Z(s) is normally distributed with mean 0 and variance s.

Nonoverlapping intervals are independently distributed. That is, Z(t+a)-Z(t) is independent of Z(t)- Z(t-b), for all a, b >0.
Z(t) is continuous.

Properties:

Z(t) is a martingale, that is, E[Z(t+s) | Z(t)] = Z(t) . This is also called a diffusion process.
Binomial approximation:

Let dt be a short change in time. Then

dZ(t) = Z(t+dt)-Z(t) = Y(t)\sqrt{dt},

where Y(t) is the binomial distribution taking values \pm 1 with probability 0.5.

quadratic variation:

\lim_{n\rightarrow \infty} \sum_{i=1}^n (Z(iT/n)-Z((i-1)T/n))^2 = T.

The quadratic variation of a Brownian motion is deterministic. This implies higher-ordered variations are zero.

total variation:

\lim_{n\rightarrow \infty} \sum_{i=1}^n |Z(iT/n)-Z((i-1)T/n)| = \infty.

Arithmetic Brownian Motion

X(t) is modified from Z(t) by introducing two parameters, \alpha and \sigma .

X(t+s)-X(t) is normally distributed with mean \alpha s and variance \sigma^2 s ; in particular, X(s) is normally distributed with mean \alpha s and variance \sigma^2 s .
For an incremental time dt,
dX(t) = X(t+dt)-X(t) = \alpha dt + \sigma dZ(t).

The term \alpha dt is called the drift in the process; it's the deterministic part of the equation and deals with the long term behavior of the process. The term \sigma dZ(t) is the random noise part, which deals with the short term behavior.

We also have

X(t)= \alpha t + \sigma Z(t).

Geometric Brownian Motion

Here the process follows

dX(t) = X(t+dt)-X(t) = \alpha X(t) dt + \sigma X(t) dZ(t),

dX(t)/X(t) = d \ln X(t) = \alpha dt + \sigma dZ(t).

We can verify that

X(t)= X(0)e^{(\alpha - 0.5 \sigma^2)t+\sigma\sqrt{t}Z(t)},

using Ito's lemma below.

A calculation shows

E(X(t)) = X(0)e^{\alpha t},

which means \alpha is the expected continuously compounded return on X.

Ito Processes

These are processes following

dX(t) = \alpha(X) dt + \sigma(X) dZ(t),

where \alpha and \sigma are functions of X. All processes above are examples.

Multiplication Rules

Basically approximations using 1st power of dt.

dt dZ = 0;

(dt)^2 =0;

(dZ)^2 = dt

Ito's Lemma

Let C(a, b) be a C^{2,1} function, and let S(t) be an Ito process. Then

d C(S, t) = C_a(S,t) dS + \frac{1}{2} C_{aa}(S, t) (dS)^2 + C_b(S,t) dt.

Use multiplication rules to simplify the result.

Useful Computation

Involving expectation
Involving variance