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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

{$$ dX(t) = X(t+dt)-X(t) = \alpha X(t) dt + \sigma X(t) dZ(t),$$} or {$$ dX(t)/X(t) = d \ln X(t) = \alpha dt + \sigma dZ(t).$$}

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

From James’s Page

Actuary: Brownian