Ito Process

It is a generalized Wiener process in which the parameters a and b are functions of the variable x and t: dx = a\times (x,t) dt + b\times (x,t) dz

This is Markov because the value of x only depends on time t. Both the drift rate and volatility rate are changing overtime.

For the change \Delta x in small time interval \Delta t,

\Delta x = a\times (x,t)\Delta t + b\times (x,t) \sqrt{\Delta t}

Ito Process

Arbitrage Theorem

Let X=(X_1, \dots, X_n) be the amount bet on each of the n experiment. Each experiment has possible outcome j=1,2, \dots, m, r_i(j) = return function of bet i when outcome is j the expected return when outcome is j: = E[X| j]= \sum_{i=1}^{n} X_i r_{i}(j)

There must be a set of probabilities P=(P_1, \dots, P_n) for which, the expected return = \sum_{j=1}^m P_j r_i(j)=0, for all i=1, \dots, n.

Otherwise, an arbitrage would exists: \sum_{i=1}^n X_i r_i(j) >0 for all j=1, \dots, m.

The set of P are called risk-neutral probabilities, resulting in all bets being fair.

Arbitrage Theorem

Generalized Wiener Process

If the change \Delta z in a small period of time \Delta t is independent, with a standardized normal distribution (0,1), then z follows the basic Wiener process (Brownian motion). \Delta z = normal(0, \Delta t); for any period T, z = normal(0,T)

A generalized Wiener process of variable x: dx = a\times dt + b\times dz, where a and b are constants

In a \Delta t, the change

\Delta x = a\times \Delta t + b\times \sqrt{\Delta t}

\Delta x has a normal distribution with an expected drift rate of a and a variance of b^2.

x in any time interval T is normally distributed with mean of change in x = aT, and variance of change in x = b^2\times T

Generalized Wiener Process