# 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 $\Delta t$,

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

# 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.

# 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 = $b^2\times T$