def. Time Series is a stochastic process indexed by discrete (integer) time points.

def. Stochastic Process is simply the sum of the sequence of random variables: :

Weiner Process

Standard Weiner Process

Motivation. Assume there are random variables as:

  1. let be a stochastic process such that:

Then we get the Weiner process as:

def. Brownian Motion. (=Weiner process) (written as ) is a set of random variables continuous-time indexed and has the following properties:

  1. Is a continuous process
    1. Thus,
  2. Any interval and where and do not overlap is independent. Notation wise, we think of , i.e. a set of random variables indexed by .

Remark. Brownian Motion can also be defined as an Ito Process that satisfies the following stochastic differential equation:

Trivially, in the case of standard brownian motion, Thus .

Properties. Brownian Motion satisfies the following properties:

  • Martingale Property: Brownian motion is martingale: where
  • Markov Property:
    • is independent of and…
    • Alternatively: where any combination
  • Scaling Invariance: If is a brownian motion, then is also a brownian motion
  • Quadratic Variation property:

With Scale and Drift

def. Weiner Process with Drift and Scaling (WPDS). Such is a Weiner process that has the following properties:

  1. Is a continuous process
  2. Any interval and where and do not overlap is independent.
  • Properties:
    • : drift; the higher, the more it climbs
    • : scaling; the higher, the more volatile (See y-axis:)

thm. Standard WP to Scale and Drift. Given:

  • is a standard Weiner process
  • is a Weiner process with drift and scaling , with initial value
  • ⇒ Then the following relationship holds (two equivalent definitions)

Geometric Brownian Motion

thm. Geometric Brownian Motion. Given is a WPDS then the following is a geometric brownian motion with initial value : (two equivalent definitions)

Geometric to WPSD

The following is equivalent:

Properties of Stochastic Processes

def. Adapted process. A stochastic process is adapted to filtration set if is a -measurable function.

Intuition. Recall that a sigma algebra can be thought of as “resolution of information”. is realized and its information is . The series of filtrations correspond to the series of random variables ; for each timestep, the information gets higher and higher resolution.

Martingale Process

Motivation. Many stochastic processes, including the standard brownian motion has the property that you can’t predict future trajectory based on information from its past trajectory. We formalize this as a martingale.

def. Martingale. A stochastic process on a filtered probability space is a martingale w.r.t. its adapted filtration if:

This is equivalent to saying

  • by taking expectation on both sides of above
    • This is a simple proof that brownian motion is a martingale;

Quadratic Variation

Motivation. Quadratic variation is not about variation of probability distributions. It’s a way to measure how “shaky” a function (in this case, a Brownian motion) is in a given interval.

def. Quadratic Variation. For a stochastic process , its quadratic variation on interval , for , is: