Martingale = Independent Increments?

Thread starter Palindrom
Start date Oct 28, 2009
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Independent martingale

In summary, a martingale is a mathematical concept used in probability theory and statistics that represents a fair game where the expected value remains constant. Independent increments refer to the property that changes in a process are not dependent on each other, and martingales have independent increments, making them fair games. Some examples of processes with independent increments include Brownian motion, Poisson process, and Levy process, which are used to model various phenomena. The concept of independent increments is important because it allows for the modeling of complex systems and phenomena and the use of powerful mathematical tools to analyze and understand these processes.

Oct 28, 2009

Palindrom

Here's a stupid question: for a Gaussian process, are these two properties equivalent?

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Oct 28, 2009

bpet

No - the increments can be independent without having zero expectation, and vice-versa.

Oct 30, 2009

Palindrom

Right, thank you, I have a centered Gaussian process in mind (should've mentioned it).

What is a martingale?

A martingale is a mathematical concept used in probability theory and statistics. It is a stochastic process that represents a fair game, meaning the expected value of the process remains constant over time.

What are independent increments?

Independent increments refer to the property that the changes or increments in a process are not dependent on each other. This means that the value of the process at any given time is not affected by the previous values of the process.

How are martingales and independent increments related?

A martingale is said to have independent increments if the increments of the process are independent from each other. This means that the value of the process at any given time is not influenced by the previous values of the process, making it a fair game.

What are some examples of processes with independent increments?

Some examples of processes with independent increments include the Brownian motion, Poisson process, and Levy process. These processes are often used in finance, physics, and other fields to model various phenomena.

Why is the concept of independent increments important?

The concept of independent increments is important because it allows for the modeling of complex systems and phenomena. It also allows for the use of powerful mathematical tools, such as martingale theory, to analyze and understand these processes.