Brownian motion is the random movement of particles in a fluid due to their collisions with other atoms or molecules. The markov and martingale properties have also been defined.

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### In particular the process is always positive one of the reasons that geometric brownian motion is used to model financial and other processes that cannot be negative.

**Brownian motion finance for dummies**. Brownian motion is also known as pedesis which comes from the greek word for leaping even though a particle may be large compared to the size of atoms and molecules in the surrounding medium it can be moved by the impact with many tiny fast moving masses. Both x is a martingale with respect to p and its own natural filtration. X is a brownian motion with respect to p i e the law of x with respect to p is the same as the law of an n dimensional brownian motion i e the push forward measure x p is classical wiener measure on c 0 0.

At this stage the rationale for stochastic calculus in regards to quantitative finance has been provided. This is a stochastic differential equation sde because it describes random movement of the stock s t. The law of motion for stocks is often based on a geometric brownian motion i e ds t mu s t.

In both articles it was stated that brownian motion would provide a model for path of an asset price over time. For x 0 in 0 infty the process x 0 x t. Note also that x 0 1 so the process starts at 1 but we can easily change this.

In this article brownian motion will be formally defined and its mathematical analogue the wiener process will be. Dt sigma s t.

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