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.
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.