## Glossary

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# Geometric Brownian motion

Geometric Brownian motion satisfies the stochastic differential equation
dX(t)=μX(t)dt+σX(t)dW(t),
X(0)=x0,
which has the solution
X(t)=x0e((μ-σ2/2)t+σW(t)).
This gives that the logarithm of X(t) is a Brownian motion with drift μ-σ2/2 and diffusion coeficient σ. Below you can see a simulation of the process where you can change the parameters.

Geometric Brownian motion is used as a model for stockprices in the so called Black-Scholes model.