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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)=x_{0},

which has the solution

X(t)=x_{0}e^{((μ-σ2/2)t+σW(t))}.

This gives that the logarithm of X(t) is a Brownian motion with drift μ-σ^{2}/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.

dX(t)=μX(t)dt+σX(t)dW(t),

X(0)=x

which has the solution

X(t)=x

This gives that the logarithm of X(t) is a Brownian motion with drift μ-σ

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

Questions: Magnus Wiktorsson

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