Geometric Brownian motion

A geometric Brownian motion GBM also so-called as exponential Brownian motion is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion also called a Wiener process with drift. it is an important example of stochastic processes satisfying a stochastic differential equation SDE; in particular, it is used in mathematical finance to expediency example stock prices in the Black–Scholes model.

Solving the SDE

For an arbitrary initial expediency S₀ the above SDE has the analytic solution under Itô's interpretation:

The derivation requires the use of Itô's formula leads to

where is the quadratic variation of the SDE.

When , converges to 0 faster than , since . So the above infinitesimal can be simplified by

Plugging the value of in the above equation together with simplifying we obtain

Taking the exponential in addition to multiplying both sides by authorises the written claimed above.