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The usual model for the time-evolution of an asset price $S(t)$ is given by the geometric Brownian motion, represented by the following stochastic differential equation:

Note that the coefficients $\mu$ and $\sigma$, representing the drift and volatility of the asset, respectively, are both constant in this model. In more sophisticated models they can be made to be functions of $t$, $S(t)$ and other stochastic processes.

The solution $S(t)$ can be found by the application of Ito's Lemma to the stochastic differential equation.

Dividing through by $S(t)$ in the above equation leads to:

This is an Ito drift-diffusion process. It is a standard Brownian motion with a drift term. Since the above formula is simply shorthand for an integral formula, we can write this as:

This is the solution the stochastic differential equation. In fact it is one of the only analytical solutions that can be obtained from stochastic differential equations.

Michael Halls-Moore

Mike is the founder of QuantStart and has been involved in the quantitative finance industry for the last five years, primarily as a quant developer and later as a quant trader consulting for hedge funds.