Geometric Brownian Motion - Solving The SDE

Solving The SDE

For an arbitrary initial value S0 the above SDE has the analytic solution (under Itō's interpretation):

To arrive at this formula, let us divide the SDE by, and write it in Itō integral form:

Of course, looks like having a lot to do with the derivative of ; however, being an Itō process, we need to use Itō calculus: by Itō's formula, we have

Plugging back to the equation we got from the SDE, we obtain

Exponentiating gives the solution claimed above.

Read more about this topic:  Geometric Brownian Motion

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