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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