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
Famous quotes containing the words solving the and/or solving:
“More than a decade after our fellow citizens began bedding down on the sidewalks, their problems continue to seem so intractable that we have begun to do psychologically what government has been incapable of doing programmatically. We bring the numbers down—not by solving the problem, but by deciding it’s their own damn fault.”
—Anna Quindlen (b. 1952)
“There are horrible people who, instead of solving a problem, tangle it up and make it harder to solve for anyone who wants to deal with it. Whoever does not know how to hit the nail on the head should be asked not to hit it at all.”
—Friedrich Nietzsche (1844–1900)