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Solutions to Problems to Work for Understanding

What is the solution of the stochastic differential equation
dY(t) = \mu Y(t) dt + \sigma Y(t) dX

This problem is adapted from
* Financial Calculus: An introduction to derivative pricing *
by M Baxter, and A. Rennie, Cambridge University Press,
1996, page 61.
We could match both the drift and volaility terms for this SDE
to the known SDE for the process \exp(\rho X(t) + \nu t) if
and only if we take \sigma = \rho and \vu = \mu -
(1/2)\sigma^2. That is, we guess the solution of the SDE to
be
Y(t) = \exp( \sigma X(t) + (\mu - (1/2) \sigma^2)t)

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