The change in a share price satisfies dS = A(S, t) dX + B(S, t) dt, for

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The change in a share price satisfies
dS = A(S, t) dX + B(S, t) dt,
for some functions A,B, what is the stochastic differential equation satisfied by f(S, t)?

Can A,B be chosen so that a function g(S) has a zero drift, but non-zero variance?

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