Solve the stochastic differential equation [ d X_{t}=h(t) X_{t} d t+sigma X_{t} d B_{t} ] where (sigma>0)

Question:

Solve the stochastic differential equation

\[
d X_{t}=h(t) X_{t} d t+\sigma X_{t} d B_{t}
\]

where \(\sigma>0\) and \(h(t)\) is a deterministic, integrable function of \(t \geqslant 0\).

Look for a solution of the form \(X_{t}=f(t) \mathrm{e}^{\sigma B_{t}-\sigma^{2} t / 2}\), where \(f(t)\) is a function to be determined, \(t \geqslant 0\).

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