Let (d S_{t}=S_{t^{-}}left(b d t+sigma d W_{t}+phi d M_{t} ight)) where (b, sigma) and (phi) are constant

Let \(d S_{t}=S_{t^{-}}\left(b d t+\sigma d W_{t}+\phi d M_{t}\right)\) where \(b, \sigma\) and \(\phi\) are constant coefficients and \(\phi>-1\). Let \(Y_{t}=\left(S_{t}\right)^{-1}\). Prove that

\[d Y_{t}=-Y_{t^{-}}\left\{\left(b-\sigma^{2}+\lambda\left(\frac{\phi}{1+\phi}-\phi\right)\right) d t+\sigma d W_{t}+\frac{\phi}{1+\phi} d M_{t}\right\}\]

The jumps of \(S\) occur when the Poisson process \(N\) jumps, and the sizes of the jumps are \(\Delta S_{t}=\phi S_{t^{-}} \Delta N_{t}\), hence \(S_{t}=S_{t^{-}}\left(1+\phi \Delta N_{t}\right)\). The coefficient of \(d M\) (or of \(d N\) ) may also be obtained by looking at the size of the jumps of the process \(S^{-1}\).