Find an integral expression for $X_t$, where $X$ is an Ornstein-Uhlenbeck-type process governed by the stochastic differential equation:
$dX_t = [\nu - \gamma\left(X_t - \nu\,t \right)]dt + \sigma\,dW_t$
I need to reverse engineer $dX_t$ using Ito's lemma, where $X_t$ has a drift of $[\nu - \gamma\left(X_t - \nu\,t \right)]$ and a volatility of $\sigma$ for constant $\nu$, $\sigma$ and $\gamma$.
Is the following approach correct? Rewriting $dX_t$:
$$dX_t = \left[\nu\,(1+\gamma\,t) - \gamma\,X_t \right]dt + \sigma\,dW_t$$
Consider $Y_t = e^{\gamma\,t}\,X_t$
By Ito, $$dY_t = \gamma\,Y_t\,dt + e^{\gamma\,t}\,dX_t$$
Substituting for $Y_t$, $dX_t$ and rearranging:
$$dY_t = \nu\,(1+\gamma\,t)e^{\gamma\,t}dt + \sigma\,e^{\gamma\,t}dW_t$$
Letting $\alpha = \nu\,(1+\gamma\,t)$ and integrating:
$$Y_t = Y_0 + \alpha\,\int_{0}^{t} e^{\gamma\,s}ds + \sigma\,\int_{0}^{t} e^{\gamma\,s}dW_s$$
Finally, substituting back for $Y_t$ and $\alpha$:
$$X_t = X_0\,e^{-\gamma\,t} + \nu\,(1+\gamma\,t),\int_{0}^{t} e^{-\gamma\,(t-s)}ds + \sigma\,\int_{0}^{t} e^{-\gamma\,(t-s)}dW_s$$