How to express the the stochastic integral into an Ito process

Question

I am facing problem while trying to express the following stochastic integral into an Ito process.

$$X_{t} = e^{-t}\int_{0}^{t}e^{u}dW_{u}$$

I would like to express the above as $dX_{t} = \Theta dt + \Delta dW_{t}$. I started with expressing the integral part into $e^{t}W_{t}-\int_{0}^{t} e^{u} W_{u}du$. However, I am stuck as if I continue with the above, I will get

$$e^{-t} \int_{0}^{t} e^{u} W_{u}du + dW_{t}-W_{t}dt$$

This is not of the form I needed. May I know which step I have done wrong and how to go from here please?

Answer 1

Recall the Itô product rule: $$d(YZ)_t = Y_t dZ_t + Z_t dY_t + dY_t dZ_t$$ Thus, for your process, \begin{align} dX_t = e^{-t} e^t dW_t -e^{-t} \left( \int_0^t e^u dW_u \right) dt= dW_t - X_t dt \end{align}