Solve the stochastic differential equation $dX_t = -X_t \, dt + e^{-t} \, dW_t$

Question

Solve the SDE

$$ dX_t=-X_tdt + e^{-t}dW_t $$

where $W_t$ is a standard Brownian process.

IC: $X(0)=X_0$.

Answer 1

Let us first forget about the stochastic part "$e^{-t} \, dW_t$" and consider the ordinary differential equation

$$dX_t = - X_t \, dt, \qquad X_0 = c$$

instead. It is well-known (and not difficult to check) that the (unique) solution is given by

$$X_t = c e^{-t}.$$

In order to solve the SDE we are interested in, we now try a variation of constants approach, i.e. we let the constant $c$ depend on $\omega$ and $t$; that is we set

$$X_t(\omega) = C_t(\omega) e^{-t} \tag{1}$$

for a suitable stochastic process $(C_t)_{t \geq 0}$. We have to check whether there exists a suitable process $(C_t)_{t \geq 0}$ such that $(1)$ solves the SDE

$$dX_t = -X_t \ dt + e^{-t} \, dW_t \tag{2}$$

To show that this is indeed true (and to determine $C_t$), apply Itô's formula to $C_t = e^t X_t$.