a) Solve the stochastic differential equation

\[

\begin{equation*}

d X_{t}=-b X_{t} d t+\sigma \mathrm{e}^{-b t} d B_{t}, \quad t \geqslant 0 \tag{4.40}

\end{equation*}

\]

where \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion and \(\sigma, b \in \mathbb{R}\).

b) Solve the stochastic differential equation

\[

\begin{equation*}

d X_{t}=-b X_{t} d t+\sigma \mathrm{e}^{-a t} d B_{t}, \quad t \geqslant 0 \tag{4.41}

\end{equation*}

\]

where \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion and \

(a,

b, \sigma>0\) are positive constants.