Let us try a systematic procedure for solving SDEs which works for a class of SDEs. Let

\begin{align*}

X(t)=a(t)\left[ x_0 + \int_0^t b(s) dB(s) \right] +c(t) \ .

\end{align*}

Assuming \(a\), \(b\), and \(c\) are differentiable, use Ito’s formula to find the equation for \(dX(t)\) of the form

\begin{align*}

dX(t)=[ F(t) X(t) + H(t)] dt + G(t)dB(t)

\end{align*}

were \(F(t)\), \(G(t)\), and \(H(t)\) are some functions of time depending on \(a,b\) and maybe their derivatives. Solve the following equations by matching the coefficients. Let \(\alpha\), \(\gamma\) and \(\beta\) be fixed numbers.

Notice that

\begin{align*}

X(t)=a(t)\left[ x_0 + \int_0^t b(s) dB(s) \right] +c(t)=F(t,Y(t)) \ .

\end{align*}

where \(dY(t)=b(t) dB(t)\). Then you can apply Ito’s formula to this definition to find \(dX(t)\).

- First consider

\[dX_t = (-\alpha X_t + \gamma) dt + \beta dB_t\]

with \(X_0 =x_0\). Solve this for \( t \geq 0\)
- Now consider

\[dY(t)=\frac{\beta-Y(t)}{1-t} dt + dB(t) ~,~~ 0\leq t < 1 ~,~~Y(0)=\alpha.\]

Solve this for \( t\in[0,1] \).
- \begin{align*}

dX_t = -2 \frac{X_t}{1-t} dt + \sqrt{2 t(1-t)} dB_t ~,~~X(0)=\alpha

\end{align*}

Solve this for \( t\in[0,1] \).