Let T > 0 be a finite and $(B_t)_{t\in[0,T]}$ a standard Brownian motion. Consider the following SDE:
$dX_t =(aE[X_t]−bX_t)dt+σdB_t, $ $ t∈[0,T]$
$X_0 =x_0 \in \mathbb{R^+}$,
where $a, b, \sigma \in \mathbb{R^+}$. I already know that this SDE has a path-wise and weakly unique strong solution which is in $L^2(\Omega)$ and continuously differentiable with respect to $x_0$.
I now want to find a solution for this SDE, but I struggle to find an ansatz.
If $\phi(t)=\Bbb E(X_t)$, then you read off the SDE $$\dot\phi(t)=(a-b)\,\phi(t),$$ so that $$\phi(t)=\phi(0)\,e^{(a-b)t}.$$ Now insert that into $$ d(e^{bt}X_t)=a\phi(t)\,e^{bt}\, dt + σ\,e^{bt}\,dB_t $$ and integrate.