Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(B_t)_{t\ge0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$
- $b:\mathbb R\to\mathbb R$ be linear
- $\sigma>0$
We know that the SDE $${\rm d}X_t=b(X_t)\:{\rm d}t+\sigma{\rm d}W_t$$ has a unique strong solution. Are we able to express this solution explicitly as it's the case for $b(x)=x$?