Linear stochastic differential equation ($d{X}_t=(a{X}_t+b)dt+(c{X}_t+d)d{W}_t$)

A stochastic differential equation with drift and diffusion coefficients affine in its state. In the constant-coefficient additive-noise case $d{X}_t=(a{X}_t+b)dt+\sigma d{W}_t$, multiplying by $e^{-at}$ givesConsequently its conditional distribution is normal, with mean $e^{a\tau }x+b(e^{a\tau }-1)/a$ and variance ${\sigma }^2(e^{2a\tau }-1)/(2a)$ when $a\ne 0$. For $a=0$, these become $x+b\tau$ and ${\sigma }^2\tau$. Additive noise permits negative states; it differs from geometric Brownian motion.