= Linear stochastic differential equation
{title2=$dX_t=(aX_t+b)\,dt+(cX_t+d)\,dW_t$}
A <stochastic differential equation> with drift and diffusion coefficients affine in its state. In the constant-coefficient additive-noise case $dX_t=(aX_t+b)dt+\sigma dW_t$, multiplying by $e^{-at}$ gives
$$
X_t=e^{at}\left(X_0+b\int_0^t e^{-au}\,du+\sigma\int_0^t e^{-au}\,dW_u\right).
$$
Consequently 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 $a\ne0$. For $a=0$, these become $x+b\tau$ and $\sigma^2\tau$. Additive noise permits negative states; it differs from <geometric Brownian motion>.
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