= Stationary law of negatively drifted reflected Brownian motion
{title2=$\mathbb P(R_\infty>b)=e^{-2db/\sigma^2}$}
The stationary law is the law of $\sup_{t\ge0}(\sigma B(t)-dt)$. Let $\kappa=2d/\sigma^2$. The <Exponential martingale for Brownian motion> $e^{\kappa(\sigma B_t-dt)}$, stopped on first hitting $b$, is bounded by $e^{\kappa b}$. On no hitting, the driving path tends to minus infinity. <Dominated convergence> and <optional stopping> therefore give $1=e^{\kappa b}\mathbb P(\text{hit }b)$. The reflected process started at zero converges to this supremum law: its value at time $t$ has the law of the driving path's maximum up to $t$, by reversing the increments. The resulting density is $(2d/\sigma^2)e^{-2dx/\sigma^2}$ for $x\ge0$.
Back to article page
