 Solution

For the drifted Brownian motion $Y_t=y+B_t+at$ on $(0,\infty )$, the diffusion generator is $\mathcal{L}=\frac{1}{2}\frac{d^2}{d{z}^2}+a\frac{d}{dz}$. Seek a bounded solution of $\mathcal{L}u=\lambda u$ with $u(0)=1$. The exponential ansatz $u(z)=e^{rz}$ givesBecause $\lambda >0$, the plus root is positive and the minus root is negative. Boundedness on $[0,\infty )$ therefore selectsIt satisfies the boundary condition and all hypotheses of part (d). HenceThis is the first-passage Laplace transform for Brownian motion with drift, with $e^{-\lambda \infty }=0$. At $a=0$ it reduces to $e^{-y\sqrt{2\lambda }}$. As $\lambda \downarrow 0$, it gives $\mathbb{P}(S<\infty )=e^{-2y\max (a,0)}$, agreeing with certainty of hitting when the drift points towards zero and a possible escape when it points away.