Last hitting time for the integral of the Brownian motion (Q1315400)

The linear Brownian motion \((B_ t)_{t\geq 0}\) starting from \(y\), and \(X_ t=\int^ t_ 0B_ sds+x\) are considered. If \(\tau^ - _{a,T}=\max\{t\in[0,T];\int^ t_ 0B_ sds+x+ty=a\}\) and \(\tau^ +_{a,T}=\min\{t\geq T;\int^ t_ 0B_ sds+x+ty=a\}\), the author defines the law of \((\tau^ -_{a,T},\tau^ +_{a,T},B_{\tau^ +_{a,T}})\). As corollaries of this main result, the laws of \((\tau^ - _{a,T},\tau^ +_{a,T})\) and of \(\tau^ -_{a,T}\) are also characterized.