# Brownian bridge to unknown via extremum

Suppose, I know what's the minimum $\min$ of a random walk $w_t$ in period $[0,\Delta t]$. I also know $w_0$ and $\sigma$. How to construct the Brownian bridge for the latter period?

I guess it's not even a bridge since I don't know $w_{\Delta t}$, so my bridge starts at known $w_0$ and goes down to $t=\Delta t$, while hitting the known minimum $\min$ on the way. In standard bridge you go from known $w_0$ to a known $w_{\Delta t}$

• "CONSTRUCTIONS OF A BROWNIAN PATH WITH A GIVEN MINIMUM" projecteuclid.org/download/pdf_1/euclid.ecp/1456938427 – Mark L. Stone Feb 13 '18 at 23:56
• I saw this paper, cant understand the damn thing in it, bloody mathematicians, I hate them – Aksakal Feb 14 '18 at 0:02
• To clarify, do you just know the minimum value, and not the time at which it occurred? – Alex R. Feb 14 '18 at 0:26
• @AlexR.yes, only the exact value, not the time. – Aksakal Feb 14 '18 at 0:32
• @MarkL.Stone, thank you, I figured out what they're doing, it turned out simpler than I thought at first. I hate the way they present results. – Aksakal Feb 14 '18 at 4:14