On the first-passage time of integrated Brownian motion

[摘要] Let(Bt;t≥0)be a Brownian motion process starting fromB0=νand defineXν(t)=∫0tBsds. Fora≥0, setτa,ν:=inf{t:Xν(t)=a}(withinf φ=∞). We study the conditional moments ofτa,νgivenτa,ν<∞. Using martingale methods, stopping-time arguments, as well as the method of dominant balance, we obtain, in particular, an asymptotic expansion for the conditional meanE(τa,ν|τa,ν<∞)asν→∞. Through a series of simulations, it is shown that a truncation of this expansion after the first few terms provides an accurate approximation to the unknown true conditional mean even for smallν.

[学科分类] 应用数学