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On the first-passage time of integrated Brownian motion. (English) Zbl 1102.60069
Author’s summary: Let \(~(B_t;t \geq 0)~\) be a Brownian motion process starting from \(~B_0=\nu~\) and define \(~X_\nu (t)= \int^t_0 B_s ds\). For \(~a\geq 0\), set \(~\tau_{a,\nu}:=\inf\{t:X_\nu(t)=a\}~\) (with \(~\inf\emptyset=\infty)\): We study the conditional moments of \(~\tau_{a,\nu}~\) given \(~\tau_{a,\nu}<\infty\). Using martingale methods, stopping-time arguments, as well as the method of dominant balance, we obtain, in particular, an asymptotic expansion for the conditional mean \(~E(\tau_{a,\nu}\mid\tau_{a,\nu}<\infty)~\) as \(~\nu \rightarrow \infty\). 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 small \(~\nu\).
MSC:
60J65 Brownian motion
60G40 Stopping times; optimal stopping problems; gambling theory
60J25 Continuous-time Markov processes on general state spaces
60J60 Diffusion processes
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