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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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