# $P(\max_{0 \leq t \leq 1} \|W(t)\| \leq x)$ has no closed-form expression… right?

$$P(\max_{0 \leq t \leq 1} \|W(t)\| \leq x)$$ shows up in a formula for computing $$p$$-values for a certain statistic, where $$W(t)$$ is a $$d$$-dimensional (standard) Wiener process. My advisor says the only way to compute this is via simulation.

Does anyone disagree?

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• In other words, you want to know the probability for a Bessel process to (not) hit $x$ before time 1. It depends what you mean by "closed form"; there are formulas in terms of infinite series of special functions. See for instance (2.7) of arxiv.org/abs/1106.6132. There are also very good bounds and asymptotics. – Nate Eldredge Dec 6 at 21:16
• @NateEldredge By closed form I basically mean that something that can be computed by non-stochastic numerical methods as opposed to needing to be simulated with an RNG. So infinite sums or expressions involving integrals are okay since they're not random and there are numerical approaches for computing them. – cgmil Dec 6 at 21:28
• @NateEldredge The paper you included appears to have what I want. Have these results been published? EDIT: Yes, they have. ams.org/journals/tran/2013-365-10/S0002-9947-2013-05799-6 – cgmil Dec 6 at 21:57