Integral of a Brownian motion

We are going to show that {M_{t} = \int_{0}^{t} B_{s} ds} is a normal distribution {\mathcal N(0, \frac 1 3 t^{3})} for each fixed {t\ge 0}, where {B} is a Brownian motion.

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Skorohod metric

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Reference: [Bil99] ([Patrick Billingsley 1999])

Goal: RCLL space is the collection of all right continuous processes with left limit exists. In this below, we will give a definition of Skorohod metric on this space. More precisely, for a {t\in (0, \infty)}, we denote by {\mathbb D^{d}_{t}} the collection of RCLL processes on {[0, t]} taking values in {\mathbb R^{d}}. We also use {\mathbb D^{d}_{\infty}} to denote the collection of RCLL processes on {[0, \infty)}.

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