A note on Brownian areas and arcsine laws

Firstly, we provide simple elementary proofs to derive the exact distributions of the areas under functions of a Brownian motion process and a Brownian bridge process. In the latter case, a solution is therefore provided to a question raised recently in the Mathematics community on StackExchange (http://math.stackexchange.com/questions/1006101). These random areas often occur in statistical applications and play an important role in, for example, financial mathematics. Comparisons are made between the variances of the two random areas, deriving interesting results that appear to be new in the statistical literature. Some illustrative examples are provided. Secondly, we derive a new arcsine law for a standard Brownian bridge process