On Exact and Asymptotic Formulas for the Distribution of the Integral of a Squared Brownian Motion with Drift

The aim of this paper is to derive a set of easily implementable formulas regarding the probability distribution of the integral of a squared Brownian motion with drift. By reestablishing the characteristic function via the Karhunen-Loeve transform, we obtain recurrence formulas for the moments as well as rapidly converging series with explicit coefficients for the probability density function and cumulative distribution function. We also perform asymptotic analyses to obtain sharp approximations for the exact formulas with small or large arguments. Extensive use is made of the parabolic cylinder function. Numerical experiments are conducted to demonstrate the applicability and efficiency of the proposed formulas as well as how they vary under the drift impact.