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-Loève 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.

中文翻译：

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带漂移的平方布朗运动积分分布的精确渐近公式

本文的目的是针对带有漂移的平方布朗运动积分的概率分布，导出一组易于实现的公式。通过Karhunen-Loève变换重新建立特征函数，我们获得了矩的递推公式以及具有显式系数的快速收敛级数的概率密度函数和累积分布函数。我们还执行渐近分析，以获取带有小或大参数的精确公式的近似值。广泛使用抛物柱面函数。进行了数值实验，以证明所提出的公式的适用性和效率以及它们在漂移影响下如何变化。