Let X be an Ornstein–Uhlenbeck process driven by a Brownian motion. We propose an expression for the joint density / distribution function of the process and its running supremum. This law is expressed as an expansion involving parabolic cylinder functions. Numerically, we obtain this law faster with our expression than with a Monte Carlo method. Numerical applications illustrate the interest of this result.

中文翻译：

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Ornstein-Uhlenbeck 过程的联合定律及其上定理

让X是由布朗运动驱动的 Ornstein-Uhlenbeck 过程。我们提出了过程的联合密度/分布函数及其运行上界的表达式。该定律表示为涉及抛物柱面函数的展开。在数值上，我们用我们的表达式比用蒙特卡洛方法更快地获得这个定律。数值应用说明了这个结果的重要性。