Abstract

We investigate the joint distribution and the multivariate survival functions for the maxima of an Ornstein-Uhlenbeck (OU) process in consecutive time-intervals. A PDE method, alongside an eigenfunction expansion, is adopted with which we first calculate the distribution and the survival functions for the maximum of a homogeneous OU-process in a single interval. By a deterministic time-change and a parameter translation, this result can be extended to an inhomogeneous OU-process. Next, we derive a general formula for the joint distribution and the survival functions for the maxima of a continuous Markov process in consecutive periods. With these results, one can obtain semi-analytical expressions for the joint distribution and the multivariate survival functions for the maxima of an OU-process, with piecewise constant parameter functions, in consecutive time periods. The joint distribution and the survival functions can be evaluated numerically by an iterated quadrature scheme, which can be implemented efficiently by matrix multiplications. Moreover, we show that the computation can be further simplified to the product of single quadratures by imposing a mild condition. Such results may be used for the modeling of heatwaves and related risk management challenges.

摘要

我们研究了连续时间间隔中 Ornstein-Uhlenbeck (OU) 过程的最大值的联合分布和多元生存函数。采用 PDE 方法以及特征函数展开，我们首先计算单个区间内齐次 OU 过程的最大值的分布和生存函数。通过确定性的时间变化和参数转换，该结果可以扩展到非均匀 OU 过程。接下来，我们推导出一个通用公式，用于连续期间连续马尔可夫过程的最大值的联合分布和生存函数。有了这些结果，我们可以得到联合分布的半解析表达式和 OU 过程最大值的多元生存函数，具有分段常数参数函数，在连续的时间段内。联合分布和生存函数可以通过迭代正交方案进行数值评估，这可以通过矩阵乘法有效地实现。此外，我们表明，通过施加温和的条件，可以将计算进一步简化为单正交的乘积。此类结果可用于热浪和相关风险管理挑战的建模。