The minimum of a stationary Markov process superimposed on a U-shaped trend

Abstract

1. This paper was motivated by some questions of Barnett and Lewis (1967) concerning extreme winter temperatures. The temperature during the winter can be hopefully regarded as generated by a stationary Gaussian process superimposed on a locally U-shaped trend. One is interested in statistical properties of the minimum of sample paths from such a process, and of their excursions below a given level. Equivalently one can consider paths from a stationary process crossing a curved boundary of the same form. Problems of this type are discussed by Cramer and Leadbetter (1967), extensively in the trend-free case and in less detail when a trend is present, following the method initiated by Rice (1945). While results on moments are easy to obtain, explicit results for the actual probability distributions are not usually available. However, in the important case when the level of values of interest is far below the mean, the asymptotic independence of up-crossing times makes it possible to derive simple approximate distributions. (See Cramer and Leadbetter (1967) page 256, Keilson (1966).)

Linear and Nonlinear Boundary Crossing Probabilities for Brownian Motion and Related Processes

“…A well-known result is that the boundary crossing probability for Brownian motion for non-linear boundary (1+ √ 1 + 8e −1/t )) is 0.479749 (see Daniels [22]). …”

Linear and Nonlinear Boundary Crossing Probabilities for Brownian Motion and Related Processes

“…A well-known result is that the BCP of Brownian motion is 0.479 749 with boundary 1 2 − t log( 1 4 (1 + √ 1 + 8e −1/t )) (see [6]). By our method, we obtained a BCP of 0.479 737 with m = 50 000.…”

“…All of the transition probability matrices of the Markov chain {Y i (m)} n i=0 have the form 6) where the fundamental matrices N i are rectangular of size…”

On first-passage-time and transition densities for strongly symmetric diffusion processes

“…[1], Giorno, V. et al [10] and references therein). However, usually the knowledge of the 'free' transition probability density function (pdf) is not sufficient; one is thus led to the more complicated task of determining transition functions in the presence of preassigned absorbing boundaries, or first-passage-time densities for time-dependent boundaries (see, for instance, Daniels, H. E. [6], [7], Giorno, V. et al [10]). Such densities are known analytically only in some special instances so that numerical methods have to be implemented in general (cf., for instance, Buonocore, A. et al [3], [4], Giorno, V. et al [11]).…”