Position and height of the global maximum of a twice differentiable stochastic process

Igor Rychlik and Eva Sjö

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,

ISSN 1403-9338
For a stochastic process /omega  with absolutely continuous sample path derivative, a formula for the joint density of (T, Z), the position and height of the global maximum of /omega in a closed interval, is given. The formula is derived using the Generalized Rice's formula. The presented result can be applied both to stationary and non-stationary processes under mild assumptions on the process. The formula for the density is explicit but involves integrals that have to be computed using numerical integration. The computation of the density is discussed and some numerical examples are given.
Key words:
Stochastic process, global maximum, supremum, generalized Rice's formula, extremes