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,
2002
ISSN 14039338

Abstract:

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 nonstationary 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