Large deviations of infinite intersections of events in Gaussian processes

Michael Mandjes, Petteri Mannersalo, Ilkka Norros (Corresponding Author), Miranda van Uitert

Research output: Contribution to journalArticleScientificpeer-review

17 Citations (Scopus)

Abstract

Consider events of the form {Zs ≥ ζ (s),s ∈ S}, where Z is a continuous Gaussian process with stationary increments, ζ is a function that belongs to the reproducing kernel Hilbert space R of process Z, and S ⊂ R is compact. The main problem considered in this paper is identifying the function β∗ ∈ R satisfying β
∗(s) ≥ ζ (s) on S and having minimal R-norm. The smoothness (mean square differentiability) of Z turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ (s) = s for s ∈ [0, 1] and Z is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process.
Original languageEnglish
Pages (from-to)1269-1293
Number of pages25
JournalStochastic Processes and their Applications
Volume116
Issue number9
DOIs
Publication statusPublished - 2006
MoE publication typeA1 Journal article-refereed

Fingerprint

Gaussian Process
Large Deviations
Intersection
Reproducing Kernel Hilbert Space
Brownian movement
Hilbert spaces
Fractional Brownian Motion
Explicit Solution
Differentiability
Mean Square
Increment
Smoothness
Norm
Form

Keywords

  • sample-path large deviations
  • dominating point
  • kernel Hilbert space
  • minimum norm problem
  • fractional brownian motion
  • busy period
  • Ornstein-Uhlenbeck process
  • Gaussian processes

Cite this

Mandjes, Michael ; Mannersalo, Petteri ; Norros, Ilkka ; van Uitert, Miranda. / Large deviations of infinite intersections of events in Gaussian processes. In: Stochastic Processes and their Applications. 2006 ; Vol. 116, No. 9. pp. 1269-1293.
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Large deviations of infinite intersections of events in Gaussian processes. / Mandjes, Michael; Mannersalo, Petteri; Norros, Ilkka (Corresponding Author); van Uitert, Miranda.

In: Stochastic Processes and their Applications, Vol. 116, No. 9, 2006, p. 1269-1293.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Large deviations of infinite intersections of events in Gaussian processes

AU - Mandjes, Michael

AU - Mannersalo, Petteri

AU - Norros, Ilkka

AU - van Uitert, Miranda

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AB - Consider events of the form {Zs ≥ ζ (s),s ∈ S}, where Z is a continuous Gaussian process with stationary increments, ζ is a function that belongs to the reproducing kernel Hilbert space R of process Z, and S ⊂ R is compact. The main problem considered in this paper is identifying the function β∗ ∈ R satisfying β∗(s) ≥ ζ (s) on S and having minimal R-norm. The smoothness (mean square differentiability) of Z turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ (s) = s for s ∈ [0, 1] and Z is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process.

KW - sample-path large deviations

KW - dominating point

KW - kernel Hilbert space

KW - minimum norm problem

KW - fractional brownian motion

KW - busy period

KW - Ornstein-Uhlenbeck process

KW - Gaussian processes

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