Level crossing probabilities for a multi-parameter Brownian process

Peggy Strait

Vol. 65 (1976), No. 1, 223–232

DOI: 10.2140/pjm.1976.65.223

Abstract

Let {ξ_s1,s2 : −∞ < s₁,< ∞,−∞ < s₂ < ∞} be a Gaussian process with ξ_s1,s2 = 0 if s₁ = 0 or s₂ = 0, mean values E(ξ_s1,s2) = 0, and covariances E(ξ_s1,s2ξ_s1′,s1′) = 1∕2min(s₁,s₁′)min(s₂,s₂′). This is the two parameter Brownian process studied by J. D. Kuelbs, W. J. Park, P. T. Strait, and J. Yeh. In this paper, upper and lower bounds for level crossing probabilities of this process are derived.

Mathematical Subject Classification 2000

Primary: 60G15

Secondary: 60J65

Milestones

Received: 19 August 1975

Published: 1 July 1976