2017/02/27 by Robert C. Dalang, Dalang, Robert C., Thomas Mountford +1
Economics, Econometrics and Finance · Mathematics · #60G15 #60G17 #60G60 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1702.08183
openalex publication_date 2017/02/27 · openalex created_date 2017/03/16 · openalex updated_date 2026/07/28
We first consider the additive Brownian motion process (X(s1,s2), (s1,s2) ∈ ℝ2) defined by X(s1,s2) = Z1(s1) - Z2 (s2), where Z1 and Z2 are two independent (two-sided) Brownian motions. We show that with probability one, the Hausdorff dimension of the boundary of any connected component of the random set \(s1,s2)∈ ℝ2: X(s1,s2) >0\ is equal to (1)/(4)(1 + √(13 + 4 √(5))) ≃ 1.421 . Then the same result is shown to hold when X is replaced by a standard Brownian sheet indexed by the nonnegative quadrant.