Symmetric Weighted Odd-Power Variations of Fractional Brownian Motion and Applications

Nualart, David; Zeineddine, Raghid

doi:10.31390/cosa.12.1.04

Cited by 2 publications

(1 citation statement)

References 15 publications

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“…In the same setting, for fixed t > 0, the power variations of x → u(t, x) were discussed in [13,30,41] for p = 2 and in [23] for γ ∈ ( 1 2 , 1] and p = 2/(2γ − 1). In the context of one-parameter stochastic processes, power variations have been investigated for semimartingales [31], fractional Brownian motion and related processes [14,15,36,37,38], and moving average processes [3,4,16], just to name a few.…”

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confidence: 99%

Power variations in fractional Sobolev spaces for a class of parabolic stochastic PDEs

Chong,

Dalang

2020

Preprint

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We consider a class of parabolic stochastic PDEs on bounded domains D ⊆ R d that includes the stochastic heat equation, but with a fractional power γ of the Laplacian. Viewing the solution as a process with values in a scale of fractional Sobolev spaces Hr, with r < γ − d/2, we study its power variations in Hr along regular partitions of the time-axis. As the mesh size tends to zero, we find a phase transition at r = −d/2: the solutions have a nontrivial quadratic variation when r < −d/2 and a nontrivial pth order variation for p = 2γ/(γ − d/2 − r) > 2 when r > −d/2. More generally, suitably normalized power variations of any order satisfy a genuine law of large numbers in the first case and a degenerate limit theorem in the second case. When r < −d/2, the quadratic variation is given explicitly via an expression that involves the spectral zeta function, which reduces to the Riemann zeta function when d = 1 and D is an interval.

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