Abstract
A tempered Hermite process modifies the power law kernel in the time domain representation of a Hermite process by multiplying an exponential tempering factor $\lambda>0$ such that the process is well defined for Hurst parameter $H>\frac{1}{2}$. A tempered Hermite process is the weak convergence limit of a certain discrete chaos process.
Highlights
The Hermite processes of order k = 1, 2, . . . are defined as multiple Wiener–Itô integrals tk ZHk (t) :=(s − yi )d+−1 ds B(dy1) . . . B(dyk), (1) Rk 0 i=1 where d = − 1−H k ∈ ( ) and
We introduce a new class of stochastic processes, which we call tempered Hermite processes
Tempered Hermite processes modify the kernel of ZHk by multiplying an exponential tempering factor λ > 0 such that they are well defined for Hermite processes are not self-similar processes, but they have a scaling property, involving both the time scale and the tempering parameter
Summary
The Hermite processes of order k = 1, 2, . Are defined as multiple Wiener–Itô integrals tk ZHk (t) :=. < 1 (the prime on the integral sign shows that one does not integrate on the diagonals xi = xj , i = j ). We introduce a new class of stochastic processes, which we call tempered Hermite processes. Tempered Hermite processes modify the kernel of ZHk by multiplying an exponential tempering factor λ > 0 such that they are well defined for Tempered. Hermite processes are not self-similar processes, but they have a scaling property, involving both the time scale and the tempering parameter. The scaling property enable us to show that the tempered Hermite processes are the weak convergence limits of certain discrete chaos processes
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