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Hermite process distributions are shown to have densities

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Research area:finance-marketsactuarial-science-risk-modeling

What the study found

The study found that, for any distinct times, the vector formed by a Hermite process has a density with respect to Lebesgue measure. In other words, its finite-dimensional distributions are absolutely continuous.

Why the authors say this matters

The authors note that the non-Gaussian case had not yet been settled, unlike the Gaussian case of fractional Brownian motion. They also say their methodology could extend to other non-Gaussian models.

What the researchers tested

The researchers studied Hermite processes of order q ≥ 1 with self-similarity parameter H in (1/2, 1). They extended a three-step approach from the Gaussian setting using Malliavin calculus, including a determinant identity for the Malliavin matrix, strong local nondeterminism at the level of Malliavin derivatives, and the Bouleau-Hirsch criterion.

What worked and what didn't

The approach worked in the sense that it led to the density result for the vector of values at distinct times. The abstract does not describe any failed steps or negative results.

What to keep in mind

The summary only states results for finite-dimensional distributions at distinct times, not for other properties of Hermite processes. Limitations beyond this scope are not described in the available abstract.

Key points

  • Hermite processes of order q ≥ 1 were shown to have finite-dimensional densities.
  • The result applies for self-similarity parameter H in (1/2, 1).
  • The proof uses Malliavin calculus and a three-step strategy adapted from the Gaussian case.
  • The abstract says the non-Gaussian case had not yet been settled before this work.
  • The authors suggest the method may extend to other non-Gaussian models.

Disclosure

Research title:
Hermite process distributions are shown to have densities
Authors:
Laurent Loosveldt, Yassine Nachit, Ivan Nourdin, Ciprian A. Tudor
Publication date:
2026-04-24
DOI:
10.1090/proc/17762
OpenAlex record:
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AI provenance: This post was generated by gpt-5.4-mini (OpenAI). The original authors did not write or review this post.

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