A pure jump Markov process with a random singularity spectrum

Abstract

We construct a non-decreasing pure jump Markov process, whose jump measure heavily depends on the values taken by the process. We determine the singularity spectrum of this process, which turns out to be random and to depend locally on the values taken by the process. The result relies on fine properties of the distribution of Poisson point processes and on ubiquity theorems.The corresponding notion of "multifractal function" was put into light by Frisch and Parisi [13], who introduced the definition of the spectrum of singularities of a function f .By monotonicity (if A ⊂ B, then D f (A, h) ≤ D f (B, h)), the limit exists and it is independent of the particular basis chosen. Clearly, a function has a homogeneous spectrum if and only if for all h ≥ 0, D f (t, h) is independent of t ≥ 0. The local spectrum allows one to recover the spectrum of all possible restrictions of f on an open interval.Lemma 4. Let f : R + → R be a locally bounded function. Then for any open interval I = (a, b) ⊂ R + , for any h ≥ 0, we have D f (I, h) = sup t∈I D f (t, h).Proof. Let thus h ≥ 0 be fixed. First, it is obvious that for any t ∈ I, D f (t, h) ≤ D f (I, h), since for (V n ) n≥1 a basis of neighborhoods of t, V n ⊂ I for n large enough. Next, set δ = D f (I, h), and consider ε > 0. We want to find t ∈ I such that for all neighborhood V

Multifractality of jump diffusion processes

“…Let us comment on the major contributions of the present work relative to what has already been achieved in the area and in particular in [10].…”

“…This new argument does not rely on the monotonicity of the sample paths, thus is applicable to more general SDEs, see Section 3. • There is a novel discussion on the extreme value of the spectrum, the latter presenting a behavior more complex than the one observed in [10], see Section 6.…”

Multifractal Analysis and Wavelets

“…In addition, we compute their exact multifractal spectrum which is not the worst possible regularity, as it is the case in the preceding section. A lot is still to be done on multifractal properties of traces of functions, and also of slices and projections of multifractal measures, 6. Some examples of multifractal wavelet series 6.1.…”

Multifractal properties of sample paths of ground state-transformed jump processes

“…However, since the drift generated by the perturbation may become rough, the challenge is to establish conditions on V under which path regularity is at least preserved. Results in [2,43,44], where L = (−∆) s(x) and V ≡ 0, i.e., stable-like processes generated by fractional Laplacians of variable order are considered, indicate that local behaviour may become very complex, and instead of an almost sure rule it can be even dependent on the individual path.…”