Results on generalized Mittag-Leffler function via Laplace transform

Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high-frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly. By nearly unstable, we mean that the $L^1$ norm of their kernel is close to unity. We study in this work such processes for which the stability condition is almost violated. Our main result states that after suitable rescaling, they asymptotically behave like integrated Cox-Ingersoll-Ross models. Thus, modeling financial order flows as nearly unstable Hawkes processes may be a good way to reproduce both their high and low frequency stylized facts. We then extend this result to the Hawkes-based price model introduced by Bacry et al. [Quant. Finance 13 (2013) 65-77]. We show that under a similar criticality condition, this process converges to a Heston model. Again, we recover well-known stylized facts of prices, both at the microstructure level and at the macroscopic scale.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1005 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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“…z n Γ(αn + β) (see, e.g., [40]), then the density φ α C of E α C is linked to this function since φ α 1 (x) = x α−1 E α,α (−x α ). 3.…”

Section: Nondegenerate Scaling Limit For Nearly Unstable Hawkes Procementioning

confidence: 99%

Limit theorems for nearly unstable Hawkes processes

Jaisson¹,

Rosenbaum²

2015

Ann. Appl. Probab.

144

169

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“…[6]). For the convergence of the series in (III.2) see [8] and also [9]. To obtain the Laplace Transform of k cos γ,α (λt) and k sin γ,α (λt) taking into account (II.10) and (II.11) we have…”

Section: Introductionmentioning

confidence: 99%