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In this paper we provide an expansion formula for Hawkes processes which involves the addition of jumps at deterministic times to the Hawkes process in the spirit of the well-known integration by parts formula (or more precisely the Mecke formula) for Poisson functional. Our approach allows us to provide an expansion of the premium of a class of cyber insurance derivatives (such as reinsurance contracts including generalized Stop-Loss contracts) or risk management instruments (like Expected Shortfall) in terms of so-called shifted Hawkes processes. From the actuarial point of view, these processes can be seen as "stressed" scenarios. Our expansion formula for Hawkes processes enables us to provide lower and upper bounds on the premium (or the risk evaluation) of such cyber contracts and to quantify the surplus of premium compared to the standard modeling with a homogeneous Poisson process. [ABSTRACT FROM AUTHOR]

In this paper we provide an Itô-Tanaka trick formula in a non semimartingale context, filling a gap in the theory of regularisation by noise. In a classical Brownian framework, the Itô-Tanaka trick links the time average of a function f along the solution to a Brownian SDE, with the solution of a Fokker-Planck PDE. Our main contribution is to provide such a link in a non-semimartingale framework, where the solution to the non-available PDE is replaced by a well-chosen random field. This allows us to improve well-posedness results for fractional SDEs with a singular drift coefficient. [ABSTRACT FROM AUTHOR]

In this paper, following Nourdin-Peccati's methodology, we combine the Malliavin calculus and Stein's method to provide general bounds on the Wasserstein distance between the law of functionals of a compound Hawkes process and the one of a Gaussian random variable. To achieve this, we rely on the Poisson imbedding representation of a Hawkes process to provide a Malliavin calculus for the Hawkes processes, and more generally for compound Hawkes processes. As an application, we close a gap in the literature by providing a quantitative Central Limit Theorem for the compound Hawkes process. [ABSTRACT FROM AUTHOR]