Search

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]

Abstract: We extend the work of Delong and Imkeller (2010)  concerning backward stochastic differential equations with time delayed generators (delay BSDEs). We give moment and a priori estimates in general -spaces and provide sufficient conditions for the solution of a delay BSDE to exist in . We introduce decoupled systems of SDEs and delay BSDEs (delay FBSDEs) and give sufficient conditions for their variational differentiability. We connect these variational derivatives to the Malliavin derivatives of delay FBSDEs via the usual representation formulas. We conclude with several path regularity results, in particular we extend the classic -path regularity to delay FBSDEs. [Copyright &y& Elsevier]

Abstract: In this paper we give a central limit theorem for the weighted quadratic variation process of a two-parameter Brownian motion. As an application, we show that the discretized quadratic variations of a two-parameter diffusion observed on a regular grid form an asymptotically normal estimator of the quadratic variation of as goes to infinity. [Copyright &y& Elsevier]

Abstract: In this paper we deal with the utility maximization problem with general utility functions including power utility with liability. We derive a new approach in which we reduce the resulting control problem to the study of a system of fully-coupled Forward–Backward Stochastic Differential Equations (FBSDEs) that promise to be accessible to numerical treatment. [Copyright &y& Elsevier]