The paper discusses multivariate self-and cross-exciting processes. We define a class of multivariate point processes via their corresponding stochastic intensity processes that are driven by stochastic jumps. Essentially, there is a jump in an intensity process whenever the corresponding point process records an event. An attribute of our modelling class is that not only a jump is recorded at each instance, but also its magnitude. This allows large jumps to influence the intensity to a larger degree than smaller jumps. We give conditions which guarantee that the process is stable, in the sense that it does not explode, and provide a detailed discussion on when the subclass of linear models is stable. Finally, we fit our model to financial time series data from the S&P 500 and Nikkei 225 indices respectively. We conclude that a nonlinear variant from our modelling class fits the data best. This supports the observation that in times of crises (high intensity) jumps tend to arrive in clusters, whereas there are typically longer times between jumps when the markets are calmer. We moreover observe more variability in jump sizes when the intensity is high, than when it is low.
The present paper discusses simulation of Lévy semistationary (LSS) processes in the context of power markets. A disadvantage of applying numerical integration to obtain trajectories of LSS processes is that such a scheme is not iterative. We address this problem by introducing and analyzing a Fourier simulation scheme for obtaining trajectories of these processes in an iterative manner. Furthermore, we demonstrate that our proposed scheme is well suited for simulation of a wide range of LSS processes, including, in particular, LSS processes indexed by a kernel function which is steep close to the origin. Finally, we put our simulation scheme to work for simulating the price of path-dependent options to demonstrate the advantages of the proposed Fourier simulation scheme.
The purpose of this paper is to investigate properties of self-exciting jump processes where the intensity is given by an SDE, which is driven by a finite activity stochastic jump process. The value of the intensity process immediately before a jump may influence the jump size distribution. We focus on properties of this intensity function, and show that the scaling limit of the intensity process equals the strong solution of the square-root diffusion process (Cox-Ingersoll-Ross process) in distribution. As a particular example, we study the case of a linear intensity process and derive explicit expressions for the expectation and variance in this case. Based on this, we show that once an appropriate scaling limit is taken, the resulting process exhibits infinite activity behaviour in distribution.
We propose a finite difference scheme to simulate solutions to a certain type of hyperbolic stochastic partial differential equation (HSPDE). These solutions can in turn estimate so called volatility modulated Volterra (VMV) processes and Lévy semistationary (LSS) processes, which is a class of processes that have been employed to model turbulence, tumor growth and electricity forward and spot prices. We will see that our finite difference scheme converges to the solution of the HSPDE as we take finer and finer partitions for our finite difference scheme in both time and space. Finally, we demonstrate our method with an example from the energy finance literature.
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