Representation and Simulation of Ambit Fields
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In this chapter we present methods for simulating ambit fields based on extending the Fourier approach presented in the temporal case in Chap. 2. In principle the approach is analogous to the simulation of volatility modulated Volterra processes, however, becoming much more technical due to the additional spatial dimension. In this technical chapter, we provide a full-blown theory for simulation using Fourier expansion with proofs. The Fourier method is based on a particular series expansion of the ambit field along a set of basis functions. We expand on this idea and view ambit fields as stochastic processes in a separable Hilbert space, where we establish a series representation of the fields as a countable sum of volatility modulated Volterra processes scaled by basis functions.
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