Abstract
This chapter focusses on stochastic simulation of volatility modulated Volterra processes. First we suggest a method based on the Fourier (or Laplace) transform, where we express the kernel function in the volatility modulated Volterra process along a basis so that the simulation task essentially becomes a summation of a weighted series of complex-valued Ornstein-Uhlenbeck processes. In some cases we can use the Laplace transform instead, providing simpler schemes. We are able to provide convergence results for the methods suggested. An alternative approach is to view the volatility modulated Volterra process as the boundary solution of a certain stochastic partial differential equation. We do this, and develop a finite difference scheme to solve for the solution of this stochastic partial differential equation, where we can read off the simulated path of the volatility modulated Volterra process. Also for this scheme we can develop convergence results.
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Notes
- 1.
Note that there is no loss in generality to assume σ ≡ 1 when σ is a constant, as we can always redefine the Lévy process L by scaling it with σ.
- 2.
We are grateful to Heidar Eyjolfsson for producing this graph.
- 3.
We are grateful to Heidar Eyjolfsson for creating these graphs.
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Barndorff-Nielsen, O.E., Benth, F.E., Veraart, A.E.D. (2018). Simulation. In: Ambit Stochastics. Probability Theory and Stochastic Modelling, vol 88. Springer, Cham. https://doi.org/10.1007/978-3-319-94129-5_2
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DOI: https://doi.org/10.1007/978-3-319-94129-5_2
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