Representation and approximation of ambit fields in Hilbert space

Abstract

We lift ambit fields to a class of Hilbert space-valued volatility modulated Volterra processes. We name this class Hambit fields, and show that they can be expressed as a countable sum of weighted real-valued volatility modulated Volterra processes. Moreover, Hambit fields can be interpreted as the boundary of the mild solution of a certain first order stochastic partial differential equation. This stochastic partial differential equation is formulated on a suitable Hilbert space of functions on the positive real line with values in the state space of the Hambit field. We provide an explicit construction of such a space. Finally, we apply this interpretation of Hambit fields to develop a finite difference scheme, for which we prove convergence under some Lipschitz conditions.

Keywords:

• Ambit fields
• Hilbert space representation
• stochastic partial differential equations
• approximation

Acknowledgements

F. E. Benth acknowledges financial support from the research projects ‘Managing Weather Risk in Energy Markets (MAWREM)’ and ‘Finance, Insurance, Energy, Weather and Stochastics (FINEWSTOCH)’, both funded by the Norwegian Research Council. H. Eyjolfsson acknowledges financial support from Finansmarkedsfondet.

Notes

No potential conflict of interest was reported by the authors.

1 Strongly measurable means that f can be approximated by simple functions, that is, , where and , such that , a.e. for when .

Additional information

Funding

F. E. Benth has financial support from the research projects ‘Managing Weather Risk in Energy Markets (MAWREM)’ and ‘Finance, Insurance, Energy, Weather and Stochastics (FINEWSTOCH)’, both funded by the Norwegian Research Council. H. Eyjolfsson has financial support from Finansmarkedsfondet.