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Generalised shot-noise representations of stochastic systems driven by non-Gaussian Lévy processes

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  21 March 2024

Simon Godsill ,
Ioannis Kontoyiannis  and
Marcos Tapia Costa Open the ORCID record for Marcos Tapia Costa [Opens in a new window]
Simon Godsill*
Affiliation:
University of Cambridge
Ioannis Kontoyiannis*
Affiliation:
University of Cambridge
Marcos Tapia Costa*
Affiliation:
University of Cambridge
*
*Postal address: Floor 9, 16–18 Prince’s Gardens, London SW7 1NE, UK.
*Postal address: Floor 9, 16–18 Prince’s Gardens, London SW7 1NE, UK.
*Postal address: Floor 9, 16–18 Prince’s Gardens, London SW7 1NE, UK.
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Abstract

We consider the problem of obtaining effective representations for the solutions of linear, vector-valued stochastic differential equations (SDEs) driven by non-Gaussian pure-jump Lévy processes, and we show how such representations lead to efficient simulation methods. The processes considered constitute a broad class of models that find application across the physical and biological sciences, mathematics, finance, and engineering. Motivated by important relevant problems in statistical inference, we derive new, generalised shot-noise simulation methods whenever a normal variance-mean (NVM) mixture representation exists for the driving Lévy process, including the generalised hyperbolic, normal-gamma, and normal tempered stable cases. Simple, explicit conditions are identified for the convergence of the residual of a truncated shot-noise representation to a Brownian motion in the case of the pure Lévy process, and to a Brownian-driven SDE in the case of the Lévy-driven SDE. These results provide Gaussian approximations to the small jumps of the process under the NVM representation. The resulting representations are of particular importance in state inference and parameter estimation for Lévy-driven SDE models, since the resulting conditionally Gaussian structures can be readily incorporated into latent variable inference methods such as Markov chain Monte Carlo, expectation-maximisation, and sequential Monte Carlo.

Keywords

Lévy processessmall jump approximationLévy simulationMonte CarloLévy state space model

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60G35: Signal detection and filtering 62M20: Prediction
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 36
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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