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Evolving Systems of Stochastic Differential Equations

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Abstract

We introduce Evolving Systems of Stochastic Differential Equations. This model generalizes the well-known stochastic differential equations with Markovian switching, enabling the countably many local systems to have solutions in regime-dependent dimension. We provide two constructions, the first one based upon general results on measure-valued processes and the second one partially inspired by recent developments of the theory of concatenation of right processes. We prove the Feller property under very mild assumptions, provide some extensions to the basic model, and show applications of our general framework to a biological model.

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Code availability

The computer codes associated with the simulations of Sect. 6 are available at https://github.com/leonardo-videla/ESDE-Codes.git.

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Acknowledgements

We would like to thank Pablo Marquet for providing us with valuable bibliographical references regarding biological models on growing domains, and for his promptness to clarify our (too often) blurred understanding of many biological phenomena. Part of the results contained in this work were developed during a research stay of the first author at Institute Élie Cartan de Lorraine, in Nancy, France, and L.V. thanks Antoine Lejay and Nicolas Champagnat for their generous conversation and for providing us with important insights regarding this ongoing research. In particular, we would like to thank Nicolas Champagnat for suggesting the paper [2] as an appropriate particular example that fits the general model exposed in this paper.

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  1. Facultad de Ingeniería, Instituto de Ingeniería Matemática, Universidad de Valparaíso, General Cruz 222, 5th floor, Valparaiso, Chile

    Leonardo Videla & Rolando Rebolledo

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  1. Leonardo Videla
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  2. Rolando Rebolledo
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Correspondence to Leonardo Videla.

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Both authors declare that they have had no financial or personal relationship with other people or organizations that could inappropriately influence or bias the content of this paper. Also, both authors declare that they have no conflict of interest.

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Leonardo Videla has been supported by CONICYT through Beca de Doctorado 21170406, Convocatoria 2017; Rolando Rebolledo has been supported by Proyecto REDES CONICYT 180018 “Biostochastic Research Network” and CIMFAV-CIDI 2019.

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Videla, L., Rebolledo, R. Evolving Systems of Stochastic Differential Equations. J Theor Probab 35, 1662–1705 (2022). https://doi.org/10.1007/s10959-021-01098-1

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