Let O be the basin of attraction of a given equilibrium of a dynamical system, whose solution is the law of large numbers limit of the solution of a Poissonian SDE as the size of the population tends to +∞. We consider the law of the exit point from O of that Poissonian SDE. We adapt the approach of Day [J. Math. Anal. Appl. 147 (1990) 134–153] who studied the same problem for an ODE with a small Brownian perturbation. For that purpose, we will use the large deviations principle for the Poissonian SDE reflected at the boundary of O, studied in our recent work Pardoux and Samegni [Stoch. Anal. Appl. 37 (2019) 836–864]. The main motivation of this work is the extension of the results concerning the time of exit from the set O established in Kratz and Pardoux [Vol. 2215 of Lecture Notes in Math.. Springer (2018) 221–327] and Pardoux and Samegni [J. Appl. Probab. 54 (2017) 905–920] to unbounded open sets O. This is done in sections 4.2.5 and 4.2.7 of Britton and Pardoux [Vol. 2255 of Lecture Notes in Math. Springer (2019) 1–120], see also The SIR model with demography subsection below.
Accepté le :
Première publication :
Publié le :
DOI : 10.1051/ps/2019031
Mots-clés : Poisson process driven SDE, law of large numbers, large deviations principle
@article{PS_2020__24_1_148_0,
author = {Pardoux, Etienne and Samegni-Kepgnou, Brice},
title = {Large deviations of the exit measure through a characteristic boundary for a {Poisson} driven {SDE}},
journal = {ESAIM: Probability and Statistics},
pages = {148--185},
publisher = {EDP-Sciences},
volume = {24},
year = {2020},
doi = {10.1051/ps/2019031},
mrnumber = {4072634},
zbl = {1434.60094},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2019031/}
}
TY - JOUR AU - Pardoux, Etienne AU - Samegni-Kepgnou, Brice TI - Large deviations of the exit measure through a characteristic boundary for a Poisson driven SDE JO - ESAIM: Probability and Statistics PY - 2020 SP - 148 EP - 185 VL - 24 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2019031/ DO - 10.1051/ps/2019031 LA - en ID - PS_2020__24_1_148_0 ER -
%0 Journal Article %A Pardoux, Etienne %A Samegni-Kepgnou, Brice %T Large deviations of the exit measure through a characteristic boundary for a Poisson driven SDE %J ESAIM: Probability and Statistics %D 2020 %P 148-185 %V 24 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2019031/ %R 10.1051/ps/2019031 %G en %F PS_2020__24_1_148_0
Pardoux, Etienne; Samegni-Kepgnou, Brice. Large deviations of the exit measure through a characteristic boundary for a Poisson driven SDE. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 148-185. doi : 10.1051/ps/2019031. http://archive.numdam.org/articles/10.1051/ps/2019031/
[1] and , Branching Processes. Vol. 196 of Grundlehren der mathematischen Wissenschaften. Springer (1972). | MR | Zbl
[2] and , Stochastic epidemics in a homogeneous community. Part I of Stochastic Epidemic Models with Inference, edited by and . Vol. 2255 of Lecture Notes in Math. Springer (2019) 1–120. | Zbl
[3] , Large deviations results for the exit problem with characteristic boundary. J. Math. Anal. Appl. 147 (1990) 134–153. | DOI | MR | Zbl
[4] and , Large deviations techniques and applications, Vol. 38 of Appplications of Mathematics. Springer (2009). | MR | Zbl
[5] and , Petites perturbations de systèmes dynamiques avec réflexion, in Séminaire de Probabilités XVII. Vol. 986 of Lecture Notes in Math. Springer (1983) 353–370. | Numdam | MR | Zbl
[6] and , Markov Processes Characterization and Convergence. J. Wiley & Sons, Inc. (1986). | DOI | MR | Zbl
[7] and , Random perturbations of dynamical systems, Vol. 260 of Grundlehren der mathematischen Wissenschaften. Springer (2012). | MR | Zbl
[8] and , Large deviations for infectious diseases models, Chapter 7 of Séminaire de Probabilités XLIX. Vol. 2215 of Lecture Notes in Math.. Springer (2018) 221–327. | MR | Zbl
[9] and , A simple vaccination model with multiple endemic states. Math. Biosci. 164 (2000) 183–201. | DOI | Zbl
[10] , Strong approximation theorems for density dependent Markov chains. Stoch. Process. Appl. 6 (1978) 223–240. | MR | Zbl
[11] and , Large deviations principle for Poisson driven SDE in epidemic models. J. Appl. Probab. 54 (2017) 905–920. | MR | Zbl
[12] and , Large deviations principle for Reflected Poisson driven SDEs in epidemic models. Stoch. Anal. Appl. 37 (2019) 836–864. | MR | Zbl
[13] , , and , The mathematical theory of optimal processes. Transl. by K. N. Trirogoff. Edited by . John Wiley & Sons (1962). | MR | Zbl
[14] , and , The minimum effort required to eradicate infections in models with backward bifurcation. J. Math. Biol. 53 (2006) 703–718. | MR | Zbl
[15] and , Large Deviations for Performance Analysis. Chapman Hall, London (1995). | MR | Zbl
[16] and , Large deviations with diminishing rates. Math. Oper. Res. 30 (2005) 281–310. | MR | Zbl
[17] , Contrôle optimal: théorie & applications. Vuibert (2008). | MR | Zbl
Cité par Sources :
