Annals of Applied Probability

Approximation of stochastic processes by nonexpansive flows and coming down from infinity

Vincent Bansaye

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Abstract

This paper deals with the approximation of semimartingales in finite dimension by dynamical systems. We give trajectorial estimates uniform with respect to the initial condition for a well-chosen distance. This relies on a nonexpansivity property of the flow and allows to consider non-Lipschitz vector fields. The fluctuations of the process are controlled using the martingale technics and stochastic calculus.

Our main motivation is the trajectorial description of stochastic processes starting from large initial values. We state general properties on the coming down from infinity of one-dimensional SDEs, with a focus on stochastically monotone processes. In particular, we recover and complement known results on $\Lambda $-coalescent and birth and death processes. Moreover, using Poincaré’s compactification techniques for flows close to infinity, we develop this approach in two dimensions for competitive stochastic models. We thus classify the coming down from infinity of Lotka–Volterra diffusions and provide uniform estimates for the scaling limits of competitive birth and death processes.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 4 (2019), 2374-2438.

Dates
Received: January 2017
Revised: September 2018
First available in Project Euclid: 23 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1563869046

Digital Object Identifier
doi:10.1214/18-AAP1456

Mathematical Reviews number (MathSciNet)
MR3984256

Zentralblatt MATH identifier
07120712

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60J75: Jump processes 60G46: Martingales and classical analysis 60F15: Strong theorems

Keywords
Approximation of stochastic processes nonexpansivity dynamical system coming down from infinity martingales scaling limits

Citation

Bansaye, Vincent. Approximation of stochastic processes by nonexpansive flows and coming down from infinity. Ann. Appl. Probab. 29 (2019), no. 4, 2374--2438. doi:10.1214/18-AAP1456. https://projecteuclid.org/euclid.aoap/1563869046


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References

  • [1] Aminzare, Z. and Sontag, E. D. (2014). Contraction Methods for Nonlinear Systems: A Brief Introduction and Some Open Problems. 53rd IEEE Conference on Decision and Control December 15-17, Los Angeles, CA.
  • [2] Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer Series in Statistics: Probability and Its Applications. Springer, New York.
  • [3] Azaïs, R., Bardet, J.-B., Génadot, A., Krell, N. and Zitt, P.-A. (2014). ESAIM: Proceedings, Vol. 44, p. 276-290 SMAI Groupe MAS—Journées MAS 2012—Session thématique.
  • [4] Bansaye, V. and Méléard, S. (2015). Stochastic Models for Structured Populations. Mathematical Biosciences Institute Lecture Series 1. Springer, Cham.
  • [5] Bansaye, V., Méléard, S. and Richard, M. (2016). Speed of coming down from infinity for birth-and-death processes. Adv. in Appl. Probab. 48 1183–1210.
  • [6] Berestycki, J., Berestycki, N. and Limic, V. (2010). The $\Lambda$-coalescent speed of coming down from infinity. Ann. Probab. 38 207–233.
  • [7] Bertoin, J. and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 147–181.
  • [8] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York.
  • [9] Cattiaux, P., Collet, P., Lambert, A., Martínez, S., Méléard, S. and San Martín, J. (2009). Quasi-stationary distributions and diffusion models in population dynamics. Ann. Probab. 37 1926–1969.
  • [10] Cattiaux, P. and Méléard, S. (2010). Competitive or weak cooperative stochastic Lotka–Volterra systems conditioned on non-extinction. J. Math. Biol. 60 797–829.
  • [11] Champagnat, N. and Villemonais, D. (2016). Exponential convergence to quasi-stationary distribution and $Q$-process. Probab. Theory Related Fields 164 243–283.
  • [12] Darling, R. W. R. and Norris, J. R. (2008). Differential equation approximations for Markov chains. Probab. Surv. 5 37–79.
  • [13] Dawson, D. A. and Li, Z. (2012). Stochastic equations, flows and measure-valued processes. Ann. Probab. 40 813–857.
  • [14] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential. B. Theory of Martingales. North-Holland Mathematics Studies 72. North-Holland, Amsterdam.
  • [15] Donnelly, P. (1991). Weak convergence to a Markov chain with an entrance boundary: Ancestral processes in population genetics. Ann. Probab. 19 1102–1117.
  • [16] Dumortier, F., Llibre, J. and Artés, J. C. (2006). Qualitative Theory of Planar Differential Systems. Universitext. Springer, Berlin.
  • [17] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York.
  • [18] Freidlin, M. I. and Wentzell, A. D. (2012). Random Perturbations of Dynamical Systems, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 260. Springer, Heidelberg.
  • [19] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24. North-Holland, Amsterdam.
  • [20] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [21] Karlin, S. and McGregor, J. L. (1957). The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85 489–546.
  • [22] Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd ed. Academic Press, New York.
  • [23] Kurtz, T. G. (1981). Approximation of Population Processes. CBMS-NSF Regional Conference Series in Applied Mathematics 36. SIAM, Philadelphia, PA.
  • [24] Le, V. and Pardoux, E. (2015). Height and the total mass of the forest of genealogical trees of a large population with general competition. ESAIM Probab. Stat. 19 172–193.
  • [25] Limic, V. (2010). On the speed of coming down from infinity for $\Xi$-coalescent processes. Electron. J. Probab. 15 217–240.
  • [26] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.
  • [27] Schweinsberg, J. (2000). A necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity. Electron. Commun. Probab. 5 1–11.
  • [28] Skorokhod, A. V., Hoppensteadt, F. C. and Salehi, H. (2002). Random Perturbation Methods with Applications in Science and Engineering. Applied Mathematical Sciences 150. Springer, New York.
  • [29] van Doorn, E. A. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. in Appl. Probab. 23 683–700.