2015/11/23 by Vincent Bansaye, Bansaye, Vincent · 1 citation
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1511.07396
openalex publication_date 2015/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We approximate stochastic processes in finite dimension by dynamical systems.\nWe provide trajectorial estimates which are uniform with respect to the initial\ncondition for a well chosen distance. This relies on some non-expansivity\nproperty of the flow, which allows to deal with non-Lipschitz vector fields. We\nuse the stochastic calculus and follow the martingale technics initiated in\nBerestycki and al [5] to control the fluctuations. Our main applications deal\nwith the short time behavior of stochastic processes starting from large\ninitial values. We state general properties on the coming down from infinity of\none-dimensional SDEs, with a focus on stochastically monotone processes. In\nparticular, we recover and complement known results on Lambda-coalescent and\nbirth and death processes. Moreover, using Poincar 'e's compactification\ntechnicsfor dynamical systems close to infinity, we develop this approach in\ntwo dimensions for competitive stochastic models. We classify the coming down\nfrom infinity of Lotka-Volterra diffusions and provide uniform estimates for\nthe scaling limits of competitive birth and death processes.\n