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Statistical and numerical analysis of jump and diffusion models in biology

Abstract : In the first part of this thesis we are studying the link between individual-based stochastic models (birth-and-death processes, Hawkes processes) and their respective continuous approximations (stochastic diffusions, partial differential equations), obtained at a larger scale. In particular, we are tackling the question of the numerical simulation of stochastic and deterministic processes with the help of splitting and implicit numerical schemes, which preserve the asymptotic behavior of the process. On the applied level, we consider mathematical models of interacting networks of biological neurons, as well as bacteria populations. In the second part of the manuscript we are dealing with statistics for stochastic differential equations, such as parametric inference for diffusions with degenerate noise, and hypothesis testing for the covariance matrix rank from discrete observations. For the parametric estimation we use quasi-maximum likelihood estimators (also known as contrast estimators), where the contrast is built on the density approximated with the local linearization scheme. For the second problem, we study a non-asymptotic regime (i.e., the case when the observations are available with a fixed time step). We consider the case when the distribution of the test statistics can be written explicitly (for example, when the drift is known and the dimension is 1 or 2). Then, we use concentration inequalities to derive the tail properties and use them to obtain a non-asymptotic control of the Type I and Type II errors.
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Contributor : Anna Melnykova Connect in order to contact the contributor
Submitted on : Thursday, February 25, 2021 - 12:16:17 PM
Last modification on : Wednesday, November 3, 2021 - 8:23:37 AM
Long-term archiving on: : Wednesday, May 26, 2021 - 6:32:46 PM


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  • HAL Id : tel-03152122, version 1


Anna Melnykova. Statistical and numerical analysis of jump and diffusion models in biology. Probability [math.PR]. Cergy Paris Université, 2020. English. ⟨tel-03152122⟩



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