arXiv subjects

Benoîte de Saporta

Publications and source records attributed to Benoîte de Saporta.

27 records · Page 2Linked to original sources

arXiv2013

Optimal stopping for partially observed piecewise-deterministic Markov processes

This paper deals with the optimal stopping problem under partial observation for piecewise-deterministic Markov processes. We first obtain a recursive formulation of the optimal filter process and derive the dynamic programming equation of the partially observed optimal stopping problem. Then, we propose a numerical method, based on the quantization of the discrete-time filter process and the inter-jump times, to approximate the value function and to compute an actual $ε$-optimal stopping time. We prove the convergence of the algorithms and bound the rates of convergence.

math.PR↗

arXiv2013

Predictive maintenance for the heated hold-up tank

We present a numerical method to compute an optimal maintenance date for the test case of the heated hold-up tank. The system consists of a tank containing a fluid whose level is controlled by three components: two inlet pumps and one outlet valve. A thermal power source heats up the fluid. The failure rates of the components depends on the temperature, the position of the three components monitors the liquid level in the tank and the liquid level determines the temperature. Therefore, this system can be modeled by a hybrid process where the discrete (components) and continuous (level, temperature) parts interact in a closed loop. We model the system by a piecewise deterministic Markov process, propose and implement a numerical method to compute the optimal maintenance date to repair the components before the total failure of the system.

math.OC↗

arXiv2012

Numerical method for expectations of piecewise-determistic Markov processes

We present a numerical method to compute expectations of functionals of a piecewise-deterministic Markov process. We discuss time dependent functionals as well as deterministic time horizon problems. Our approach is based on the quantization of an underlying discrete-time Markov chain. We obtain bounds for the rate of convergence of the algorithm. The approximation we propose is easily computable and is flexible with respect to some of the parameters defining the problem. Two examples illustrate the paper.

math.PR↗

arXiv2011

Asymmetry tests for Bifurcating Auto-Regressive Processes with missing data

We present symmetry tests for bifurcating autoregressive processes (BAR) when some data are missing. BAR processes typically model cell division data. Each cell can be of one of two types \emph{odd} or \emph{even}. The goal of this paper is to study the possible asymmetry between odd and even cells in a single observed lineage. We first derive asymmetry tests for the lineage itself, modeled by a two-type Galton-Watson process, and then derive tests for the observed BAR process. We present applications on both simulated and real data.

math.ST↗

arXiv2011

Parameters estimation for asymmetric bifurcating autoregressive processes with missing data

We estimate the unknown parameters of an asymmetric bifurcating autoregressive process (BAR) when some of the data are missing. In this aim, we model the observed data by a two-type Galton-Watson process consistent with the binary tree structure of the data. Under independence between the process leading to the missing data and the BAR process and suitable assumptions on the driven noise, we establish the strong consistency of our estimators on the set of non-extinction of the Galton-Watson, via a martingale approach. We also prove a quadratic strong law and the asymptotic normality.

math.PR↗

arXiv2011

Numerical method for impulse control of Piecewise Deterministic Markov Processes

This paper presents a numerical method to calculate the value function for a general discounted impulse control problem for piecewise deterministic Markov processes. Our approach is based on a quantization technique for the underlying Markov chain defined by the post jump location and inter-arrival time. Convergence results are obtained and more importantly we are able to give a convergence rate of the algorithm. The paper is illustrated by a numerical example.

math.PR↗

arXiv2011

Numerical methods for the exit time of a piecewise-deterministic Markov process

We present a numerical method to compute the survival function and the moments of the exit time for a piecewise-deterministic Markov process (PDMP). Our approach is based on the quantization of an underlying discrete-time Markov chain related to the PDMP. The approximation we propose is easily computable and is even flexible with respect to the exit time we consider. We prove the convergence of the algorithm and obtain bounds for the rate of convergence in the case of the moments. An academic example and a model from the reliability field illustrate the paper.

math.PR↗

arXiv2011

Optimal stopping for the predictive maintenance of a structure subject to corrosion

We present a numerical method to compute the optimal maintenance time for a complex dynamic system applied to an example of maintenance of a metallic structure subject to corrosion. An arbitrarily early intervention may be uselessly costly, but a late one may lead to a partial/complete failure of the system, which has to be avoided. One must therefore find a balance between these too simple maintenance policies. To achieve this aim, we model the system by a stochastic hybrid process. The maintenance problem thus corresponds to an optimal stopping problem. We propose a numerical method to solve the optimal stopping problem and optimize the maintenance time for this kind of processes.

math.PR↗

arXiv2010

Numerical method for optimal stopping of piecewise deterministic Markov processes

We propose a numerical method to approximate the value function for the optimal stopping problem of a piecewise deterministic Markov process (PDMP). Our approach is based on quantization of the post jump location---inter-arrival time Markov chain naturally embedded in the PDMP, and path-adapted time discretization grids. It allows us to derive bounds for the convergence rate of the algorithm and to provide a computable $ε$-optimal stopping time. The paper is illustrated by a numerical example.

math.PR↗