Variational inequalities for the optimal stopping with continuous control of piecewise deterministic Markov processes

1999 ◽  
Keyword(s):  
Variational Inequalities ◽  
Markov Processes ◽  
Optimal Stopping ◽  
Continuous Control ◽  
Piecewise Deterministic Markov Processes

scholarly journals Optimal stopping for measure-valued piecewise deterministic Markov processes

2020 ◽  
Vol 57 (2) ◽  
pp. 497-512
Author(s):  
Bertrand Cloez ◽  
Benoîte de Saporta ◽  
Maud Joubaud
Keyword(s):  
Markov Processes ◽  
Optimal Stopping ◽  
Small Population ◽  
Individual Characteristics ◽  
Locally Finite ◽  
Counter Example ◽  
Random Horizon ◽  
Piecewise Deterministic Markov Processes ◽  
Point Measure ◽  
Finite Measures

AbstractThis paper investigates the random horizon optimal stopping problem for measure-valued piecewise deterministic Markov processes (PDMPs). This is motivated by population dynamics applications, when one wants to monitor some characteristics of the individuals in a small population. The population and its individual characteristics can be represented by a point measure. We first define a PDMP on a space of locally finite measures. Then we define a sequence of random horizon optimal stopping problems for such processes. We prove that the value function of the problems can be obtained by iterating some dynamic programming operator. Finally we prove via a simple counter-example that controlling the whole population is not equivalent to controlling a random lineage.

scholarly journals The Vanishing Discount Approach for the Average Continuous Control of Piecewise Deterministic Markov Processes

2009 ◽  
Vol 46 (04) ◽  
pp. 1157-1183 ◽  
Author(s):  
O. L. V. Costa ◽  
F. Dufour
Keyword(s):  
Optimal Control ◽  
Markov Processes ◽  
Sufficient Conditions ◽  
Relative Difference ◽  
Continuous Control ◽  
Value Functions ◽  
Optimal Control Strategy ◽  
Long Run ◽  
Piecewise Deterministic Markov Processes ◽  
Vanishing Discount Approach

This work is concerned with the existence of an optimal control strategy for the long-run average continuous control problem of piecewise-deterministic Markov processes (PDMPs). In Costa and Dufour (2008), sufficient conditions were derived to ensure the existence of an optimal control by using the vanishing discount approach. These conditions were mainly expressed in terms of the relative difference of the α-discount value functions. The main goal of this paper is to derive tractable conditions directly related to the primitive data of the PDMP to ensure the existence of an optimal control. The present work can be seen as a continuation of the results derived in Costa and Dufour (2008). Our main assumptions are written in terms of some integro-differential inequalities related to the so-called expected growth condition, and geometric convergence of the post-jump location kernel associated to the PDMP. An example based on the capacity expansion problem is presented, illustrating the possible applications of the results developed in the paper.

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