2015/11/20 by Julien Claisse, Claisse, Julien · 3 citations
Economics, Econometrics and Finance · Mathematics · #49L25 #60J60 #60J70 #60J80 #60J85 #FOS: Mathematics #Mathematical Biology Tumor Growth #Optimization and Control (math.OC) #Primary 93E20 #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #secondary 49L20
paper · pdf · doi:10.48550/arxiv.1511.06809
openalex publication_date 2015/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we aim to develop the theory of optimal stochastic control for\nbranching diffusion processes where both the movement and the reproduction of\nthe particles depend on the control. More precisely, we study the problem of\nminimizing a criterion that is expressed as the expected value of the product\nof individual costs penalizing the final position of each particle. In this\nsetting, we show that the value function is the unique viscosity solution of a\nnonlinear parabolic PDE, that is, the Hamilton-Jacobi-Bellman equation\ncorresponding to the problem. To this end, we extend the dynamic programming\napproach initiated by Nisio to deal with the lack of independence between the\nparticles as well as between the reproduction and the movement of each\nparticle. In particular, we exploit the particular form of the optimization\ncriterion to recover a weak form of the branching property. In addition, we\nprovide a precise formulation and a detailed justification of the adequate\ndynamic programming principle.\n