## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 28, Number 1 (2018), 1-34.

### Optimal control of branching diffusion processes: A finite horizon problem

#### Abstract

In this paper, we aim to develop the stochastic control theory of branching diffusion processes where both the movement and the reproduction of the particles depend on the control. More precisely, we study the problem of minimizing the expected value of the product of individual costs penalizing the final position of each particle. In this setting, we show that the value function is the unique viscosity solution of a nonlinear parabolic PDE, that is, the Hamilton–Jacobi–Bellman equation corresponding to the problem. To this end, we extend the dynamic programming approach initiated by Nisio [*J. Math. Kyoto Univ.* **25** (1985) 549–575] to deal with the lack of independence between the particles as well as between the reproduction and the movement of each particle. In particular, we exploit the particular form of the optimization criterion to derive a weak form of the branching property. In addition, we provide a precise formulation and a detailed justification of the adequate dynamic programming principle.

#### Article information

**Source**

Ann. Appl. Probab., Volume 28, Number 1 (2018), 1-34.

**Dates**

Received: September 2016

Revised: December 2016

First available in Project Euclid: 3 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1520046082

**Digital Object Identifier**

doi:10.1214/17-AAP1290

**Mathematical Reviews number (MathSciNet)**

MR3770871

**Zentralblatt MATH identifier**

06873678

**Subjects**

Primary: 93E20: Optimal stochastic control 60J60: Diffusion processes [See also 58J65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Secondary: 49L20: Dynamic programming method 49L25: Viscosity solutions 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60J85: Applications of branching processes [See also 92Dxx]

**Keywords**

Stochastic control branching diffusion process dynamic programming principle Hamilton–Jacobi–Bellman equation viscosity solution

#### Citation

Claisse, Julien. Optimal control of branching diffusion processes: A finite horizon problem. Ann. Appl. Probab. 28 (2018), no. 1, 1--34. doi:10.1214/17-AAP1290. https://projecteuclid.org/euclid.aoap/1520046082