## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 28, Number 1 (2018), 204-249.

### Diffusion approximations for controlled weakly interacting large finite state systems with simultaneous jumps

Amarjit Budhiraja and Eric Friedlander

#### Abstract

We consider a rate control problem for an $N$-particle weakly interacting finite state Markov process. The process models the state evolution of a large collection of particles and allows for multiple particles to change state simultaneously. Such models have been proposed for large communication systems (e.g., ad hoc wireless networks) but are also suitable for other settings such as chemical-reaction networks. An associated diffusion control problem is presented and we show that the value function of the $N$-particle controlled system converges to the value function of the limit diffusion control problem as $N\to\infty$. The diffusion coefficient in the limit model is typically degenerate; however, under suitable conditions there is an equivalent formulation in terms of a controlled diffusion with a uniformly nondegenerate diffusion coefficient. Using this equivalence, we show that near optimal continuous feedback controls exist for the diffusion control problem. We then construct near asymptotically optimal control policies for the $N$-particle system based on such continuous feedback controls. Results from some numerical experiments are presented.

#### Article information

**Source**

Ann. Appl. Probab., Volume 28, Number 1 (2018), 204-249.

**Dates**

Received: March 2016

First available in Project Euclid: 3 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1520046087

**Digital Object Identifier**

doi:10.1214/17-AAP1303

**Mathematical Reviews number (MathSciNet)**

MR3770876

**Zentralblatt MATH identifier**

06873683

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60H30: Applications of stochastic analysis (to PDE, etc.) 93E20: Optimal stochastic control

Secondary: 60J28: Applications of continuous-time Markov processes on discrete state spaces 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60K25: Queueing theory [See also 68M20, 90B22] 91B70: Stochastic models

**Keywords**

Mean field approximations diffusion approximations asymptotic optimality stochastic networks stochastic control propagation of chaos rate control ad hoc wireless networks

#### Citation

Budhiraja, Amarjit; Friedlander, Eric. Diffusion approximations for controlled weakly interacting large finite state systems with simultaneous jumps. Ann. Appl. Probab. 28 (2018), no. 1, 204--249. doi:10.1214/17-AAP1303. https://projecteuclid.org/euclid.aoap/1520046087