Probabilistic Max-Plus Schemes for Solving Hamilton-Jacobi-Bellman Equations

Abstract

We consider fully nonlinear Hamilton-Jacobi-Bellman equations associated to diffusion control problems involving a finite set-valued (or switching) control and possibly a continuum-valued control. In previous works (Akian and Fodjo, A probabilistic max-plus numerical method for solving stochastic control problems. In: 55th Conference on Decision and Control (CDC 2016), Las Vegas, 2016 and From a monotone probabilistic scheme to a probabilistic max-plus algorithm for solving Hamilton-Jacobi-Bellman equations. In: Kalise, D., Kunisch, K., Rao, Z. (eds.) Hamilton-Jacobi-Bellman Equations. Radon Series on Computational and Applied Mathematics, vol. 21. De Gruyter, Berlin 2018), we introduced a lower complexity probabilistic numerical algorithm for such equations by combining max-plus and numerical probabilistic approaches. The max-plus approach is in the spirit of the one of McEneaney et al. (Idempotent method for continuous-time stochastic control and complexity attenuation. In: Proceedings of the 18th IFAC World Congress, 2011, pp 3216–3221. Milano, Italie, 2011), and is based on the distributivity of monotone operators with respect to suprema. The numerical probabilistic approach is in the spirit of the one proposed by Fahim et al. (Ann Appl Probab 21(4):1322–1364, 2011). A difficulty of the latter algorithm was in the critical constraints imposed on the Hamiltonian to ensure the monotonicity of the scheme, hence the convergence of the algorithm. Here, we present new probabilistic schemes which are monotone under rather weak assumptions, and show error estimates for these schemes. These estimates will be used in further works to study the probabilistic max-plus method.