Singular limit of BSDES and optimal control of two scale systems with jumps in infinite dimensional spaces

¹ University of Bologna, Department of Mathematics, piazza di Porta San Donato 5, 40126 Bologna, Italy
² Politecnico di Milano, Department of Mathematics, via Bonardi 9, 20133 Milan, Italy
³ University of Milano-Bicocca, Department of Mathematics and its Applications, via Roberto Cozzi 55, 20125 Milan, Italy

^* Corresponding author: elena.bandini7@unibo.it

Received: 15 January 2024
Accepted: 3 March 2025

Abstract

The paper is devoted to a stochastic optimal control problem for a two scale, infinite dimensional, stochastic system. The state of the system consists of “slow” and “fast” component and its evolution is driven by both continuous Wiener noises and discontinuous Poisson-type noises. The presence of discontinuous noises is the main feature of the present work. We use the theory of backward stochastic differential equations (BSDEs) to prove that, as the speed of the fast component diverges, the value function of the control problem converges to the solution of a reduced forward-backward system that, in turn, is related to a reduced stochastic optimal control problem. The results of this paper generalize to the case of discontinuous noise the ones in [Guatteri and Tessitore, Appl. Math. Optim. 83 (2021) 1025–1051] and [Świ¸ech, ESAIM Control Optim. Calc. Var. 27 (2021) Paper No. 6, 34].