| Abstract: |
| In this talk, we discuss the convergence of the value function of a linear-quadratic stochastic optimal control problem as the ratio between the speed of the two-scale state dynamics diverge. More specifically, we represent the value function in terms of a Riccati representation and show that it converges towards a reduced backwards equation. In the simplified one dimensional case, we show that the reduced backwards equation is the Riccati equation of a classical linear-quadratic stochastic control problem. Connections to ergodic control will also be discussed. This work is inspired by the study of the nonlinear case done by Alvarez - Bardi [1] and Guatteri - Tessitore [2], via HJB and BSDE methods respectively. It should be noted that both papers adopt Lipschitz and boundedness assumptions which do not cover the linear - quadratic case. This is a joint work Beniamin Goldys and Gianmario Tessitore.
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[1] Alvarez, O., \& Bardi, M. (2002). Viscosity solutions methods for singular perturbations in deterministic and stochastic control. SIAM journal on control and optimization, 40(4), 1159-1188.\
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[2] Guatteri, G., \& Tessitore, G. (2018). Singular limit of BSDEs and Optimal control of two scale stochastic systems in infinite dimensional spaces. arXiv preprint arXiv:1803.05908. |
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