Singular Limit of BSDEs and Optimal Control of Two Scale Stochastic Systems in Infinite Dimensional Spaces
In this paper we study by probabilistic techniques the convergence of the value function for a two-scale, infinite-dimensional, stochastic controlled system as the ratio between the two evolution speeds diverges. The value function is represented as the solution of a backward stochastic differential...
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| Published in: | Applied mathematics & optimization Vol. 83; no. 2; pp. 1025 - 1051 |
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| Format: | Journal Article |
| Language: | English |
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01.04.2021
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| ISSN: | 0095-4616, 1432-0606 |
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| Abstract | In this paper we study by probabilistic techniques the convergence of the value function for a two-scale, infinite-dimensional, stochastic controlled system as the ratio between the two evolution speeds diverges. The value function is represented as the solution of a
backward stochastic differential equation
(BSDE) that it is shown to converge towards a
reduced
BSDE. The noise is assumed to be additive both in the slow and the fast equations for the state. Some non degeneracy condition on the slow equation is required. The limit BSDE involves the solution of an
ergodic
BSDE and is itself interpreted as the value function of an auxiliary stochastic control problem on a reduced state space. |
|---|---|
| AbstractList | In this paper we study by probabilistic techniques the convergence of the value function for a two-scale, infinite-dimensional, stochastic controlled system as the ratio between the two evolution speeds diverges. The value function is represented as the solution of a
backward stochastic differential equation
(BSDE) that it is shown to converge towards a
reduced
BSDE. The noise is assumed to be additive both in the slow and the fast equations for the state. Some non degeneracy condition on the slow equation is required. The limit BSDE involves the solution of an
ergodic
BSDE and is itself interpreted as the value function of an auxiliary stochastic control problem on a reduced state space. In this paper we study by probabilistic techniques the convergence of the value function for a two-scale, infinite-dimensional, stochastic controlled system as the ratio between the two evolution speeds diverges. The value function is represented as the solution of a backward stochastic differential equation (BSDE) that it is shown to converge towards a reduced BSDE. The noise is assumed to be additive both in the slow and the fast equations for the state. Some non degeneracy condition on the slow equation is required. The limit BSDE involves the solution of an ergodic BSDE and is itself interpreted as the value function of an auxiliary stochastic control problem on a reduced state space. |
| Author | Guatteri, Giuseppina Tessitore, Gianmario |
| Author_xml | – sequence: 1 givenname: Giuseppina orcidid: 0000-0002-9116-6270 surname: Guatteri fullname: Guatteri, Giuseppina organization: Dipartimento di Matematica, Politecnico di Milano – sequence: 2 givenname: Gianmario orcidid: 0000-0001-9893-3703 surname: Tessitore fullname: Tessitore, Gianmario email: gianmario.tessitore@unimib.it organization: Dipartimento di Matematica e Applicazioni, Università Milano-Bicocca |
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| CitedBy_id | crossref_primary_10_1016_j_spa_2021_07_006 crossref_primary_10_1137_21M1408488 crossref_primary_10_1007_s00245_023_10018_0 |
| Cites_doi | 10.1137/S0363012900366741 10.1137/140976091 10.1007/978-1-4612-4482-0 10.1111/1467-9965.00022 10.1007/978-3-540-34514-5 10.1080/104S1120290024856 10.1007/978-3-662-13242-5 10.1137/07069849X 10.1007/978-3-319-53067-3 10.1007/BF01211749 10.1007/s00245-004-0814-x 10.1214/aop/1029867132 10.1016/j.spa.2010.11.009 10.1017/CBO9780511662829 10.1007/BFb0007334 |
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| DOI | 10.1007/s00245-019-09577-y |
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| SubjectTerms | Calculus of Variations and Optimal Control; Optimization Control Convergence Differential equations Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical and Computational Physics Optimal control Probability Simulation Stochastic processes Stochastic systems Systems Theory Theoretical |
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| Title | Singular Limit of BSDEs and Optimal Control of Two Scale Stochastic Systems in Infinite Dimensional Spaces |
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