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

Elena Bandini, Giuseppina Guatteri, Gianmario Tessitore

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.

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

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.
Paper Structure (12 sections, 14 theorems, 163 equations)

This paper contains 12 sections, 14 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. The forward system
  4. Well-posedness results
  5. Limit equation and convergence of singular BSDEs
  6. The ergodic parametrized BSDE
  7. Main Convergence Result
  8. The two scale control problem
  9. Formulation of the problem
  10. Solution of the problem: the BSDE approach
  11. Control interpretation of the limit forward-backward system
  12. Girsanov Theorems

Key Result

Lemma 3.1

Let assumptions (HAB), (HRG), (HF), (HF+B), and (H$\beta^B$) hold. For any $T < \infty$, $\tau \in [0,\,T]$ and any $\mathcal{F}_\tau$-measurable square integrable random variable $\tilde{X}_\tau \in H$, has a unique (up to modification) mild solution $(X_t^{\tau, \tilde{X}_\tau})_{t \in [\tau, T]}$ with a càdlàg version. More precisely, there exists a unique process $(X_t)_{t \in [\tau, T]} \in

Theorems & Definitions (19)

  • Remark 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 4.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • ...and 9 more