Ergodic control of infinite-dimensional stochastic differential equations with degenerate noise

Abstract

The present paper is devoted to the study of the asymptotic behavior of the value functions of both finite and infinite horizon stochastic control problems and to the investigation of their relation with suitable stochastic ergodic control problems. Our methodology is based only on probabilistic techniques, as for instance the so-called randomization of the control method, thus avoiding completely analytical tools from the theory of viscosity solutions. We are then able to treat with the case where the state process takes values in a general (possibly infinite dimensional) real separable Hilbert space and the diffusion coefficient is allowed to be degenerate.

Singular Limit of Two Scale Stochastic Optimal Control Problems in Infinite Dimensions by Vanishing Noise Regularization

“…The key issue, at this level, is to construct this new control problem in such a way that its running cost has enough regularity to allow the final passage to the limit as the small noise regularization vanishes and, eventually, to get the main result of this paper (see Theorem 6.3). In addition the value function of our reduced control problem can be shown to coincide with the minimal solution of a Backward Stochastic Differential Equation with constraints on the martingale term (see Remark 6.5 here and [13], [6]). The paper is organized as follows: in Section 2 we introduce general notation, in Section 3 we formulate the control problems introducing the weak formulation that will be used throughout the paper, in Section 4 we introduce the small noise regularization of the system, in Section 5 we prove that we can change order between the limit with respect to the speed ratio parameter ε and the small noise parameter η, finally in Section 6 we prove our mail result.…”

“…Remark 6.5 Taking advantage of our main result (see Theorem 6.3) we can further represent the singular limit V (x 0 ) as the value at time 0 of the minimal solution of a BSDE with constrints on the martingale term (see [13] for the definition for and [6] for the infinite dimensional case). The bridge is given by the results in [6] allowing to represent the value function of a control problem by such a constrained BSDE without using viscosity solutions of the related HJB equation.…”

“…First of all, to fit the framework in [6] we notice that the control problem introduced in Theorem 6.3 can be rewritten considering an unbounded set of controls. This is readily (and rather obviously) done by introduciong the class S 1 H of 6-uples U H = (Ω, F, (F t ), P, (W 1 t ), (u t )) identical to the ones in S 1 H with the only difference that here u are not required to be bounded by M + 1 and noticing that:…”

“…Then we just have to remind the construction and the results in [6]. Given x 0 ∈ H we consider the following system of forward-backward stochastic differential equations:…”

BSDE representation and randomized dynamic programming principle for stochastic control problems of infinite-dimensional jump-diffusions

“…Notice however that the current results require a special structure of the controlled state equations, namely that the diffusion coefficient σ = σ(t, x) is uncontrolled and the drift has the following specific form b = b 1 (t, x) + σ(t, x)b 2 (t, x, a). Up to our knowledge, only the recent paper [6], which is devoted to the study of ergodic control problems, applies the BSDEs techniques to a more general class of infinite-dimensional controlled state processes; in [6] the drift has the general form b = b(x, a), however the diffusion coefficient is still uncontrolled and indeed constant, moreover the space of control actions Λ is assumed to be a real separable Hilbert space (or, more generally, according to Remark 2.2 in [6], Λ has to be the image of a continuous surjection ϕ defined on some real separable Hilbert space). Finally, [6] only addresses the non-path-dependent (or Markovian) case, and does not treat the Hamilton-Jacobi-Bellman (HJB) equation related to the stochastic control problem.…”