Backward SDEs for optimal control of partially observed path-dependent stochastic systems: A control randomization approach

Abstract

We introduce a suitable backward stochastic differential equation (BSDE) to represent the value of an optimal control problem with partial observation for a controlled stochastic equation driven by Brownian motion. Our model is general enough to include cases with latent factors in Mathematical Finance. By a standard reformulation based on the reference probability method, it also includes the classical model where the observation process is affected by a Brownian motion (even in presence of a correlated noise), a case where a BSDE representation of the value was not available so far. This approach based on BSDEs allows for greater generality beyond the Markovian case, in particular our model may include path-dependence in the coefficients (both with respect to the state and the control), and does not require any non-degeneracy condition on the controlled equation. We also discuss the issue of numerical treatment of the proposed BSDE.We use a randomization method, previously adopted only for cases of full observation, and consisting, in a first step, in replacing the control by an exogenous process independent of the driving noise and in formulating an auxiliary ("randomized") control problem where optimization is performed over changes of equivalent probability measures affecting the characteristics of the exogenous process. Our first main result is to prove the equivalence between the original partially observed control problem and the randomized problem. In a second step we prove that the latter can be associated by duality to a BSDE, which then characterizes the value of the original problem as well.

Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs

“…Following [2], we next introduce a variant of the optimal switching problem formulated in the new setting (Ω,F,P,Ŵ ). We define a new filtration, denoted FŴ ,∞ = (FŴ ,∞ t ) t≥0 , as follows: we first introduce…”

“…We follow closely [2], making use in particular of the basic Proposition 4.2 in that paper. Let (Ω, F, P, W ) be a setting for the optimal switching problem formulated in section 2.2.…”

“…It follows that J R (ν) ≤υ ∞ 0 , where the latter was defined in (4.5). From the arbitrariness ofν we deduce that supν ∈V inf > 0 J R (ν) ≤υ In this proof we borrow some constructions from [21] and [2], but the proofs need substantial extensions. This is due to the cost J 2 (α) in (2.9) related to switching: in the various approximations and convergence arguments in Lemmas 4.2, 4.4, 4.5, this term requires a special treatment, in particular because we do not require boundedness of the costs c t (x, a, a ′ ) nor a uniform bound on the number N T of switchings.…”

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

“…Let (Ω, F, P; x 0 , W, π; X; A) be the setting of the original stochastic control problem in Section 3.1. Proceeding along the same lines as at the beginning of Section 4.1 in [1], we construct an atomless finite measure λ ′ 0 on (R, B(R)) and a surjective Borel-measurable map π : R → Λ such that λ 0 = λ ′ 0 • π −1 . Let (Ω ′ , F ′ , P ′ ) be the completion of the canonical probability space of a Poisson random measure…”

“…for all 0 ≤ t ≤ T , with (Tν 0 ,ην 0 ) := (0, a 0 ), belongs toĀḠ and (4.23) holds. Finally, concerning the existence of a sequence (Tν n ,ην n ) n≥1 satisfying (i)-(ii)-(iii)-(iv), we do not report the proof of this result as it can be done proceeding along the same lines as in the proof of Lemma 4.3 in [1], the only difference being that the filtration F W in [1] (notice that in [1] W denotes a finite dimensional Brownian motion) is now replaced by F x 0 ,W,π : this does not affect the proof of Lemma 4.3 in [1]. ✷ Remark 4.2 Let (Ω,F ,P;x 0 ,W ,π,θ;Ī,X;V) andḠ be respectively the probabilistic setting for the randomized control problem and the σ-algebra mentioned in Lemma 4.2.…”