Stochastic Near-Optimal Controls for Path-Dependent Systems

Abstract

In this article, we present a general methodology for control problems driven by the Brownian motion filtration including non-Markovian and non-semimartingale state processes controlled by mutually singular measures. The main result of this paper is the development of a concrete pathwise method for characterizing and computing near-optimal controls for abstract controlled Wiener functionals. The theory does not require ad hoc functional differentiability assumptions on the value process and elipticity conditions on the diffusion components. The analysis is pathwise over suitable finite dimensional spaces and it is based on the weak differential structure introduced by Leão, Ohashi and Simas [31] jointly with measurable selection arguments. The theory is applied to stochastic control problems based on path-dependent SDEs where both drift and possibly degenerated diffusion components are controlled. Optimal control of drifts for path-dependent SDEs driven by fractional Brownian motion is also discussed. We finally provide an application in the context of financial mathematics. Namely, we construct near-optimal controls in a non-Markovian portfolio optimization problem.

Weak differentiability of Wiener functionals and occupation times

“…s for every t ∈ [0, T ] and k ≥ 1, where we recall N k (t) is given by (2.2). We now notice that Lemma 5.2 in [34] yields…”

“…This shows sup k≥1 E[J k , J k ](T ) < ∞ and we conclude J = {J k ; k ≥ 1} is an imbedded discrete structure for Y s (B s )dg and B1 is fulfilled. Moreover, (4.8) jointly with Lemma 5.2 in [34] allow us to state that B6 holds true. Then, J(B) satisfies the assumptions of Proposition 4.1.…”

“…In [33], the authors show that a large class of Wiener functionals admits a stable imbedded discrete structure which has to be interpreted as a rather weak concept of continuity w.r.t Brownian motion driving noise. One major advantage of this theory is the possibility of computing the sensitivities of X w.r.t B via simple and explicit differential-type operators based on D. For a concrete application of this theory, we refer the reader to [34,35,5].…”

Discrete-type approximations for non-Markovian optimal stopping problems: Part I

“…Let us fix 0 < β < 1 and ζ > 1. A direct application of Corollary 6.1 in [29] jointly with Assumptions I-II and Lemma 2.2 in [28] yields , σ, β). In the sequel, C is a constant which may differ from line to line.…”

“…}, where we recall N k (1) is given by (2.3). By using Lemma 6.2 in [29], Assumption I, Jensen and Hölder's inequality for ζ > 2, we get (5.9)…”