Me, myself and I: a general theory of non-Markovian time-inconsistent stochastic control for sophisticated agents

Abstract

We develop a theory for continuous-time non-Markovian stochastic control problems which are inherently time-inconsistent. Their distinguishing feature is that the classical Bellman optimality principle no longer holds. Our formulation is cast within the framework of a controlled non-Markovian forward stochastic differential equation, and a general objective functional setting. We adopt a game-theoretic approach to study such problems, meaning that we seek for sub-game perfect Nash equilibrium points. As a first novelty of this work, we introduce and motivate a new definition of equilibrium that allows us to establish rigorously an extended dynamic programming principle, in the same spirit as in the classical theory. This in turn allows us to introduce a system of backward stochastic differential equations analogous to the classical HJB equation. We prove that this system is fundamental, in the sense that its well-posedness is both necessary and sufficient to characterise the value function and equilibria. As a final step we provide an existence and uniqueness result. Some examples and extensions of our results are also presented.

Infinite horizon backward stochastic Volterra integral equations and discounted control problems

“…For example, time-inconsistent dynamic risk measures and time-inconsistent recursive utilities can be modelled by the solutions to Type-I BSVIEs (see [42,18,2,31]). Time-inconsistent stochastic control problems related to Type-I BSVIEs were studied by Wang and Yong [32], Hernández and Possamaï [14], and the author [13]. Also, Beissner and Rosazza Gianin [5] applied Type-I BSVIEs to arbitrage-free asset pricing via a path of EMMs called an EMM-string.…”

Nonlocal Fully Nonlinear Parabolic Differential Equations Arising in Time-Inconsistent Problems

“…σ(s, y, a) = σ(s, y), which would lead to the u yy (s, s, y)-independence of the equilibrium control policy e(s, y) (7) and the equilibrium HJB equation (8). The similar restriction is inherited to the subsequent works, e.g., [14,15,16]. However, without controls in the diffusion of the state, we can hardly control the risk (noises) from the stochastic systems, which are crucial in many problems such as portfolio management, inventory control, etc.…”

“…In fact, only when the controls could or would take effect on the magnitude of uncertainty, the stochastic problems differ from the deterministic ones. In the related works of [10,11,17,14,15,18], all the authors admit that the existence and uniqueness problems of nonlocal fully nonlinear parabolic PDEs with the term u yy (s, s, y) is a complicated open problem. Hence, if we can establish the well-posedness of nonlocal fully nonlinear parabolic PDEs (1), the well-posedness of the subgame perfect equilibrium is resolved, so are some open problems listed in the discussion of [9].…”

Quasilinearization Methods for Nonlocal Fully-Nonlinear Parabolic Systems

“…The systems above characterize a series of problems indexed by t while they are connected via its dependence on (∂ I u) |I|≤2r (s, s, y). This kind of PDE problems have drawn attention of many researchers but they are limited to some special types of nonlocal second-order systems; see [24,23,7,16] and the next section for the technical explanations on their specialties via a comprehensive introduction of different types of nonlocal parabolic systems. To the best of our knowledge, only [12,13] are concerned about the nonlocal fully-nonlinear parabolic PDEs and systems, investigated with a linearization approach.…”

“…which is studied in [24,23,7,16,21,5]. Since there is no highest-order terms at (s, s, y) in (2.5), i.e.…”