Subgame-perfect equilibrium strategies for time-inconsistent recursive stochastic control problems

We study time-inconsistent recursive stochastic control problems. Since for this class of problems classical optimal controls may fail to exist or to be relevant in practice, we focus on subgame-perfect equilibrium policies. The approach followed in our work relies on the stochastic maximum principle: we adapt the classical spike variation technique to obtain a characterization of equilibrium strategies in terms of a generalized second-order Hamiltonian function defined through a pair of backward stochastic differential equations. The theoretical results are applied in the financial field to finite horizon investment-consumption policies with non-exponential actualization.Comment: arXiv admin note: substantial text overlap with arXiv:2105.0147

Optimal Controls for Forward-Backward Stochastic Differential Equations: Time-Inconsistency and Time-Consistent Solutions

This paper is concerned with an optimal control problem for a forward-backward stochastic differential equation (FBSDE, for short) with a recursive cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). It is found that such an optimal control problem is time-inconsistent in general, even if the cost functional is reduced to a classical Bolza type one as in Peng [50], Lim-Zhou [41], and Yong [74]. Therefore, instead of finding a global optimal control (which is time-inconsistent), we will look for a time-consistent and locally optimal equilibrium strategy, which can be constructed via the solution of an associated equilibrium Hamilton-Jacobi-Bellman (HJB, for short) equation. A verification theorem for the local optimality of the equilibrium strategy is proved by means of the generalized Feynman-Kac formula for BSVIEs and some stability estimates of the representation for parabolic partial differential equations (PDEs, for short). Under certain conditions, it is proved that the equilibrium HJB equation, which is a nonlocal PDE, admits a unique classical solution. As special cases and applications, the linear-quadratic problems, a mean-variance model, a social planner problem with heterogeneous Epstein-Zin utilities, and a Stackelberg game are briefly investigated. It turns out that our framework can cover not only the optimal control problems for FBSDEs studied in [50,41,74], and so on, but also the problems of the general discounting and some nonlinear appearance of conditional expectations for the terminal state, studied in Yong [75,77] and Bj\"{o}rk-Khapko-Murgoci [7]

Time-Consistent Mean-Variance Portfolio Selection in Discrete and Continuous Time

It is well known that mean-variance portfolio selection is a time-inconsistent optimal control problem in the sense that it does not satisfy Bellman's optimality principle and therefore the usual dynamic programming approach fails. We develop a time- consistent formulation of this problem, which is based on a local notion of optimality called local mean-variance efficiency, in a general semimartingale setting. We start in discrete time, where the formulation is straightforward, and then find the natural extension to continuous time. This complements and generalises the formulation by Basak and Chabakauri (2010) and the corresponding example in Bj\"ork and Murgoci (2010), where the treatment and the notion of optimality rely on an underlying Markovian framework. We justify the continuous-time formulation by showing that it coincides with the continuous-time limit of the discrete-time formulation. The proof of this convergence is based on a global description of the locally optimal strategy in terms of the structure condition and the F\"ollmer-Schweizer decomposition of the mean-variance tradeoff. As a byproduct, this also gives new convergence results for the F\"ollmer-Schweizer decomposition, i.e. for locally risk minimising strategies

A tutorial on recursive models for analyzing and predicting path choice behavior

The problem at the heart of this tutorial consists in modeling the path choice behavior of network users. This problem has been extensively studied in transportation science, where it is known as the route choice problem. In this literature, individuals' choice of paths are typically predicted using discrete choice models. This article is a tutorial on a specific category of discrete choice models called recursive, and it makes three main contributions: First, for the purpose of assisting future research on route choice, we provide a comprehensive background on the problem, linking it to different fields including inverse optimization and inverse reinforcement learning. Second, we formally introduce the problem and the recursive modeling idea along with an overview of existing models, their properties and applications. Third, we extensively analyze illustrative examples from different angles so that a novice reader can gain intuition on the problem and the advantages provided by recursive models in comparison to path-based ones