We study n-player games of portfolio choice in general common Ito-diffusion markets under relative performance criteria and time monotone forward utilities. We, also, consider their continuum limit which gives rise to a forward mean field game with unbounded controls in both the drift and volatility terms. Furthermore, we allow for general (time monotone) preferences, thus departing from the homothetic case, the only case so far analyzed. We produce explicit solutions for the optimal policies, the optimal wealth processes and the game values, and also provide representative examples for both the finite and the mean field game.

In this paper, we present a probabilistic numerical method for a class of forward utilities in a stochastic factor model. For this purpose, we use the representation of forward utilities using the ergodic Backward Stochastic Differential Equations (eBSDEs) introduced by Liang and Zariphopoulou in [27]. We establish a connection between the solution of the ergodic BSDE and the solution of an associated BSDE with random terminal time $\tau $, defined as the hitting time of the positive recurrent stochastic factor. The viewpoint based on BSDEs with random horizon yields a new characterization of the ergodic cost $\lambda $ which is a part of the solution of the eBSDEs. In particular, for a certain class of eBSDEs with quadratic generator, the Cole-Hopf transformation leads to a semi-explicit representation of the solution as well as a new expression of the ergodic cost $\lambda $. The latter can be estimated with Monte Carlo methods. We also propose two new deep learning numerical schemes for eBSDEs. Finally, we present numerical results for different examples of eBSDEs and forward utilities together with the associated investment strategies.

Predictable forward performance processes (PFPPs) are stochastic optimal control frameworks for an agent who controls a randomly evolving system but can only prescribe the system dynamics for a short period ahead. This is a common scenario in which a controlling agent frequently re-calibrates her model. We introduce a new class of PFPPs based on rank-dependent utility, generalizing existing models that are based on expected utility theory (EUT). We establish existence of rank-dependent PFPPs under a conditionally complete market and exogenous probability distortion functions which are updated periodically. We show that their construction reduces to solving an integral equation that generalizes the integral equation obtained under EUT in previous studies. We then propose a new approach for solving the integral equation via theory of Volterra equations. We illustrate our result in the special case of conditionally complete Black-Scholes model.

Classical indifference valuation, a widely studied approach in incomplete markets, uses critically the a priori knowledge of the characteristics (arrival, maturity, payoff structure) of the projects in consideration. This assumption, however, may not accommodate realistic scenarios in which projects, not initially anticipated, arrive at later times. To accommodate this, we employ forward indifference valuation criteria, which by construction are flexible enough to adapt to such “non-anticipated” cases while yielding time-consistent indifference prices. We consider and analyze in detail two representative cases: valuation adjustments due to incoming non-anticipated project and the relative forward indifference valuation of new projects in relation to existing ones.

We propose a forward approach to study the performance of liquidation strategies under sequential model parameter updates. The forward liquidation program consists of pasting forward in time and in a time-consistent fashion a series of optimal liquidation problems. They are triggered at the parameter shift instances, thus entirely eliminating model error, and last at most till the next parameter update. However, due to the nature of the model dynamics, solutions may cease to exist in finite time, even before the subsequent parameter update. Furthermore, forward liquidation strategies may never lead to full liquidation, even though they maximize the average utility of revenue and always preserve time-consistency. In juxtaposition, the traditional approach delivers full liquidation at the sought horizon but encounters considerable model error, generates value erosion, and is time-inconsistent.