Predictable forward performance processes: Infrequent evaluation and applications to human‐machine interactions

Abstract

We study discrete‐time predictable forward processes when trading times do not coincide with performance evaluation times in a binomial tree model for the financial market. The key step in the construction of these processes is to solve a linear functional equation of higher order associated with the inverse problem driving the evolution of the predictable forward process. We provide sufficient conditions for the existence and uniqueness and an explicit construction of the predictable forward process under these conditions. Furthermore, we find that these processes are inherently myopic in the sense that optimal strategies do not make use of future model parameters even if these are known. Finally, we argue that predictable forward preferences are a viable framework to model human‐machine interactions occurring in automated trading or robo‐advising. For both applications, we determine an optimal interaction schedule of a human agent interacting infrequently with a machine that is in charge of trading.

Optimal investment and consumption with forward preferences and uncertain parameters

“…and is not a singleton, i.e., without the element of consumption but with the model uncertainty, the definition reduces to the robust forward investment preference, which was recently introduced by Källblad et al in [27], in which they studied extensively the dual representation of robust forward investment preference. See also [31] by Lin et al…”

“…The forward investment preference, forward investment and consumption preferences, and robust forward investment preference, were characterized by SPDEs in, respectively, [38], [26], and [27], with the novel element of volatility processes as a model input. Instead of endogenously being determined by the dynamic programming principle as in the classical framework, the volatility of forward preferences is exogenously chosen by the agent, and is regarded as their belief on how their preference in the performance criterion is going to evolve in the future.…”

“…In reality, agents often confront ambiguity regarding financial market model uncertainty. In forward theory, Källblad et al in [27] initiated the study of the agent's robust forward investment preference (see Definition 1 below, with and being a non-singleton) and their optimal investment strategy, focusing on the dual representation of robust forward investment preference. However, relatively little is known about forward investment and consumption preferences with uncertainty parameters, with an exception found in [30].…”

“…Different from [30], the financial market considered in this paper may be incomplete, stemming from general investment portfolio constraints. Consequently, the model includes all the previous cases studied in [5,14,26,27,30] as special cases.…”

<i>G</i> -forward performance process and representation of homothetic case via ergodic quadratic <i>G</i> -BSDE

“…In this paper, we propose a new class of robust forward performance processes under model uncertainty, called G -forward performance processes, using the notion of G -expectation space , which is a dynamically consistent sublinear expectation space proposed by Peng [37,38]. The canonical process on this space is the so-called G -Brownian motion, and the corresponding G -Itô's calculus theory is also established (see, among others, [7,15,16,20,28,39]). It was shown by Denis, Hu and Peng [7] that the G -expectation can be written as an upper expectation over a family of linear expectations, namely,…”