Predictable Forward Performance Processes: The Binomial Case

TL;DR

This paper introduces a new class of predictable forward performance processes within a binomial market model, focusing on dynamically updating utility functions through a single-period inverse investment problem.

Contribution

It develops a framework for constructing endogenous, predictable forward performance processes in a binomial setting with random, evolving parameters.

Findings

Solution reduces to a functional equation

Existence and uniqueness conditions established

Framework applicable to dynamic market environments

Abstract

We introduce a new class of forward performance processes that are endogenous and predictable with regards to an underlying market information set and, furthermore, are updated at discrete times. We analyze in detail a binomial model whose parameters are random and updated dynamically as the market evolves. We show that the key step in the construction of the associated predictable forward performance process is to solve a single-period inverse investment problem, namely, to determine, period-by-period and conditionally on the current market information, the end-time utility function from a given initial-time value function. We reduce this inverse problem to solving a functional equation and establish conditions for the existence and uniqueness of its solutions in the class of inverse marginal functions.