# Temporal and Spatial Turnpike-Type Results Under Forward Time-Monotone Performance Criteria

## TL;DR

This paper investigates the asymptotic behavior of the risk tolerance function under forward performance criteria in incomplete markets, revealing that temporal and spatial limits differ and depend on the measure's support.

## Contribution

It introduces novel turnpike-type results for forward performance criteria, highlighting differences from classical models and analyzing the influence of measure support.

## Key findings

- Temporal and spatial limits of risk tolerance do not coincide.
- Limits depend on the support of the underlying measure.
- Examples include both discrete and continuous measure cases.

## Abstract

We present turnpike-type results for the risk tolerance function in an incomplete market setting under time-monotone forward performance criteria. We show that, contrary to the classical case, the temporal and spatial limits do not coincide. We also show that they depend directly on the left- and right-end of the support of an underlying measure, which is used to construct the forward performance criterion. We provide examples with discrete and continuous measures, and discuss the asymptotic behavior of the risk tolerance for each case.