Structure conditions under progressively added information

TL;DR

This paper investigates how the addition of a non-adapted random time affects the mathematical structure conditions necessary for the existence of optimal portfolios in semi-martingale market models, providing conditions under which these structures are preserved.

Contribution

It introduces a detailed analysis of the impact of arbitrary random times on structure conditions in market models, including preservation criteria and illustrative examples.

Findings

Structure conditions can remain valid under mild assumptions.

Certain random time models preserve structure conditions universally.

Examples highlight the importance of assumptions for structure preservation.

Abstract

It has been understood that the "local" existence of the Markowitz' optimal portfolio or the solution to the local-risk minimization problem is guaranteed by some specific mathematical structures on the underlying assets price processes known in the literature as "{\it Structure Conditions}". In this paper, we consider a semi-martingale market model, and an arbitrary random time that is not adapted to the information flow of the market model. This random time may model the default time of a firm, the death time of an insured, or any the occurrence time of an event that might impact the market model somehow. By adding additional uncertainty to the market model, via this random time, the {\it structures conditions} may fail and hence the Markowitz's optimal portfolio and other quadratic-optimal portfolios might fail to exist. Our aim is to investigate the impact of this random time on the structures conditions from different perspectives. Our analysis allows us to conclude that under some mild assumptions on the market model and the random time, these structures conditions will remain valid on the one hand. Furthermore, we provide two examples illustrating the importance of these assumptions. On the other hand, we describe the random time models for which these structure conditions are preserved for any market model. These results are elaborated separately for the two contexts of stopping with the random time and incorporating totally a specific class of random times respectively.