Conditional indicators

TL;DR

This paper introduces a broad class of conditional indicators on probability spaces, exploring their properties, characterizations, and applications in finance and operator theory, extending the concept of conditional expectation.

Contribution

It defines and characterizes conditional indicators, a new class of positive mappings related to conditional expectation, with examples relevant to finance and operator theory.

Findings

Conditional indicators share properties with conditional expectation.

Several characterizations of conditional indicators are provided.

Examples demonstrate applications in finance and Riesz space operators.

Abstract

In this paper, we introduce a large class of (so-called) conditional indicators, on a complete probability space with respect to a sub $\sigma$-algebra. A conditional indicator is a positive mapping, which is not necessary linear, but may share common features with the conditional expectation, such as the tower property or the projection property. Several characterizations are formulated. Beyond the definitions, we provide some non trivial examples that are used in finance and may inspire new developments in the theory of operators on Riesz spaces.