Derivative Preserving Conditions in Conditional Expectation Operator

TL;DR

This paper investigates conditions under which higher order derivatives are preserved in the conditional expectation operator and introduces higher order derivatives of Expected Shortfall for nonlinear portfolio losses.

Contribution

It presents new conditions for derivative preservation in conditional expectations and defines higher order derivatives of Expected Shortfall for nonlinear risks.

Findings

Identifies specific conditions for derivative preservation in conditional expectation.

Defines higher order derivatives of Expected Shortfall for nonlinear portfolio losses.

Provides theoretical framework for analyzing derivatives of risk measures.

Abstract

In this paper, we consider conditions that a higher order derivative preserve in conditional expectation operator for a generic nonlinear random variable. Also, the paper introduces higher order derivatives of the Expected Shortfall for a generic nonlinear portfolio loss random variable.