[Paper Review] Optimization of conditional convex risk measures

Tiexin Guo|arXiv (Cornell University)|Oct 14, 2019

Risk and Portfolio Optimization37 references4 citations

TL;DR

This paper establishes existence and uniqueness conditions for optimizing conditional convex risk measures in dynamic portfolio selection using random functional analysis. It provides a necessary and sufficient condition for conditional monotone mean–variance optimization and proves existence and uniqueness for conditional entropic risk measures via $L^0$–convex compactness under bounded return constraints.

ABSTRACT

Optimization of conditional convex risk measure is a central theme in dynamic portfolio selection theory, which has not yet systematically studied in the previous literature perhaps since conditional convex risk measures are neither random strictly convex nor random coercive. The purpose of this paper is to give some basic results on the existence and uniqueness on this theme, in particular our results for conditional monotone mean--variance and conditional entropic risk measures are complete and deep. As the basis for the work of this paper, this paper first begins with a brief introduction to random functional analysis, including the historical backgrounds for its birth and some important advances. This paper then further surveys some recent progress in random convex analysis and its applications to conditional convex risk measures. Finally, based on these, we establish a concise sufficient and necessary condition for a return to be a solution to the optimization problem of conditional monotone mean--variance. We also make use of the recently developed theory of $L^0$--convex compactness to establish the existence of the optimization problem of conditional entropic risk measure when the conditional mean of returns is given and the returns fall within a random closed ball. Besides, the related uniqueness problems are also solved.

Motivation & Objective

• Address the lack of systematic study on optimization of conditional convex risk measures, which are neither random strictly convex nor coercive.
• Establish foundational results in random functional analysis, particularly $L^0$–convex compactness, to support optimization under dynamic risk measures.
• Provide a complete and deep analysis of conditional monotone mean–variance and entropic risk measure optimization problems.
• Resolve existence and uniqueness issues in portfolio selection under conditional risk measures using novel theoretical tools.

Proposed method

• Leverages random functional analysis, particularly $L^0$–convex compactness in random normed modules, to handle non-coercive and non-strictly convex risk measures.
• Applies the Hahn–Banach theorem for almost surely bounded random linear functionals in random reflexive $RN$ modules.
• Uses the separation theorem between a point and a closed $L^0$–convex set in $RLC$ modules to derive existence results.
• Employs the random strict convexity of $RIP$ modules to prove uniqueness via the equality condition in the Cauchy–Schwarz inequality.
• Introduces a constrained set $G = \{x \in M : \pi(x) = 1, E[x|\mathcal{F}] = w, \|x\|_p \leq r\}$ to ensure existence when standard sets fail.
• Applies the representation of conditional entropic risk measures via conditional expectations and logarithmic transforms to derive uniqueness.

Experimental results

Research questions

• RQ1 Under what conditions does an optimal portfolio exist for conditional monotone mean–variance risk measures?
• RQ2 How can existence be guaranteed for conditional entropic risk measure optimization when the risk measure lacks coercivity?
• RQ3 What is the necessary and sufficient condition for a return to be optimal in conditional monotone mean–variance optimization?
• RQ4 When is the solution to the conditional entropic risk measure optimization problem unique?
• RQ5 How can $L^0$–convex compactness be applied to overcome the lack of standard convexity and coercivity in conditional risk optimization?

Key findings

• A necessary and sufficient condition for optimality in conditional monotone mean–variance risk measure optimization is established via $L^0$–convex compactness and separation theorems.
• Existence of an optimal solution for conditional entropic risk measure optimization is proven when returns are constrained within a random closed ball and the conditional mean is fixed.
• Uniqueness of the optimal solution for conditional entropic risk measure optimization is established using the equality condition in the random Cauchy–Schwarz inequality.
• The solution to the conditional entropic risk measure problem is unique if two solutions yield equal risk, implying they are almost surely equal via logarithmic transformation and conditional expectation.
• The constraint set $G = \{x : \pi(x) = 1, E[x|\mathcal{F}] = w, \|x\|_p \leq r\}$ ensures existence where standard sets fail due to lack of coercivity.
• The proof of uniqueness relies on showing $x = e^{-\gamma u/2} = \alpha y = e^{-\gamma v/2}$, leading to $\alpha = 1$ and thus $u = v$ almost surely.