Abstract
The envelope formula is obtained for optimization problems with positively homogeneous convex functionals defined on a space of random variables. Those problems include linear regression with general error measures and optimal portfolio selection with the objective function being either a general deviation measure or a coherent risk measure subject to a constraint on the expected rate of return. The obtained results are believed to be novel even for Markowitz's mean-variance portfolio selection but are far more general and include explicit envelope relationships for the rates of return of portfolios that minimize lower semivariance, mean absolute deviation, deviation measures of ${\cal L}^p$-type and semi-${\cal L}^p$ type, and conditional value-at-risk. In each case, the envelope theorem yields explicit estimates for the absolute value of the difference between deviation/risk of optimal portfolios with the unperturbed and perturbed asset probability distributions in terms of a norm of the perturbation.
Highlights
Financial markets, being inherently uncertain, require elaborate stochastic models for solving a variety of problems ranging from predicting asset prices, setting interest rates, and selecting appropriate portfolios to making any kind of short-term and longterm decisions
The envelope relationship (Theorem 3.1) is obtained for optimization problems whose objectives and constraints involve arbitrary positively homogeneous convex functionals defined on the space of random variables
When problem solution sets and corresponding subdifferentials are singletons, it simplifies to random-parameter version (27) of the ordinary envelope relationship
Summary
Financial markets, being inherently uncertain, require elaborate stochastic models for solving a variety of problems ranging from predicting asset prices, setting interest rates, and selecting appropriate portfolios to making any kind of short-term and longterm decisions. For convex functionals on L 2(Ω), which are finite and continuous but not positively homogeneous, the dual characterization (6) does not hold and Theorem 5 in [38] does not seem to be directly applicable.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
