Minimizing shortfall risk and applications to finance and insurance problems
type: Item
instance of: scholarly article
title: Minimizing shortfall risk and applications to finance and insurance problems
zbMATH Open document ID: 1015.93071
author: Huyên Pham
published in: The Annals of Applied Probability
publication date: 2003-05-06T00:00:00Z
review text: The author studies the problem of minimizing the shortfall risk defined as the expectation of the shortfall \((B-X_{T}^{x,\theta})_+\) (weighted by some loss function) with the following controlled process: \[ X_t^{x,\theta}=x+\int_0^t\theta_u dS_u+H_t^{\theta}, \] where \(B\) is a given nonnegative measurable random variable, \(S\) is a semimartingale, \(\Theta\) is the set of the control process, \(\theta\) is a convex subset of \(L(S)\) and \((H^{\theta}:\theta\in \Theta)\) is a concave family of adapted processes with finite variation, \(t\in [0,T].\) An existence result to this optimization problem is stated, and some qualitative properties of the associated value function are shown. A verification theorem in terms of a dual control problem is established. Some applications to hedging problems in constrained portfolios, large investor and reinsurance models are given. The approach for solving these control problems uses probabilistic methods rather than PDE methods via the Bellman equation. This allows relaxing the assumption of a Markov state process required in the PDE approach.
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 49K45
Mathematics Subject Classification ID: 91G80
zbMATH DE Number: 1906199
zbMATH Keywords: shortfall risk minimization
zbMATH Keywords: semimartingales
zbMATH Keywords: optional decomposition under constraints
zbMATH Keywords: duality theory
zbMATH Keywords: finance and insurance
zbMATH Keywords: dual control
zbMATH Keywords: hedging
zbMATH Keywords: constrained portfolios
reviewed by: Anatoliy Swishchuk
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1015961159
OpenAlex ID: W2059498346
cites work: Optimal consumption and equilibrium prices with portfolio constraints and stochastic income
cites work: Optimal consumption choices for a `large' investor
cites work: Minimizing Expected Loss of Hedging in Incomplete and Constrained Markets
cites work: On dynamic measure of risk
cites work: Erratum to: ``Utility maximization in incomplete markets with random endowment''
cites work: A general version of the fundamental theorem of asset pricing
cites work: Optimization of consumption with labor income
cites work: Dynamic Programming and Pricing of Contingent Claims in an Incomplete Market
cites work: Q4032143
cites work: Optional decompositions under constraints
cites work: Quantile hedging
cites work: Efficient hedging: cost versus shortfall risk
cites work: Q4086524
cites work: Calcul stochastique et problèmes de martingales
cites work: Q4368791
cites work: Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets
cites work: The asymptotic elasticity of utility functions and optimal investment in incomplete markets
cites work: Optimal Control of Favorable Games with a Time Limit
cites work: Dynamic L p-Hedging in Discrete Time under Cone Constraints
cites work: Approximating random variables by stochastic integrals
cites work: Maximizing the probability of a perfect hedge
cites work: Optimal insurance demand under marked point processes shocks.
DOI: 10.1214/AOAP/1015961159
description: scientific article; zbMATH DE number 1906199