Modeling and Implementation of Risk-Averse Preferences in Stochastic Programs Using Risk Measures

2006 ◽  
pp. 95-116 ◽  
Author(s):  
Pavlo A. Krokhmal ◽  
Robert Murphey
Keyword(s):  
Risk Measures ◽  
Risk Averse ◽  
Stochastic Programs

Consistency of Sample Estimates of Risk Averse Stochastic Programs

2013 ◽  
Vol 50 (02) ◽  
pp. 533-541 ◽  
Author(s):  
Alexander Shapiro
Keyword(s):  
Stochastic Programming ◽  
Law Of Large Numbers ◽  
Risk Measures ◽  
Regularity Conditions ◽  
Distributed Data ◽  
Risk Averse ◽  
Asymptotic Consistency ◽  
Stochastic Programs ◽  
Large Numbers ◽  
Independent Identically Distributed

In this paper we study asymptotic consistency of law invariant convex risk measures and the corresponding risk averse stochastic programming problems for independent, identically distributed data. Under mild regularity conditions, we prove a law of large numbers and epiconvergence of the corresponding statistical estimators. This can be applied in a straightforward way to establish convergence with probability 1 of sample-based estimators of risk averse stochastic programming problems.

scholarly journals Consistency of Sample Estimates of Risk Averse Stochastic Programs

2013 ◽  
Vol 50 (2) ◽  
pp. 533-541 ◽  
Author(s):  
Alexander Shapiro
Keyword(s):  
Stochastic Programming ◽  
Law Of Large Numbers ◽  
Risk Measures ◽  
Regularity Conditions ◽  
Distributed Data ◽  
Risk Averse ◽  
Asymptotic Consistency ◽  
Stochastic Programs ◽  
Large Numbers ◽  
Independent Identically Distributed

In this paper we study asymptotic consistency of law invariant convex risk measures and the corresponding risk averse stochastic programming problems for independent, identically distributed data. Under mild regularity conditions, we prove a law of large numbers and epiconvergence of the corresponding statistical estimators. This can be applied in a straightforward way to establish convergence with probability 1 of sample-based estimators of risk averse stochastic programming problems.

Technical Note—Time Inconsistency of Optimal Policies of Distributionally Robust Inventory Models

2020 ◽  
Vol 68 (5) ◽  
pp. 1576-1584 ◽  
Author(s):  
Alexander Shapiro ◽  
Linwei Xin
Keyword(s):  
Decision Making ◽  
Large Class ◽  
Risk Measures ◽  
Technical Note ◽  
Time Inconsistency ◽  
Base Stock ◽  
Risk Averse ◽  
Inventory Models ◽  
Optimal Policies ◽  
Distributionally Robust

The authors extend previous studies of time inconsistency to risk averse (distributionally robust) inventory models and show that time inconsistency is not unique to robust multistage decision making, but may happen for a large class of risk averse/distributionally robust settings. In particular, they demonstrate that if the respective risk measures are not strictly monotone, then there may exist infinitely many optimal policies which are not base-stock and not time consistent. This is in a sharp contrast with the risk neutral formulation of the inventory model where all optimal policies are base-stock and time consistent.

scholarly journals Risk-Averse Stochastic Programming: Time Consistency and Optimal Stopping

2021 ◽  
Author(s):  
Alois Pichler ◽  
Rui Peng Liu ◽  
Alexander Shapiro
Keyword(s):  
Optimal Stopping ◽  
Risk Measures ◽  
Time Consistency ◽  
Risk Averse ◽  
Principle Of Optimality ◽  
Optimal Policies ◽  
Natural Dynamic ◽  
Programming Time ◽  
Over Time ◽  
Numerical Approaches

This paper addresses time consistency of risk-averse optimal stopping in stochastic optimization. It is demonstrated that time-consistent optimal stopping entails a specific structure of the functionals describing the transition between consecutive stages. The stopping risk measures capture this structural behavior and allow natural dynamic equations for risk-averse decision making over time. Consequently, associated optimal policies satisfy Bellman’s principle of optimality, which characterizes optimal policies for optimization by stating that a decision maker should not reconsider previous decisions retrospectively. We also discuss numerical approaches to solving such problems.

A Central Limit Theorem and Hypotheses Testing for Risk-averse Stochastic Programs

2018 ◽  
Vol 28 (2) ◽  
pp. 1337-1366 ◽  
Author(s):  
Vincent Guigues ◽  
Volker Krätschmer ◽  
Alexander Shapiro
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Hypotheses Testing ◽  
Risk Averse ◽  
Stochastic Programs

scholarly journals Problem-driven scenario generation: an analytical approach for stochastic programs with tail risk measure

2019 ◽  
Author(s):  
Jamie Fairbrother ◽  
Amanda Turner ◽  
Stein W. Wallace
Keyword(s):  
Portfolio Selection ◽  
Random Vector ◽  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Monte Carlo Sampling ◽  
Conditional Value At Risk ◽  
Scenario Generation ◽  
Tail Risk ◽  
Stochastic Programs

AbstractScenario generation is the construction of a discrete random vector to represent parameters of uncertain values in a stochastic program. Most approaches to scenario generation are distribution-driven, that is, they attempt to construct a random vector which captures well in a probabilistic sense the uncertainty. On the other hand, a problem-driven approach may be able to exploit the structure of a problem to provide a more concise representation of the uncertainty. In this paper we propose an analytic approach to problem-driven scenario generation. This approach applies to stochastic programs where a tail risk measure, such as conditional value-at-risk, is applied to a loss function. Since tail risk measures only depend on the upper tail of a distribution, standard methods of scenario generation, which typically spread their scenarios evenly across the support of the random vector, struggle to adequately represent tail risk. Our scenario generation approach works by targeting the construction of scenarios in areas of the distribution corresponding to the tails of the loss distributions. We provide conditions under which our approach is consistent with sampling, and as proof-of-concept demonstrate how our approach could be applied to two classes of problem, namely network design and portfolio selection. Numerical tests on the portfolio selection problem demonstrate that our approach yields better and more stable solutions compared to standard Monte Carlo sampling.

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