Consistent modeling of risk averse behavior with spectral risk measures: Wächter/Mazzoni revisited

AbstractWe study the minimization of a spectral risk measure of the total discounted cost generated by a Markov Decision Process (MDP) over a finite or infinite planning horizon. The MDP is assumed to have Borel state and action spaces and the cost function may be unbounded above. The optimization problem is split into two minimization problems using an infimum representation for spectral risk measures. We show that the inner minimization problem can be solved as an ordinary MDP on an extended state space and give sufficient conditions under which an optimal policy exists. Regarding the infinite dimensional outer minimization problem, we prove the existence of a solution and derive an algorithm for its numerical approximation. Our results include the findings in Bäuerle and Ott (Math Methods Oper Res 74(3):361–379, 2011) in the special case that the risk measure is Expected Shortfall. As an application, we present a dynamic extension of the classical static optimal reinsurance problem, where an insurance company minimizes its cost of capital.

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Consistency of Sample Estimates of Risk Averse Stochastic Programs

Journal of Applied Probability ◽

10.1017/s0021900200013541 ◽

2013 ◽

Vol 50 (02) ◽

pp. 533-541 ◽

Cited By ~ 1

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.

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

Operations Research ◽

10.1287/opre.2019.1932 ◽

2020 ◽

Vol 68 (5) ◽

pp. 1576-1584 ◽

Cited By ~ 1

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.

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