Results 1  10
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100
Conditional Risk Mappings
, 2004
"... We introduce an axiomatic definition of a conditional convex risk mapping and we derive its properties. In particular, we prove a representation theorem for conditional risk mappings in terms of conditional expectations. We also develop dynamic programming relations for multistage optimization probl ..."
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Cited by 54 (13 self)
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We introduce an axiomatic definition of a conditional convex risk mapping and we derive its properties. In particular, we prove a representation theorem for conditional risk mappings in terms of conditional expectations. We also develop dynamic programming relations for multistage optimization problems involving conditional risk mappings.
Coherent approaches to risk in optimization under uncertainty
 In Tutorials in Operations Research INFORMS
, 2007
"... Keywords Decisions often need to be made before all the facts are in. A facility must be built to withstand storms, floods, or earthquakes of magnitudes that can only be guessed from historical records. A portfolio must be purchased in the face of only statistical knowledge, at best, about how marke ..."
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Cited by 39 (3 self)
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Keywords Decisions often need to be made before all the facts are in. A facility must be built to withstand storms, floods, or earthquakes of magnitudes that can only be guessed from historical records. A portfolio must be purchased in the face of only statistical knowledge, at best, about how markets will perform. In optimization, this implies that constraints may need to be envisioned in terms of safety margins instead of exact requirements. But what does that really mean in model formulation? What guidelines make sense, and what are the consequences for optimization structure and computation? The idea of a coherent measure of risk in terms of surrogates for potential loss, which has been developed in recent years for applications in financial engineering, holds promise for a far wider range of applications in which the traditional approaches to uncertainty have been subject to criticism. The general ideas and main facts are presented here with the goal of facilitating their transfer to practical work in those areas. optimization under uncertainty; safeguarding against risk; safety margins; measures of risk; measures of potential loss; measures of deviation; coherency; valueatrisk; conditional valueatrisk; probabilistic constraints; quantiles; risk envelopes; dual representations; stochastic programming 1.
Analysis of Stochastic Dual Dynamic Programming Method
"... Abstract. In this paper we discuss statistical properties and rates of convergence of the Stochastic Dual Dynamic Programming (SDDP) method applied to multistage linear stochastic programming problems. We assume that the underline data process is stagewise independent and consider the framework wher ..."
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Cited by 36 (2 self)
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Abstract. In this paper we discuss statistical properties and rates of convergence of the Stochastic Dual Dynamic Programming (SDDP) method applied to multistage linear stochastic programming problems. We assume that the underline data process is stagewise independent and consider the framework where at first a random sample from the original (true) distribution is generated and consequently the SDDP algorithm is applied to the constructed Sample Average Approximation (SAA) problem.
Constructing risk measures from uncertainty sets
, 2005
"... We propose a unified theory that links uncertainty sets in robust optimization to risk measures in portfolio optimization. We illustrate the correspondence between uncertainty sets and some popular risk measures in finance, and show how robust optimization can be used to generalize the concepts of t ..."
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Cited by 19 (1 self)
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We propose a unified theory that links uncertainty sets in robust optimization to risk measures in portfolio optimization. We illustrate the correspondence between uncertainty sets and some popular risk measures in finance, and show how robust optimization can be used to generalize the concepts of these measures. We also show that by using properly defined uncertainty sets in robust optimization models, one can in fact construct coherent risk measures. Our approach to creating coherent risk measures is easy to apply in practice, and computational experiments suggest that it may lead to superior portfolio performance. Our results have implications for efficient portfolio optimization under different measures of risk.
THE FUNDAMENTAL RISK QUADRANGLE IN RISK MANAGEMENT, OPTIMIZATION AND STATISTICAL ESTIMATION
, 2013
"... Random variables that stand for cost, loss or damage must be confronted in numerous situations. Dealing with them systematically for purposes in risk management, optimization and statistics is the theme of this presentation, which brings together ideas coming from many different areas. Measures of r ..."
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Cited by 15 (7 self)
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Random variables that stand for cost, loss or damage must be confronted in numerous situations. Dealing with them systematically for purposes in risk management, optimization and statistics is the theme of this presentation, which brings together ideas coming from many different areas. Measures of risk can be used to quantify the hazard in a random variable by a single value which can substitute for the otherwise uncertain outcomes in a formulation of constraints and objectives. Such quantifications of risk can be portrayed on a higher level as generated from penaltytype expressions of “regret ” about the mix of potential outcomes. A tradeoff between an upfront level of hazard and the uncertain residual hazard underlies that derivation. Regret is the mirror image of utility, a familiar concept for dealing with gains instead of losses, but regret concerns hazards relative to a benchmark. It bridges between risk measures and expected utility, thereby reconciling those two approaches to optimization under uncertainty Statistical estimation is inevitably a partner with risk management in handling hazards, which may be known only partially through a data base. However, a much deeper connection has come to light with statistical theory itself, in particular regression. Very general measures of error can
A soft robust model for optimization under ambiguity
, 2008
"... In this paper, we propose a framework for robust optimization that relaxes the standard notion of robustness by allowing the decisionmaker to vary the protection level in a smooth way across the uncertainty set. We apply our approach to the problem of maximizing the expected value of a payoff funct ..."
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Cited by 14 (3 self)
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In this paper, we propose a framework for robust optimization that relaxes the standard notion of robustness by allowing the decisionmaker to vary the protection level in a smooth way across the uncertainty set. We apply our approach to the problem of maximizing the expected value of a payoff function when the underlying distribution is ambiguous and therefore robustness is relevant. Our primary objective is to develop this framework and relate it to the standard notion of robustness, which deals with only a single guarantee across one uncertainty set. First, we show that our approach connects closely to the theory of convex risk measures. We show that the complexity of the this approach is equivalent to that of solving a small number of standard robust problems. We then investigate the conservatism benefits and downside probability guarantees implied by this approach and compare to the standard robust approach. Finally, we illustrate the methodology on an asset allocation example consisting of historical market data over a 25year investment horizon and find in every case we explore that relaxing standard robustness with soft robustness yields a seemingly favorable riskreturn tradeoff: each case results in a higher outofsample expected return for a relatively minor degradation of outofsample downside performance.
Subdifferential representation of risk measures
 Mathematical Programming
, 2007
"... Measures of risk appear in two categories: Risk capital measures serve to determine the necessary amount of risk capital in order to avoid ruin if the outcomes of an economic activity are uncertain and their negative values may be interpreted as acceptability measures (safety measures). Pure risk me ..."
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Cited by 13 (1 self)
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Measures of risk appear in two categories: Risk capital measures serve to determine the necessary amount of risk capital in order to avoid ruin if the outcomes of an economic activity are uncertain and their negative values may be interpreted as acceptability measures (safety measures). Pure risk measures (risk deviation measures) are natural generalizations of the standard deviation. While pure risk measures are typically convex, acceptability measures are typically concave. In both cases, the convexity (concavity) implies under mild conditions the existence of subgradients (supergradients). The present paper investigates the relation between the subgradient (supergradient) representation and the properties of the corresponding risk measures. In particular, we show how monotonicity properties are reflected by the subgradient representation. Once the subgradient (supergradient) representation has been established, it is extremely easy to derive these monotonicity properties. We give a list of Examples.
On Kusuoka representation of law invariant risk measures
"... Abstract. In this paper we discuss representations of law invariant coherent risk measures in a form of integrals of the Average ValueatRisk measures. We show that such integral representation exists iff the dual set of the considered risk measure is generated by one of its elements, and this repr ..."
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Cited by 11 (2 self)
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Abstract. In this paper we discuss representations of law invariant coherent risk measures in a form of integrals of the Average ValueatRisk measures. We show that such integral representation exists iff the dual set of the considered risk measure is generated by one of its elements, and this representation is uniquely defined. On the other hand, representation of risk measures as maximum of such integral forms is not unique.