## Optimization with stochastic dominance constraints

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Venue: | SIAM Journal on Optimization |

Citations: | 38 - 5 self |

### BibTeX

@ARTICLE{Dentcheva_optimizationwith,

author = {Darinka Dentcheva and Andrzej Ruszczyński},

title = {Optimization with stochastic dominance constraints},

journal = {SIAM Journal on Optimization},

year = {},

pages = {548--566}

}

### Years of Citing Articles

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### Abstract

We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return. We develop optimality and duality theory for these models. We construct equivalent optimization models with utility functions. Numerical illustration is provided.

### Citations

3466 | Convex Analysis
- Rockafellar
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Citation Context ...[18, 26]). It follows from (9) and (4) that { G(α, R(x)) = sup η = sup η αη − E [ ] (η − R(x))+ } { } αη − F2(R(x); η) . Therefore G(·, R(x)) is the Fenchel conjugate of the function F2(R(x); ·) (see =-=[28]-=- for the theory of conjugate duality). The second order dominance relation (7) is equivalent to F2(R(x); η) ≤ F2(Y ; η) for all η ∈ R, which implies that G(α, R(x)) ≥ G(α, Y ) for all α ∈ (0, 1]. (12)... |

3180 |
Probability and Measure
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Citation Context ...4.3). Let us nowprove the converse. If u ∈U1, then the left derivative of u, u ′ −(t) = lim τ↑t [u(t) − u(τ)]/(t − τ), is well-defined, nonincreasing, and continuous from the left. By Theorem 12.4 of =-=[2]-=-, after an obvious adaptation, there exists a unique regular nonnegative measure µ satisfying µ([t, b]) = u ′ −(t). Thus the correspondence between nonnegative measures in rca([a, b]) and functions in... |

1441 |
Portfolio Selection
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Citation Context ...k, utility functions. 1 Introduction The problem of optimizing a portfolio of finitely many assets is a classical problem in theoretical and computational finance. Since the seminal work of Markowitz =-=[19, 20, 21]-=- it is generally agreed that portfolio performance should be measured in two distinct dimensions: the mean describing the expected return, and the risk which measures the uncertainty of the return. In... |

1117 |
Theory of games and economic behavior
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Citation Context ...blem (1.1) is to consider the set C of random variables X such that, for some z ∈ Z, one has X(ω) ≤ ϕ(z,ω) a.s. Then we can write the model as max X∈C E[X]. Von Neumann and Morgenstern, in their book =-=[36]-=-, introduced the expected utility hypothesis: for every rational decision maker there exists a utility function u(·) such that she prefers outcome X over outcome Y if and only if E[u(X)] > E[u(Y )]. T... |

626 |
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- 1997
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Citation Context ...s involving two-stage and multistage decisions has been developed in [37, 38] and in [30, 31, 32]. A comprehensive treatment of the theory and numerical methods for expectation models can be found in =-=[3]-=-. Models involving constraints on probability were introduced [5, 18, 24]. The book [25] discusses in detail the theory and numerical methods for linear models with one ∗Received by the editors Decemb... |

455 |
Convex Analysis and Minimization Algorithms
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Citation Context ...n is a sum of T + 2 convex polyhedral functions with known domains. Moreover, their subgradients are readily available. Therefore the dual problem can be solved by nonsmooth optimization methods (see =-=[13, 12]-=-). We have developed a specialized version of the regularized decomposition method described in [31]. This approach is particularly suitable, because the dual function is a sum of very many polyhedral... |

405 |
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Citation Context ...the following optimization problem max x∈X E[ u(R(x)) ] . (3) It is usually required that the function u(·) is concave and nondecreasing, thus representing preferences of a risk-averse decision maker =-=[7, 8]-=-. The challenge here is to select the appropriate utility function that represents well our preferences and whose application leads to non-trivial and meaningful solutions of (3). Another way to incor... |

330 |
J.E.: Increasing risk: I. A definition
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- 1969
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Citation Context ...t of stochastic dominance is related to models of riskaverse preferences [7]. It originated from the theory of majorization [11, 22] for the discrete case, was later extended to general distributions =-=[27, 9, 10, 30]-=-, and is now widely used in economics and finance [17]. The usual (first order) definition of stochastic dominance gives a partial order in the space of real random variables. More important from the ... |

286 | Perturbation Analysis of Optimization Problems - Bonnans, Shapiro - 2000 |

232 |
Stochastic Programming
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Citation Context ...0, 31, 32]. A comprehensive treatment of the theory and numerical methods for expectation models can be found in [3]. Models involving constraints on probability were introduced [5, 18, 24]. The book =-=[25]-=- discusses in detail the theory and numerical methods for linear models with one ∗Received by the editors December 28, 2002; accepted for publication (in revised form) June 16, 2003; published electro... |

231 |
Comparison Methods for Stochastic Models and Risks
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- 2002
(Show Context)
Citation Context ...n the second order, which we write as X ≽ (2) Y , if E[u(X)] ≥ E[u(Y )] for every concave nondecreasing function u(·), for which these expected values are finite. We refer the reader to the monograph =-=[20]-=- for a modern viewon the stochastic dominance relation and other comparison methods for random outcomes. The main objective of this paper is to introduce a newstochastic optimization model involving d... |

230 | Optimization of conditional value-at-risk
- Rockafellar, Uryasev
(Show Context)
Citation Context ...non-trivial and meaningful solutions of (3). Another way to incorporate risk-aversion into the model is the use of Value at Risk (VaR) constraints [6] and Conditional Value at Risk (CVaR) constraints =-=[29]-=-. In this paper we propose an alternative approach, by introducing a comparison to a reference return into our optimization problem. The comparison is based on the stochastic dominance relation. More ... |

203 | Conditional value-at-risk for general loss distributions
- Rockafellar, Uryasev
(Show Context)
Citation Context ...en the optimal outcome ˆ X is not stochastically dominated by any other feasible outcome (see [21, 22, 23]). Other stochastic optimization models involving general risk functionals were considered by =-=[35, 13, 27, 29]-=-. Our model (1.2)–(1.4) is a newway to formulate a stochastic optimization problem. } .550 DARINKA DENTCHEVA AND ANDRZEJ RUSZCZY ŃSKI Example 1. Let R1,...,RN ∈L 1 (Ω,F,P) be random returns of assets... |

156 |
Linear programming under uncertainty
- Dantzig
- 1955
(Show Context)
Citation Context ...citly, or via some constraints that may involve the elementary event ω and must hold with some prescribed probability. The first stochastic optimization models with expected values were introduced in =-=[1, 6]-=-. Mathematical theory of expectation models involving two-stage and multistage decisions has been developed in [37, 38] and in [30, 31, 32]. A comprehensive treatment of the theory and numerical metho... |

154 |
Rules for Ordering Uncertain Prospects
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- 1969
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Citation Context ...pt of stochastic dominance is related to models of riskaverse preferences [6]. It originated from the theory of majorization [9, 18] for the discrete case, was later extended to general distributions =-=[23, 7, 8, 25]-=-, and is now widely used in economics and finance [14]. The usual (first order) definition of stochastic dominance gives a partial order in the space of real random variables. More important from the ... |

151 | Linear Programming: Foundations and Extensions - Vanderbei - 1996 |

127 |
Methods of Measuring the Concentration of Wealth
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- 1905
(Show Context)
Citation Context ... consider for a real random variable V the function G(α, V ) = αCVaRα(V ), α ∈ (0, 1]. For α = 0 we set G(0, V ) = 0. The function G(·, V ) is frequently referred to as the absolute Lorenz curve (see =-=[18, 26]-=-). It follows from (9) and (4) that { G(α, R(x)) = sup η = sup η αη − E [ ] (η − R(x))+ } { } αη − F2(R(x); η) . Therefore G(·, R(x)) is the Fenchel conjugate of the function F2(R(x); ·) (see [28] for... |

117 | A Mean-Absolute Deviation Portfolio Optimization Model and - Konno, Yamazaki - 1991 |

100 |
Stochastic dominance and expected utility: survey and analysis
- Levy
- 1992
(Show Context)
Citation Context ... parametric optimization problem, and it facilitates the trade-off analysis between mean and risk. Another theoretical approach to the portfolio selection problem is that of stochastic dominance (see =-=[23, 35, 17]-=-). The concept of stochastic dominance is related to models of riskaverse preferences [7]. It originated from the theory of majorization [11, 22] for the discrete case, was later extended to general d... |

90 |
On minimizing a convex function subject to linear inequalities
- Beale
- 1955
(Show Context)
Citation Context ...citly, or via some constraints that may involve the elementary event ω and must hold with some prescribed probability. The first stochastic optimization models with expected values were introduced in =-=[1, 6]-=-. Mathematical theory of expectation models involving two-stage and multistage decisions has been developed in [37, 38] and in [30, 31, 32]. A comprehensive treatment of the theory and numerical metho... |

86 |
Methods of Descent for Nondifferentiable Optimization
- Kiwiel
(Show Context)
Citation Context ...n is a sum of T + 2 convex polyhedral functions with known domains. Moreover, their subgradients are readily available. Therefore the dual problem can be solved by nonsmooth optimization methods (see =-=[13, 12]-=-). We have developed a specialized version of the regularized decomposition method described in [31]. This approach is particularly suitable, because the dual function is a sum of very many polyhedral... |

78 |
Increasing Risk
- Rothschild, Stiglitz
- 1970
(Show Context)
Citation Context ...rimary, 90C15, 90C46, 90C48; Secondary, 46N10, 60E15, 91B06 DOI. S1052623402420528 1. Introduction. The relation of stochastic dominance is a fundamental concept of decision theory and economics (see =-=[11, 12, 26, 33]-=-). A random variable X dominates another random variable Y in the second order, which we write as X ≽ (2) Y , if E[u(X)] ≥ E[u(Y )] for every concave nondecreasing function u(·), for which these expec... |

70 |
Decision and Value Theory
- Fishburn, Fishburn
- 1964
(Show Context)
Citation Context ... Another theoretical approach to the portfolio selection problem is that of stochastic dominance (see [23, 35, 17]). The concept of stochastic dominance is related to models of riskaverse preferences =-=[7]-=-. It originated from the theory of majorization [11, 22] for the discrete case, was later extended to general distributions [27, 9, 10, 30], and is now widely used in economics and finance [17]. The u... |

69 | A.: From stochastic dominance to mean-risk models: semideviations as risk measures
- Ogryczak, Ruszczyński
- 1999
(Show Context)
Citation Context ...een the mean and the risk. The general question of constructing mean–risk models which are in harmony with the stochastic dominance relations has been the subject of the analysis of the recent papers =-=[24, 25, 26, 32]-=-. We have identified there several primal risk measures, most notably central semideviations, and dual risk measures, based on the Lorenz curve, which are consistent with the stochastic dominance rela... |

64 | Dual stochastic dominance and related mean–risk models
- Ogryczak, Ruszczýnski
(Show Context)
Citation Context ...een the mean and the risk. The general question of constructing mean–risk models which are in harmony with the stochastic dominance relations has been the subject of the analysis of the recent papers =-=[24, 25, 26, 32]-=-. We have identified there several primal risk measures, most notably central semideviations, and dual risk measures, based on the Lorenz curve, which are consistent with the stochastic dominance rela... |

55 |
A regularized decomposition method for minimizing a sum of polyhedral functions
- Ruszczyński
- 1986
(Show Context)
Citation Context ...dily available. Therefore the dual problem can be solved by nonsmooth optimization methods (see [13, 12]). We have developed a specialized version of the regularized decomposition method described in =-=[31]-=-. This approach is particularly suitable, because the dual function is a sum of very many polyhedral functions. After the dual problem is solved, we obtain not only the optimal dual solution ( ˆ β, ˆ ... |

55 |
Theory of Vector Optimization
- Luc
- 1989
(Show Context)
Citation Context ... (see, e.g., [19] and the references therein). Partial orders appear in abstract optimization problems when the values of the objective operator are elements of a topological vector space (see, e.g., =-=[15]-=-). It is usually assumed that the partial order is generated by a convex cone. The stochastic dominance orders in L k (Ω,F,P) are not generated by cones in this space, as we shall see in Proposition 2... |

47 |
The efficiency analysis of choices involving risk
- Hanoch, Levy
- 1969
(Show Context)
Citation Context ...t of stochastic dominance is related to models of riskaverse preferences [7]. It originated from the theory of majorization [11, 22] for the discrete case, was later extended to general distributions =-=[27, 9, 10, 30]-=-, and is now widely used in economics and finance [17]. The usual (first order) definition of stochastic dominance gives a partial order in the space of real random variables. More important from the ... |

45 |
Stochastic programs with fixed recourse: The equivalent deterministic program
- Wets
- 1974
(Show Context)
Citation Context .... The first stochastic optimization models with expected values were introduced in [1, 6]. Mathematical theory of expectation models involving two-stage and multistage decisions has been developed in =-=[37, 38]-=- and in [30, 31, 32]. A comprehensive treatment of the theory and numerical methods for expectation models can be found in [3]. Models involving constraints on probability were introduced [5, 18, 24].... |

44 |
Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil
- Charnes, Cooper, et al.
- 1958
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Citation Context ... in [37, 38] and in [30, 31, 32]. A comprehensive treatment of the theory and numerical methods for expectation models can be found in [3]. Models involving constraints on probability were introduced =-=[5, 18, 24]-=-. The book [25] discusses in detail the theory and numerical methods for linear models with one ∗Received by the editors December 28, 2002; accepted for publication (in revised form) June 16, 2003; pu... |

32 |
Mean-Variance Analysis
- Markowitz
- 1987
(Show Context)
Citation Context ...k, utility functions. 1 Introduction The problem of optimizing a portfolio of finitely many assets is a classical problem in theoretical and computational finance. Since the seminal work of Markowitz =-=[19, 20, 21]-=- it is generally agreed that portfolio performance should be measured in two distinct dimensions: the mean describing the expected return, and the risk which measures the uncertainty of the return. In... |

30 | On Stochastic Dominance and Mean-Semideviation Models, Rutcor Research Report 7-98
- Ogryczak, Ruszczynski
- 1998
(Show Context)
Citation Context ...een the mean and the risk. The general question of constructing mean–risk models which are in harmony with the stochastic dominance relations has been the subject of the analysis of the recent papers =-=[24, 25, 26, 32]-=-. We have identified there several primal risk measures, most notably central semideviations, and dual risk measures, based on the Lorenz curve, which are consistent with the stochastic dominance rela... |

29 |
Admissibility and Measurable Utility Functions
- Quirk, Saposnik
- 1962
(Show Context)
Citation Context ...t of stochastic dominance is related to models of riskaverse preferences [7]. It originated from the theory of majorization [11, 22] for the discrete case, was later extended to general distributions =-=[27, 9, 10, 30]-=-, and is now widely used in economics and finance [17]. The usual (first order) definition of stochastic dominance gives a partial order in the space of real random variables. More important from the ... |

29 | Quantitative stability in stochastic programming: the method of probability metrics
- Rachev, Römisch
(Show Context)
Citation Context ...en the optimal outcome ˆ X is not stochastically dominated by any other feasible outcome (see [21, 22, 23]). Other stochastic optimization models involving general risk functionals were considered by =-=[35, 13, 27, 29]-=-. Our model (1.2)–(1.4) is a newway to formulate a stochastic optimization problem. } .550 DARINKA DENTCHEVA AND ANDRZEJ RUSZCZY ŃSKI Example 1. Let R1,...,RN ∈L 1 (Ω,F,P) be random returns of assets... |

27 |
Ordered Families of Distributions
- Lehmann
- 1955
(Show Context)
Citation Context ...ion is defined as the right-continuousPortfolio Optimization with Stochastic Dominance Constraints 5 cumulative distribution function of V : F (V ; η) = P{V ≤ η} for η ∈ R. A random return V is said =-=[16, 27]-=- to stochastically dominate another random return S to the first order, denoted V ≽ F SD S, if F (V ; η) ≤ F (S; η) for all η ∈ R. The second performance function F2 is given by areas below the distri... |

25 | On probabilistic constrained programming
- Prékopa
- 1970
(Show Context)
Citation Context ... in [37, 38] and in [30, 31, 32]. A comprehensive treatment of the theory and numerical methods for expectation models can be found in [3]. Models involving constraints on probability were introduced =-=[5, 18, 24]-=-. The book [25] discusses in detail the theory and numerical methods for linear models with one ∗Received by the editors December 28, 2002; accepted for publication (in revised form) June 16, 2003; pu... |

24 |
Portfolio selection. Journal of Finance 7
- Markowitz
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(Show Context)
Citation Context ...k, utility functions. 1 Introduction The problem of optimizing a portfolio of finitely many assets is a classical problem in theoretical and computational finance. Since the seminal work of Markowitz =-=[19, 20, 21]-=- it is generally agreed that portfolio performance should be measured in two distinct dimensions: the mean describing the expected return, and the risk which measures the uncertainty of the return. In... |

22 | A Linear Programming Approximation for the General Portfolio Selection Problem - SHARPE - 1971 |

20 |
eds.): Stochastic Dominance: An Approach to Decision–Making Under
- Whitmore, Findlay
- 1978
(Show Context)
Citation Context ... parametric optimization problem, and it facilitates the trade-off analysis between mean and risk. Another theoretical approach to the portfolio selection problem is that of stochastic dominance (see =-=[23, 35, 17]-=-). The concept of stochastic dominance is related to models of riskaverse preferences [7]. It originated from the theory of majorization [11, 22] for the discrete case, was later extended to general d... |

18 |
Optimality and duality theory for stochastic optimization with nonlinear dominance constraints
- Dentcheva, Ruszczyński
- 2004
(Show Context)
Citation Context ...oncave functions u(·) for which these expected values are finite. Thus, no risk-averse decision maker will prefer a portfolio with return Y over a portfolio with return R. In our earlier publications =-=[2, 3, 4, 5]-=- we have introduced a new stochastic optimization model with stochastic dominance constraints. In this paper we show how this theory can be used for risk-averse portfolio optimization. We add to the p... |

15 | 2003): “Frontiers of stochastically nondominated portfolios
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Citation Context |

14 |
Stochastic convex programming: relatively complete recourse and induced feasibility
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Citation Context ...hastic optimization models with expected values were introduced in [1, 6]. Mathematical theory of expectation models involving two-stage and multistage decisions has been developed in [37, 38] and in =-=[30, 31, 32]-=-. A comprehensive treatment of the theory and numerical methods for expectation models can be found in [3]. Models involving constraints on probability were introduced [5, 18, 24]. The book [25] discu... |

13 |
Stochastic convex programming: basic duality
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- 1976
(Show Context)
Citation Context ...hastic optimization models with expected values were introduced in [1, 6]. Mathematical theory of expectation models involving two-stage and multistage decisions has been developed in [37, 38] and in =-=[30, 31, 32]-=-. A comprehensive treatment of the theory and numerical methods for expectation models can be found in [3]. Models involving constraints on probability were introduced [5, 18, 24]. The book [25] discu... |

11 |
Chance-constrained programming with joint constraints
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- 1965
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Citation Context ... in [37, 38] and in [30, 31, 32]. A comprehensive treatment of the theory and numerical methods for expectation models can be found in [3]. Models involving constraints on probability were introduced =-=[5, 18, 24]-=-. The book [25] discusses in detail the theory and numerical methods for linear models with one ∗Received by the editors December 28, 2002; accepted for publication (in revised form) June 16, 2003; pu... |

10 |
Programming under uncertainty: The equivalent convex program
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(Show Context)
Citation Context .... The first stochastic optimization models with expected values were introduced in [1, 6]. Mathematical theory of expectation models involving two-stage and multistage decisions has been developed in =-=[37, 38]-=- and in [30, 31, 32]. A comprehensive treatment of the theory and numerical methods for expectation models can be found in [3]. Models involving constraints on probability were introduced [5, 18, 24].... |

7 |
der Vlerk, Integrated chance constraints: reduced forms and an algorithm
- Haneveld, van
(Show Context)
Citation Context ...en the optimal outcome ˆ X is not stochastically dominated by any other feasible outcome (see [21, 22, 23]). Other stochastic optimization models involving general risk functionals were considered by =-=[35, 13, 27, 29]-=-. Our model (1.2)–(1.4) is a newway to formulate a stochastic optimization problem. } .550 DARINKA DENTCHEVA AND ANDRZEJ RUSZCZY ŃSKI Example 1. Let R1,...,RN ∈L 1 (Ω,F,P) be random returns of assets... |

6 |
Stochastic convex programming: Singular multipliers and extended duality singular multipliers and duality
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- 1976
(Show Context)
Citation Context ...hastic optimization models with expected values were introduced in [1, 6]. Mathematical theory of expectation models involving two-stage and multistage decisions has been developed in [37, 38] and in =-=[30, 31, 32]-=-. A comprehensive treatment of the theory and numerical methods for expectation models can be found in [3]. Models involving constraints on probability were introduced [5, 18, 24]. The book [25] discu... |

5 |
Haneveld, Duality in Stochastic Linear and Dynamic
- Klein
- 1986
(Show Context)
Citation Context ...ance Constraints 8 The CVaR constraint for the portfolio problem can be formulated as follows: CVaRα(R(x)) ≥ −ω. (10) It is the financial counterpart of the integrated chance constraint introduced in =-=[14]-=-. Theorem 1 The second order stochastic dominance constraint (7) is equivalent to the continuum of CVaR constraints: CVaRα(R(x)) ≥ CVaRα(Y ) for all α ∈ (0, 1]. (11) Proof. Let us consider for a real ... |

5 |
Conditional value-at-risk: optimization approach. Stochastic optimization: algorithms and applications
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- 2001
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Citation Context |

2 |
Optimization under linear stochastic dominance, Comptes Rendus de l’Academie Bulgare des Sciences 56
- Dentcheva, Ruszczyński
(Show Context)
Citation Context ...oncave functions u(·) for which these expected values are finite. Thus, no risk-averse decision maker will prefer a portfolio with return Y over a portfolio with return R. In our earlier publications =-=[2, 3, 4, 5]-=- we have introduced a new stochastic optimization model with stochastic dominance constraints. In this paper we show how this theory can be used for risk-averse portfolio optimization. We add to the p... |