Deviation Measures on Banach Spaces and Applications

In this article we generalize the notion of the deviation measure, which were initially defined on spaces of squarely integrable random variables, as an extension of the notion of standard deviation. We extend them both under a frame which requires some elements from the theory of partially ordered linear spaces and also under a frame which refers to some closed subspace, whose elements are supposed to have zero deviation. This subspace denotes in general a set of risk-less assets, since in finance deviation measures may replace standard deviation as a measure of risk. In the last sections of the article we treat the minimization of deviation measures over a set of financial positions as a zero-sum game between the investor and the nature and we determine the solution of such a minimization problem via min-max theorems.


Introduction
Consider two time-periods of economic activity, denoted by and 1 .The time-period is the time-period in which all the individuals make their own decisions under uncertainty, while the time-period is the one in which they enjoy the effects of these decisions, in which the true state of the economy is revealed.Let us consider a Banach space , which is supposed to be the space of financial positions, denoting the total value of a portfolio of assets selected at time-period , when time-period comes. is usually a space of random variables, namely 0 0 is the space of the -measurable random variables      defined on the probability space  , ,   of the economy, where denotes the set of states of the world, the   -algebra denotes the observable events of the economy and   denotes a probability measure on the set of events .We also consider the riskless asset , being the random variable for which 1 .
The deviation risk measures according to what is initially introduced in (Rockafellar, Uryasev and Zabarankin, 2003) is a class of risk measures which generalizes the notion of standard deviation on the space of squarely integrable financial posi- tions for any 2 X L  and for any c   , where is the constant random variable with Another class of risk measures which is connected to the deviation measures in (Rockafellar, Uryasev and Zabarankin, 2003) is the one of expectation-bounded risk measures, which are defined as follows: Definition 1.2.An expectation-bounded risk measure satisfies the following properties: for any 2 X L  and for any c   , where is the constant random variable with is an expectation-bounded risk measure, while 2  L is partially ordered by the usual partial ordering (denoted by ) and implies , then R is coherent in the classical sense of (Artzner, Delbean, Eber, & Heath, 1999).The seminal survey (Rockafellar, Uryasev and Zabarankin, 2003) contains a lot of themes, such as examples of deviation and expectation-bounded risk measures (see Example 2 in (Rockafellar, Uryasev, & Zabarankin, 2003), Example 5 in (Rockafellar, Uryasev, & Zabarankin, 2003)), dual representation (see Theorem 3 of (Rockafellar, Uryasev, & Zabarankin, 2003)) and portfolio optimization results (see Theorem 4 in (Rockafellar, Uryasev, & Zabarankin, 2003), Theorem 5 in (Rockafellar, Uryasev, & Zabarankin, 2003)).Equilibrium in CAPM-like models in which deviation measures are used is studied in (Rockafellar, Uryasev, & Zabarankin, 2007).Also, results of quantile representation of law-invariant deviation  satisfies the following properties: C. E. KOUNTZAKIS measures are proved in (Grechuk, Molyboha, & Zabarankin, 2009).
The deviation measures were also studied in the published article (Rockafellar, Uryasev, & Zabarankin, 2006a).Since the properties of a deviation measure are similar to the ones of standard deviation (and this is the explanation for their name), there is also a connection of their properties to those of the class of expectation-bounded risk measures, see for example Theorem 1 in (Rockafellar, Uryasev, & Zabarankin, 2003).Expectation-bounded measures are a greater class than coherent risk measures (coherent risk measures are mainly studied in (Artzner, Delbean, Eber, & Heath, 1999), (Delbaen, 2002), (Jaschke & Küchler, 2001)).Hence we may say that deviation measures is a "bridge" which unifies an "older" and a "newer" aspect on risk functionals.Many of the main results of (Rockafellar, Uryasev, & Zabarankin, 2003) are transfered to (Rockafellar, Uryasev, & Zabarankin, 2006a).The major addition of the material contained in (Rockafellar, Uryasev, & Zabarankin, 2006a) compared to (Rockafellar, Uryasev, & Zabarankin, 2003) is the Paragraph 4, which is devoted to the error functionals and their relation to deviation measures.Specifically, (Rockafellar, Uryasev, & Zabarankin, 2006a) contains the above definition of deviation measures (Definition 1 in (Rockafellar, Uryasev, & Zabarankin, 2006a), while continuity and dual representation results are proved (Proposition 2 of (Rockafellar, Uryasev, & Zabarankin, 2006a), Theorem 1 of (Rockafellar, Uryasev, & Zabarankin, 2006a)).The relation between coherent and deviation measures is studied via the class of expectated-bounded risk measures (Theorem 2 of (Rockafellar, Uryasev, & Zabarankin, 2006a)).The last Theorem indicates that the values of an expectation-bounded measure R on the financial position   X L  , defines an expectation-bounded risk measure R .This Theorem is similar to the corresponding generalizations contained in the present article.We extend the content of the Paragraph 4 of (Rockafellar, Uryasev, & Zabarankin, 2006a) about deviation from error expressions in what we mention in this article about the relation between deviation measures in Banach spaces and seminorms.
The standard one-period problem of minimizing the deviation is studied in (Rockafellar, Uryasev, & Zabarankin, 2006b).The random variable is the linear combination of in which i are the rate of return variables of assets in L and is a portfolio vector which lies in a polyhedral set of constraints.The problem which arises here is the one of minimizing deviation subject to the polyhedral constraints.The problem is solved through subgradients which arise from the dual representation of the deviation measures in 2 L (see Theorem 1 of (Rockafellar, Uryasev, & Zabarankin, 2006a)).Optimal portfolios are discriminated according to the sum of their coefficients and the financial positions they provide are called master funds.Master funds are either of positive type, or of negative type, or of threshold type, see Theorem 5 in (Rockafellar, Uryasev, & Zabarankin, 2006b).For all sorts of master funds, CAPM-like relations are deduced, see Definition 3 of (Rockafellar, Uryasev, & Zabarankin, 2006b).In (Rockafellar, Uryasev, & Zabarankin, 2006c), the random variable is the convex combination of are the rate of return variables of assets in L , where denotes the risk-free asset rerurn.The problem which arises here is the one of minimizing deviation 0 r   D X subject to a threshold contraint which indicates that the return of the portfolio  at the time-period must be more than 0 1 r  


, where denotes an amount of money, denoting a risk premium.The existence of some solution to the above problem which is characterized initially either whether the price of the portfolio of the risky assets' price is negative, positive, or equal to 0, see Theorem 2 of (Rockafellar, Uryasev, & Zabarankin, 2006c).Master funds are also introduced in this case and efficiency frontiers of expectationdeviation type are studied, related to these master funds, see Paragraph 5 in (Rockafellar, Uryasev, & Zabarankin, 2006c).


We don't cope with master-funds' portfolio theory in this article.On the contrary, we propose a saddle-point scheme for the minimization of the deviation risk for the choices of an investor which belong to a set which is either bounded or unbounded.We consider different min-max Theorems (like the one mentioned in Corollary 3.4 of (Barbu & Precupanu, 1986), or like the one mentioned in p. 10 of (Delbaen, 2002), in order to prove the existence of solution to the problem of deviation risk minimization for reflexive and non-reflexive spaces.Finally we prove the existence of solution to the general minimization problem with convex contraints' set for the wellknown deviation measure The portfolio selection problem we study in this article may be compare with the ones contained in (Grechuk, Molyboha, & Zabarankin, 2011).In the Section 2.2 of (Grechuk, Molyboha, & Zabarankin, 2011) a cooperative portfolio selection problem is considered which is directly compared to the Markowitz portfolio selection problem in the case of a single investor.The difference is the use of deviation measures.
In the case of the single investor, the Markowitz' type problems-especially the risk minimization over a set of financial positions-is widely studied in our article.We have to mention that throughout the article, we refer to classes of deviation measures defined on Banach spaces whose partial ordering is not the pointwise one in order to indicate the generality of our results.Moreover, as we have also mentioned in (Kountzakis, 2011), the wedge E  (which may be actually a cone) by which the partial ordering of E is defined, is a way to interpret "the less and the more", or else when a financial position x is "of greater payoff" than the financial position whether y . Then a rational question is "Who thinks that E x y  "?A possible answer is "All (Some) of the investors of the market do".Let us denote the set of these investors by , which is a wedge of E , according to the properties of a coherent risk measure.Namely, the investors may decide to use a deviation measuse but previously they may have pre-determined by the way of comparing the financial positions according to their initial "risk preferences" indicated by an individual coherent acceptance set . Finally, the deviation measures are connected to actuarial science applications and the actuarial approach provided the main motivation about the definition of deviation measures on general Banach spaces.The random variable of the surplus of an insurance company at a future date is in general a heavy-tailed one, hence either the positive part , , E L p     may be naturally considered as the Banach space of the surplus positions, if the distribution of is such that leads to this result.Another motivation for this generalization is the actuarial definition of Solvency Capital, as it is mentioned in (Dhaene, Goovaerts, Kaas, Tang, Vanduffel, & Vyncke, 2003).In this review on risk measures and the notion of solvency the following definition of capital requirement functional for an insurance company is given: if represents the time-T liabilities of an insurance company and X  X  K X is the economic capital associated with these liabilites, while is the value of them calculated either by a quantile method, or by an additional margin method, or by a replicationg portfolio method, then if the risk measure used is which is a liability variable.If X    , or else the pricing functional is considered without the margin term, then we may take that In these case the insurance company calculates its own Solvency Capital with respect to a generalized risk measure (for example a deviation one), so that it may be acceptable by the regulator.Since the liability variable is a heavy-tail dis-X tributed one, the moments L to be the model in which we work.This is a motivation for the use of deviation measures on Banach spaces except 2 L .Another motivation related to financial applications is the class of p G L -spaces, which are actually Banach spaces related to G-expectation, see in (Peng, 2007).We may suppose that the variables which denote the value of the portfolios at a certain future date , belong to such spaces, since martingale theory according to the G-expectation is related to the T 2 G L space, as (Soner, Touzi, & Zhang, 2011) indicates.Hence, we may consider the case of definition of deviation measures on this class of Banach spaces.Also, a reference about considering stochastic models of markets under model uncertainty is (Denis & Martini, 2006).But the definition and the study of deviation measures on p G L -spaces should require a separate article.

Deviation Measures on Banach Spaces
First we remind the definitions of convex and coherent risk measures associated to the Monotonocity property related to the partial ordering defined on E by some wedge A of it.
In the following we refer to the notions of the   which satisfies the properties  satisfies the following properties: x for any x E  and for any c   and any for any x E  and for any 0 The definition of the K -expectation-bounded risk measures is the following: Definition 2.4.A K -expectation-bounded risk measure   satisfies the following properties:  and for any 0 risk measure, while is partially ordered by the usual partial ordering induced by (denoted by P ) and  for any x E  and for any 0   .This property also holds due to the definition of D R and the equivalent property of as a deviation measure, namely for any , x x E   .By the same way, we have that , where for for any x E  and for any 0   .From the definition of R D and the equivalent property of the -expectation bounded risk measure K R , we have and for every 0 By the same way we have that Let us see some examples, classes of deviation risk measures which are defined on partially ordered Banach spaces by using coherent risk measures, which are actually expectation-bounded risk measures.
Corollary 2.7.Suppose that such that the functionals of are strictly positive functionals of and x K  and any .On the other hand, for any , then the functional , where for every 0 and the definition of the risk measure R , we have that then . On the other hand, if x E K   , then there is some such that we may repeat the same argument for B is the base defined by e on .Then which is defined by where and e B is the base defined by on .Then which is defined by (Kountzakis, 2011)), it is also a -expectation bounded risk measure, since .Hence, by the Proposition 2.6 (Rockafellar, Uryasev and Zabarankin, 2003) the deviation measures are defined on 2 L spaces, we may state and prove similar Corollaries for the usual (component-wise partial ordering) of p L spaces with . 1 p   We rely on the unified dual representation Theorem 2.9 of (Kaina & Rüschendorf, 2009) in order to state the following: Corollary 2.10.
where is such that q 1 1 1  p q  , while 1 M denotes the set of  -continuous probability measures on the measurable space .Let us denote by Q the functional lying in the base defined on q L  by 1 .Here we refer to the case where .,1 and for any , where . From the definition of c   c   D  and the Translation Invariance of the risk measure  , we have that 1 p   and for any 0   .From the definition of D  and the Positive Homogeneity of  , we have By the same way we have that , then there is some and this implies that , then we may repeat the same argument for x E K    .Another example of K -deviation measures arises if we depart from the component-wise partial ordering of p L -spaces.Example 2.11.
and we may suppose that is partially ordered by the wedge .Then every  is weakly closed, is represented in the way that Theorem 3.5 of (Kountzakis, 2011) indicates: We have to verify that also the subspace of the constant random variables.
It suffices to prove that satisfies the properties of a D K -deviation risk measure. 1) and for any c   , where c   .From the definition of and the Translation Invariance of the risk measure for every 0 . By the same way we have that , which implies that in this case.Also, we may notice that if x y k   1 for some 0 k   .But from the first property , hence it suffices to prove that , then there is some y , then apply again the previous argument for . 0 Since the value of a risk measure at any financial position has both the financial and the actuarial interpretation of the premium, the term   x  corresponds to a standard term of the risk premium, which is related to the geometry of the acceptance set.When the acceptance set is the positive cone 2 L  of the space of the square-integrable risks, then this standard term is the mean value , since in this case.The last case is the usual attitude towards risk, under which a non-risky position is a position whose outcomes are positive The subspace K mentioned in the Definition 2.3 above, may be considered to be a subspace of non-risky assets.For this reason, the addition of such an asset does not affect the premium calculation, according to the first property of the Kdeviation measures.However, the whole theory of -deviation measures can be developed without reference to the partial ordering.

K
Consider a proper subspace of assets (which are considered to be the non-risky ones), denoted by K .
Definition 2.12.A K -deviation risk measure   : 0, D E   satisfies the following properties: 1) for any x E  and for any c   , where , where and for for any x E  and for any 0 The definition of the K -expectation-bounded risk measures is the following: Definition 2.13.A K expectation-bounded risk measure for any for any x K  and for any 0 Proof.The conclusion is immediate, since by property is positively homogeneous and by property is subadditive, hence it is sublinear, according to Definition 5.32 of (Aliprantis and Border, 1999).This implies by Lemma 5.33 of (Aliprantis and Border, 1999) that the function defined by by the properties of maximum of real numbers.Hence D g is Subadditive.Also, by Homogeneity Property of , we have that   and also by well-known properties of maximum of real numbers, for the same reason.Also, if 0 x for any x E  and any    .
The same proof may be repeated for K -deviation measures defined on partially ordered spaces. Corollary It suffices to prove that satisfies the properties of a -deviation measure.
for any x E  and for any k K  .This holds due to the subadditivity property of the seminorm according to which, p for any x E  and for any 0 for any    and any for any , x x E   , from the subadditivity of the seminorm p .

4)
for any , then it belongs to some co-set of the form 0 , where .Then for some .This implies .
Again, by the above Proposition, we obtain another Corollary for the deviation measures which were initially defined on , is actually a K -deviation measure.
The same proof may be repeated for -deviation measures defined on partially ordered spaces in the sense we defined them before, hence we obtain the following , where e B   , is actually a -deviation measure.K Proof.In both of cases of 2 L and the case of the above Corollary, we repeat the proof of Proposition 2.17 In the case of 2 L we replace by c , while in the case of an ordered Banach space we replace by , where .
, where : For the subadditivity of I p we have that from the well-known properties of the suprema of subsets of real numbers.Also, about the positive homogeneity of I p we have that . For the inverse inclusion, suppose that . Then, I p is actually a -deviation measure.

Support Functionals and the Dual Characterization of -Deviation Measures K
In this Section we extend the duality characterization Theorem Theorem 1 of (Rockafellar, Uryasev, & Zabarankin, 2003) which is proved in the case where the space of financial positions is 2 L in the case of -deviation measures being defined on Banach spaces.
where F E   is non-empty, weak-star closed and convex, E    is a linear functional which corresponds to a "standard premium term" for any x E  , , where is a wedge of 0 is finite-valued then this is equivalent to the fact that D F is bounded.Proof.Since is a lower semicontinuous -deviation measure, by Theorem 5.104 of (Aliprantis & Border, 1999) is the support functional of the weak-star closed, convex subset of The last Theorem implies that . But also for the inverse inclusion, we get that if If we suppose that the functional provides a standard "premium term", we define F is also a weak-star closed, convex subset of E  .Then in terms of F we also take the following dual representation: If D F is a bounded set then is a bounded set and this implies that is finite-valued, because , where 0 M  is an upper bound for the norms of the elements f from the Uniform Boundedness Principle and this implies that This is actually a characterization of K -deviation risk measures defined on a Banach space E .For the inverse direction of the proof, suppose that the functional where F E   is non-empty, weak-star closed.Then is a lower semicontinuous D K -deviation measure, where Let us verify the properties of these risk measures: 1) from the properties of supremum. 3)  , and this holds from the definition of K .On the other hand if x E K   then there is some 0 we have and .

 
Also, is a lower semicontinuous function defined on because it is the supremum of a family of lower semicontinuous functions on .The family is the set of linear functionals

The Min-Max Approach on the Risk Minimization for Deviation Risk Measures in L 2
In this section we consider the following risk-minimization portfolio-payoff selection problem: where  is a risk measure (not necessarily coherent) and is a portfolio-payoff selection set.


The subject of this section is to investigate the saddle-value form of the solution for the problem 1, if  is some deviation measure in the sense defined in (Rockafellar, Uryasev and Zabarankin, 2003).
It is well-known that the portfolio selection problem 1 is a part of the efficient portfolio selection theory and practice, see (Markowiz, 1952), (Kroll, Levy, & Markowitz, 1984).
We remind that the classic form of a zero-sum game between two players has as payoff function the bilinear form of a dual pair , X X  and the strategy set of the one player may be identified by a set X   , while the strategy set of the other player may be identified by some . The payoff , x x  is understood to be a reward paid from the first player to the second.By selecting x   , the first players' maximum loss is max , . By choosing a proper strategy , he may 0 x   achieve to pay to the second player no more than the minimum of these losses, which is equal to 0 min max , , if this quantity is well-defined.On the other hand, for any strategy of the second player the minimum payoff he earns is and by choosing a proper strategy x    , he may achieve to receive from the first player at least the maximum of these earnings, which is equal to 0 max min , , if this quantity is well-defined.
holds and if the equality holds, then the common value is called saddle-value, while the pair   which is the solution point of the game, is called saddle-point.We may replace the bilinear form ,   by another payoff function F defined on and the notions are repeated in the same form.For a brief explanation on zero-sum games which leads to the min-max theorems, see in (Luenberger, 1969).Also, a primal reference for zero-sum games is (von Neumann, 1928).The saddle value , can be interpreted as the value of a zero sum game between two players.The one player minimizes   , F x y over  supposing that the other player follows the strategy x , while the other player maximizes

 
, y  y F x over supposing that the other player follows the strategy , see also (Kountzakis, 2011).

closed and convex sets, F is an upper-lower semicontinuous, concave-convex function on
, then Also, we give the following definitions of the payoff functions: Definition 4.1.A function for all .y   2) for every and  are convex sets for every and .
is concave in the first variable and convex in the second variable.
According to Theorem 3 in (Rockafellar, Uryasev, & Zabarankin, 2003), by considering some set of elements consisted by random variables such that is a subset of the base of 2 L  2 defined by the constant random variable which is a strictly positive functional of it.The set as it is mentioned in p. 17 of (Rockafellar, Uryasev and Zabarankin, 2003) is considered to be a convex and closed of the base defined by on of a deviation measure if  is convex, closed and bou- nded and we consider some financial positions' choice set for an investor denoted by , which has the same properties and it is a subset of L , drives us wonder whether Corollary 3.7 of (Barbu & Precupanu, 1986) and its game-theoretic implication can be applied in the case of the risk minimization problem.The boundedness of  in this case simplifies the saddle- value solution of the problem.
Actually, we suppose that we have the following version of the risk minimization problem 1: Apart from the Proposition 2 in (Rockafellar, Uryasev, & Zabarankin, 2003) which indicates that finite-valued deviation measures on 2 L being lower semicontinuous are norm-continuous, we prove a stronger result than Proposition 2 in (Rockafellar, Uryasev, & Zabarankin, 2003), since it indicates that they are Lipschitz continuous in the case we consider.
Proposition 4.4.Any deviation measure , where is a convex and bounded subset of But if is a norm-bounded set, this implies that  is a Lipschitz function.This is true because for two families of functions such that  , : and for any 2 , X Y L  that satisfy the above finite suprema conditions, By the same way we have that , where 1 Q  because since  is convex, closed and bounded subset of a reflexive space, it is a weakly compact subset of it and the supremum in for some upper bound of the norms of the elements of . Proposition 4.5.If we suppose that and are convex, closed and bounded, the problem 2 has a solution.

 
Proof.Since  is a norm-continuous function, then the problem 2 has a solution, since is also weakly lower semicontinuous and is a weakly compact set.
Since the problem 2 has a solution, it has an optimal value.We will investigate whether this optimal value is a saddle value, according to Corollary 3.7 of (Barbu & Precupanu, 1986).
The duality form of , implies that the candidate twovariable function for the application of Corollary 3.7 of (Barbu & Precupanu, 1986) is For this function we have the following.Proposition 4.6.The function satisfies the properties of Corollary 3.7 of (Barbu & Precupanu, 1986), hence the optimal value of the risk minimization problem 2 is a value of the function Proof.F is upper-lower semicontinuous, because it is norm-continuous in both of its variables.Moreover, it is linear in both of its variables, which implies that it is concave-convex.Hence the conclusion is true from Corollary 3.7 of (Barbu & Precupanu, 1986).
The economic interpretation of the fact that the risk minimization problem is solved through determining a saddle-point of the function F is the following: The minimization of risk corresponds to a zero-sum game between the investor and the market.The payoff function of the game-the one which is minimized by the investor as a cost function for a given "valuation" density over the set of financial positions is the partial function .The function being maximized as a "value" function for a specific financial position . The value of the game, which is also the optimal value of the risk minimization problem 2 is achieved at a saddle point  0 0 , Q X .This meets the notion of a "two-person zero-sum game" for one more reason, because the market can be viewed as a whole to which the monetary cost of the risk minimization is paid (the one player) and the investor can be viewed as the other player who earns the monetary payoff concerning a certain financial position X , which is formulated by the market as the value of it.To be more accurate, suppose that the set of strategies of the market is the set of the valuation measures , while the set of strategies of the investor is the set of the financial positions .If we select some . By choosing a proper strategy 1 , she may achieve to pay to the second player (to the market) no more than the minimum of the above costs, being , if this quantity is well-defined.On the other hand, the market for any strategy of it, the minimum payoff that it earns from the investor is and by choosing a proper strategy 1 , it may achieve to receive from the investor at least the maximum of these earnings which is X is a solution to the deviation minimization problem 2. For a similar explanation on saddle-value form that minimization of convex risk measures may take, see also in (Kountzakis, 2011).

The Risk Minimization for Deviation Measures on
Reflexive Spaces: Bounded Sets In this section we prove the existence of solution to the problem of minimization of deviation if the deviation measure comes from a certain class of coherent risk measures.
Specifically, if we transfer the above results to the frame of the commodity-price duality , E E  , where the space E denotes a reflexive space in which the financial positions lie in, then we get a saddle-point solution result for the following minimization problem where is a convex, closed, bounded subset of , is closed and and , and holds for any π B  , while for any . The functional is defined as follows:   , : In order to apply Corollary 3.7 of (Barbu & Precupanu, 1986) in this case, we have to determine the payoff function : are convex, closed and bounded subsets of the reflexive spaces , X Y , B and has to be a concave-convex and upper-lower semicontinuous function.We notice that . F is concaveconvex and upper-lower semicontinuous.Then a saddle-point x According to the saddle-point conditions for  

The Minimization of Deviation Measures in Banach Spaces: Unbounded Sets
The question which arises is whether the above min-max approach for the minimization of deviation measures can be generalized in the case of an unbounded choice set of financial (risk) positions.The answer is affirmative due to an alternative min-max theorem reminded in p. 10 of (Delbaen, 2002).We also focus on the classes of deviation measures related to the coherent measures arising from ordering cones with non-empty interior.
Specifically, the statement of the previously mentioned min-max theorem is the following: Let K be a compact, convex subset of a locally convex space .Let Y L be a convex subset of an arbitrary vector space X .Suppose that is a bilinear function Then we have the following Theorem 5.1.Suppose that is a reflexive space.Consider the problem where is a convex, unbounded subset of  E , is closed and P intP   , and e intP  , .The closed subspace is such that for any is defined as follows: Proof.If we apply the previous min-max theorem, we have that Y E   endowed with the weak topology, is the one specified by assumptions.Also, for any   , the partial function   , : and hence for weakly and for eac y parti h specific es that an al function , x u x E x   .This impli  is continuous, which is also valid for any x so, u is a bilinear function as it arises from its defi .Since base e weakly the  niti  on .Al B of the cone P is convex and weakly compact, the set B i sio s weakly comp t and convex, too.Also, the set  is convex and the conditions for the validity of the conclun of the previous min-max theorem hold.Hence the minmax equation holds for u , which implies the existence of a saddle-point

 
Then the al and e B  a is the base defined by e on P .Then 5 has solution.
Proof.I the problem x theorem, f we apply tha the previous min-ma e we hav t Y E   endowed with the weak-star topology, and for each specific plies that artial function , x u x E x   .This im  is weak-star continuous, which is also valid for any x   .Also, u is a bilinear function as it arises from its defi Also, ce the base e nition.sin B  is weak-star compact and convex base of the cone P , the the set 1 n B is a weak-star compact and convex subset of E  and the set  is a convex subset of E , then the condit s for the vali y of the conclusion of previous min-max theorem hold.Hence the min-max equation holds for u , which implies the existence of a saddle-point ion dit the , .
nt is The existence of a sadd i implied by Proposition 3.1 at of we (Barbu & Precupanu, 1986) which says that a function satisfies the min-max equality if and only if it has a saddle-point.
Remark 5.3.We remind for the sake of completeness of wh proved in the last two Theorems that the fact that if e intP  then 0 P has a -compact base is men-Propo on 13.8.1 eson, 1970).The weakstar compactness of bases defined by elements of tioned in siti 2 in (Jam E on cones of E  is implied in non-reflexive spaces from the roposition Proposition 2.4 of (Kountzakis, 2011), which is actually a reference to Theorem 39 of (Xanthos, 2009)  res, according to the results containing in (Kusuoka, 2001).These properties of a CVaR may make it very attractive in applications, since it replace a VaR .Also, as it is mentioned in (Rockafellar, Uryasev, & Zabar 2003), a shortfall relative to expectation is more adequate in practice.A very interesting application of the saddle-point method in order to verify the existence of solution to the minimization of deviation risk is also by the use of min-max Theorem mentioned in p. 10 of (Delbaen, 2002) in the case of the "deviation which arises from expeced shortfall", which is defined as the functional As it is well-known from Acerbi and Tasche, 2002) and (Tasche, 2002) the expected shortfall   a ES x for a financial position and a level of significance   0,1 is defined in Definition 2.6 of (Acerbi & Tasche, 2002) as the negative of tail-mean of a  x at the level a , being equal to . But for the probability measures of the repr  -compact due to Lemma 7.54 of (Aliprantis & 99).We also have to prove that a Z is weak-star . We have to prove that f is a Radon-Nikodym derivative of some measure with respect to and the definition of , the fact that a y characteristic function . We may also refer to the Monotone Convergence Theorem (11.17 in (Aliprantis & Border, 1999), where the restriction of the f on the set ∪ is the integrable function which is mentioned in the Theorem, while n f is the restriction of f on a set of the form   (Rockafellar, Uryasev, & Zabarankin, 2003), see p. 7 in (Rockafellar, Uryasev, & Zabarankin, 2003).
Hence, we have the following risk minimization problem The existence of solution to the risk minimization problem 6 does not depend on the fact whether the set positions which is the selection set of the investor is bo of financial  unded or not.Theorem 5.5.
, then the deviation risk minimization problem 6 has a solution.
Proof.We will apply the min-max theorem reminded in p. 10 of ndowed w (Delbaen, 2002).We have that Y L   e ith the weak-star topology, The existence of a saddle-point is implied Proposition 3.1 of (Barbu & Precupanu, 1986) which says that a function satisfies the min-max equality if and only if it has a saddle-point.All the previously mentioned notions and related propositions concerning partially ordered linear spaces are contained in (Jameson, 1970).
A topological linear space is is boundedly order complete if for every bounded increasing net in the space E E X , the supremum of the elements of it exists.A cone of a linear topological space P E is called Daniell cone if every increasing net of E which is upper bounded converges to its supremum.
Note that every well-based cone in a Banach space which has a base defined by a continuous linear functional.Every closed, well-based cone in a Banach space is a Daniell cone.Every Banach space partially ordered by a closed, well-based cone is a boundedly order-complete space.
A  Kountzakis, 2011).The family of these cones in a normed linear space is the following: A proof for the existence of interior points in these cones is contained in p. 127 of (Jameson, 1970).
Another family of cones with non-empty interior is the family of Henig Dilating cones.These cones are defined as follows: Consider a closed, well-based cone in the normed linear space C E , which has a base B , such that where   convex risk measures (where is partially ordered by the partial ordering relation induced by the wedge E A of it), whose definitions are the following: Definition 2.1.A function

K
Proposition 2.21.If a -deviation measure is of the form K I p indicated in the Example 2.20, it is Lipschitz-continuous. Proof.According to what is indicated in the Example 2 The problem 3 has a solution via saddlepoints.
a q x denotes er quantile of the a -lowx .T Z he deviatio sure D is introduced in Example 4 of (Rockafellar,Uryasev, & abarankin, 2003).Also, expected shor- .e. and   0 f   ,  -a.e.In order to show L  , all the conditions of the min-max Theorem reminded in p.10 of (Delba valid.Hence th in-max equation holds for , whic plies the existence of a saddle-point  en, 200 .
aES is a coherent risk measure on  , then E is called normed lattice.Finally, we remind that the usual partial ordering of an C ,