#### Abstract

We prove a general dual representation form for restricted coherent risk measures, and we apply it to a minimization problem of the required solvency capital for an insurance company.

#### 1. Overview of the Paper

The aim of this paper is to propose a min-max scheme for the specification of the solvency capital for an insurance company. We actually suppose that the initial surplus of the company is equal to . According to the evolution of the premiums and the investment payoffs of the company both with the evolution of the cumulative claim obligations till a certain time-period , the final surplus of the company is formulated by the corresponding stochastic process . In order to anticipate the probable risk which arises from the variability of at the different states of the world, the insurance company needs a capital being expressed either in units of money or in an amount of some shares of a numeraire asset. In order to measure this risk, we use a risk measure, which is in general one of the form , it is more rational to define the risk measure on the set , being the set of financial positions which represent the final surplus variables for the company, than defining the risk measure on the entire space (which is in general an ordered normed linear space), where these financial positions lie in. Also, the value is in general excluded from the duality representations we deduce, hence we refer to risk measures . Classical properties of coherent and convex risk measures, like translation invariance, are not transferred in the exact form they have in articles of unrestricted risk measures. Relative definitions are given in the following sections. In the present paper, we deduce a dual representation result for such restricted coherent risk measures and we also propose a min-max method of solution to the above solvency capital optimization problem, which is rightly stated if we suppose that the capital constraint holds, where represents the initial capital of the insurance company. We use the dual representation of the corresponding risk measure which provides inf-compactness in this optimization problem. This is the main point of the paper. The proof of a dual representation theorem for restricted coherent risk measures which are defined on wedges of reflexive spaces is given. The domain of the risk measure we desire to minimize is a subspace of random variables, as we prove in the following. The specification of the surplus position which minimizes a coherent risk measure of this kind, under the constraint implies a threshold of solvency capital needed by this way for any feasible (in the sense that holds) surplus position. Namely, the optimal value of the problem is the part of the initial capital that the insurance company expends in order to anticipate the risk, no matter what will happen (and which will be the final surplus of the company). The existence of a minimum point is also deduced. The calculation of the solvency capital—or else of this threshold—indicates a decision criterion for the insurance company, because its optimal value is going to be spent by the company. The sole calculation of the risk for any surplus variable with respect to does not involve any decision criterion. Hence a decision criterion for the solvency of the insurance company described by the model below may be indicated by the solution of the minimization of the risk subject to the constraint .

#### 2. Previous Results and Essential Notions

First, we remind two general definitions in which is a wedge of the space .

Definition 2.1. A real-valued function which satisfies the properties(i)  (Translation Invariance),(ii)  for any   (Convexity) and, (iii)  implies   (-Monotonicity),where is called -convex risk measure.

Definition 2.2. A real-valued function is a -coherent risk measure if it is an -convex risk measure and it satisfies the following property: for any and any (Positive Homogeneity).
In [1, Theorem  2.12], the coherent or convex (see the Definition 5.8) risk measures which are defined on a domain which is a proper subset of are called restricted. We have to remind that in [1], the restricted convex risk measures are studied on cash-invariant subsets of -spaces, where . A dual representation theorem is proved there [1, Theorem  2.12] for -convex risk measures. A primal reference for the study of restricted convex risk measures is [2], in which such risk measures are studied in and mainly by means of theory of conjugate convex functions defined on normed linear spaces. A similarity between the frame of [2] and the frame we propose is that authors consider a convex cone which implies a partial ordering for a locally convex topological vector space and a numeraire . The examples of [2] focus on normed linear spaces and moreover on spaces ordered by their usual (componentwise) partial ordering. In [2, Lemmas  3.5 and  3.6], the conditions are directly related to the boundedness of the base that the numeraire asset, (see also [3, Theorem  4.4.4]). It is valid only in the case where , because is an interior point of . In the present paper, we are going to define restricted risk measures on wedges of ordered normed linear spaces containing the numeraire asset , which may be proper subspaces of reflexive spaces-like the spaces . We define the restricted measures on such spaces, because this arises from our financial model (Itô processes correspond to spaces according to Itô Isometry). As it is well known the -spaces are associated to the existence of the moments for , where is a -measurable random variable. If all these moments are real numbers. Of course, the model of risk measurement would be set in a different mode in which the insurance company examines the effects of its decisions continuously, and it calculates the equivalent solvency capital according to the evolution of its investment portfolio, by using dynamic risk measures (see, e.g., [46]). But the intention of the present paper is to provide some results on static, restricted coherent risk measures and emphasizing in the application we discussed above. However, as it also quoted in [7] the time moments and cannot be very far, because the control of the decision effects of an insurer, even if it is set in a static place, cannot refers to a time interval which is great enough. Moreover in practice, the insurance companies do not pay their clients as far as the claims arrive, but after a certain time in which they can anticipate the total risk that they adopted by those claims partially by their investment portfolio and partially by the solvency capital they shaped. However, is indicated as the solvency capital under is indicated also for example in [8, page  3]. For more details about an actuarial definition of Solvency Capital Requirement functionals, we refer to [9]. If represents the time- liabilities of an insurance company and is the economic capital associated with these liabilities, while is the value of them calculated either by a quantile method, or by an additional margin method, or by a replicating portfolio method, then if the risk measure used is , the solvency capital for is equal to . The functionals are connected to by the identity for any which is a liability variable. This notion of the solvency capital partially excuses a static frame than a dynamic one, but a dynamic one is however accurate, too. We may say that the frame of the present paper is exactly the one of [7] expressed in the case of restricted coherent risk measures. However, the presentation of the results is autonomous since it is related to the solution of the problem (7.1). This problem is related to the determination of the solvency capital for a set of financial positions (which are liability variables of an insurance company) which are not consisted of a sole element. The minimization of a coherent risk measure over a set of positions as a problem is however studied for example in [10, 11], mainly for . The Duality Representation Theorem for restricted coherent risk measures on wedges (see Theorem 6.6) also relies both on the strong separation theorem for convex sets in locally convex spaces and on the Hahn-Banach extension theorem for linear functionals, by which we reach the primal [12, Theorem  2.3], which refers to the case of coherent risk measures defined on . In this case, is ordered by its usual partial ordering and the numeraire asset is 1, which is an interior point of . The important is that without the assumption of the existence of interior points in the positive cone which was provided from the idea of the application of the Krein-Rutman Theorem [3, Corollary  1.6.2] and the extension result [3, Proposition  3.1.8]. This is a point of similarity to the equivalent result in [7]. This is the cause of presence of the classes of cones in Examples 3.1 and 3.2 in this paper.

#### 3. A Financial Model for the Surplus Evolution of an InsuranceCompany

The model we consider is described as follows. We assume a closed interval for some constant representing the time-horizon in our model and an infinite set of states of the world . We suppose that all the transaction decisions take place at time . The state faced by the investors is contained in some event , where represents some -algebra of subsets of which provides information about the states being available at time . We denoted by the objective probability measure on the measurable space and we consider the probability space . The information evolution along time-horizon is represented by a filtration of .

These integrability conditions are implied from [15, Definition  2.1, page  7]. We also may suppose that for any the random variables are not necessarily stochastically independent, since the insurance company may borrow an amount of money during an infinitesimal time and this money can be invested for buying an amount of shares of the risky asset. The model presented here is an extension of the model described in [14]. Moreover, the claim payment process may be represented by an Itô process where is the initial total claim and is a one-dimensional Brownian motion with respect to the same filtration , being independent from . Hence, the stochastic process of the surplus of the insurance company is an Itô process (by consideration of the Itô Lemma) and its dynamics are demonstrated in the form of a stochastic differential equation: where . With notation of for the stochastic differential of the 2-dimensional Brownian motion ( are independent) and , , while , , the stochastic differential equation for is obtained in the form where . For the risk model we propose, we may refer to [16] in which both the capital of the insurance company and the total liabilities of the company are modelled as simple diffusion processes. The stochastic process is a 2-dimensional Brownian motion with respect to the filtration generated by it, that means . Hence, the surplus process can be considered as a financial asset whose drift is and whose volatility process with respect to the 2-dimensional Brownian motion is , , with realizations as vectors of . It is not restrictive to require where denotes the Lebesgue measure on , for each portfolio strategy . It implies that This holds when for example If we consider the financial market consisted by the surplus asset and the “bank account” asset whose dynamics are driven by the stochastic differential equation where denotes the short-term rate process, to conclude (see [17, Theorems  12.1.8, and  12.2.5]) that the market is incomplete. This criterion of incompleteness was established in [15, Theorem  6.6, page  24].

The attainable final surplus variables are in this case a subspace denoted by . This set is actually The fact that is a subspace under the strategies is deduced as follows: if for some corresponding strategies and also for some corresponding strategies , then where .

The set of the attainable surplus variables of the specific market is not equal to for any Equivalent Martingale Measure of it. Hence we consider as the space in which the positions lie in, any of the spaces for any Equivalent Martingale Measure of the market consisted by the surplus asset and the bank account asset. Every such is defined as follows: The market prices of risk for the surplus-bond market are given by the following process equation for : The equivalent Radon-Nikodym derivative of with respect to is

The spaces are reflexive; therefore, we can refer to the same frame of ordered normed linear spaces which appears in [7], where the main results are proved under the assumptions that is an ordered linear space with nonempty cone interior. As in [7], the order structure of is given by a closed wedge which is not necessarily the positive cone of the pointwise partial ordering if . It is in general a wedge whose elements may be financial positions whose outcomes are negative at some of the states of the world. The geometry of the positive wedge reflects the joint beliefs of the investors about the payoffs of these positions. The beliefs of the investors determine the set of the positions which are ‘‘jointly considered to be the nice investments.’’ Actually, we mean that the positive cone is itself an acceptance set in the sense of the definition contained in [18]. Moreover, we mention two families of cones with nonempty interior, since the results of the present paper join this frame. An element of these families may represent the positive cone of .

Example 3.1. A family of cones in normed linear spaces having nonempty cone-interior are the Bishop-Phelps cones (see in [7]). The family of these cones in a normed linear space is the following: An explanation for the existence of interior points in these cones is found in [3, page  127].

Example 3.2. Another family of cones with nonempty 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 , which has a base , such that . Let be such that where denotes the closed unit ball in . If then , is a cone for any ,. About these cones, see for example [19, Lemma  2.1]. For example, a Bishop-Phelps cone in a reflexive space which is a well-based cone as the construction of the requires, provides a set of interior points of the cone . If we consider the base defined by , this base is a closed set where . Hence there is a such that for any . can be selected to be such that , hence . By setting , we may construct a sequence of approximating cones , since we can set , . We remind that if is a convex set, then the set is a wedge and by , we denote its norm (or weak) closure.
We remind that in general, the acceptance set of a risk measure (where the infinity value may be omitted) is the set . If the risk measure is restricted, namely it is of the form , where , the acceptance set of is .

#### 4. Some Quotes on the Content of the Paper

The insurance company determines a numeraire asset which corresponds to an insurable claim and belongs to an acceptance set . This set , namely demonstrates the exposure limits of risk for this insurance company. If , then the company can anticipate the risk arising from the variability of among the states of . is a wedge of , because it should satisfy the main properties of the acceptance sets of the coherent risk measures introduced in [18]:(i). (ii).

We remind that a set satisfying and for any is called wedge. A wedge for which is called cone. The coherence of the acceptance set is related to the property of Monotonicity. In [18, Ax. 2.1], a coherent acceptance set is supposed to contain the cone of the space in which the financial positions lie in. In the case where the risk measure is defined in the whole subspace , this property can be expressed in the following way: which indicates that the cone of the induced partial ordering of the subspace of is contained in the acceptance set. The acceptance set is not necessarily a subset of because the risk exposure limits are rather independent from the specific surplus process . Also, if we talk about convex risk measures which are not coherent is an unbounded convex subset of containing .

If is a subspace of , this subspace becomes a partially ordered linear space itself by using the order structure of . In this case, we call an ordered subspace of whose positive wedge is . For the rest notions related to the partially ordered linear spaces, the reader may append to the Appendix of [7] or to the brief Appendix at the end of this article.

If , then the value can be understood as the solvency capital for the financial position under or else as the minimum amount of shares of that allows jointly with these shares of to belong to the acceptance set of ,

On the other hand, the solvency position that the insurance company needs in order to be insured in this case is . Since the numeraire asset may be normalized according to a price-functional , namely a strictly positive functional of such that , then the solvency capital of with respect to is equal to . Also note that if the space is reflexive, then the numeraire asset defines a base on the cone , because . This base is weakly compact due to the fact that is an interior point of and also due to [3, Theorem  3.8.6(i)] which indicates that if a wedge of a locally convex space has an interior point, then has a -compact base, which is actually the base defined by this interior point. This is due to [3, Theorem  3.8.12] which indicates that If is an ordered normed linear space with positive wedge , then a linear functional of is uniformly monotonic if and only if it is an interior point of with respect to the norm topology of . We may come to the same conclusion by using [20, Theorem  4]. About the notation—even it is explained in the Appendix—we have that . This is the reason of considering families of cones with nonempty interior. They actually provide the conditions for the solution of the optimization problem (7.1), which is mentioned below.

#### 5. Restricted Coherent and Restricted Convex Risk Measures

Restrictions in cash invariance or actually in numeraire invariance may appear in the attempt of the insurance company to draw the limits of its risk exposure. The sorts of invariance are described in the following definitions.

Definition 5.1. The subset is - translation invariant (cash translation invariant) if () and for any and any ).

Definition 5.2. The subset is weak - translation invariant (weak cash translation invariant) if there is some such that for any ), where .

Lemma 5.3. A wedge of is weak -translation invariant with respect to any element of it.

Proof. If , then for any , . Hence is weak -translation invariant.

The definition of translation invariance for a risk measure defined on a wedge takes a more general form.

Definition 5.4. Let be a wedge of and be a function . is -translation invariant if for any and any such that .

Definition 5.5. Let be a wedge of , be a function , and be a cone of . Then is -monotone (or equivalently it satisfies the -monotonicity), if the inclusion holds.

Definition 5.6. Let be a wedge of and the function satisfy the properties of -Translation Invariance, Subadditivity, Positive Homogeneity and -Monotonicity. Then we call it -restricted coherent.

Definition 5.7. A risk measure satisfies the Convexity property if it is a convex function for any and .

Definition 5.8. If is a wedge of the ordered linear space and a function satisfies the properties of -translation invariance, convexity, and -monotonicity, then it is called -restricted convex.
The rest of results concerning the primal insurance and the reinsurance companies’ risk measures are simple implications of the following results.

Remark 5.9. The main condition under which the following results hold, is that is a (closed) wedge of an ordered normed space (which is reflexive if it is required and equal to some appropriately defined -space, too) such that is a nonempty wedge of containing the numeraire .

#### 6. Dual Representation of Restricted Coherent Risk Measures

The following Theorem relies on Hahn-Banach extension theorem for linear functionals defined on subspaces of linear spaces.

Theorem 6.1 ((Krein-Rutman) [3, Corollary  1.6.2]). Let be an ordered linear space and be a linear subspace of it containing an order unit of . Then a positive linear functional defined on has a positive extension to .

The following Proposition assures the continuity of this extension.

Proposition 6.2 ([3, Pr. 3.1.8]). Let be an ordered locally convex space with positive wedge and suppose that . Let be a linear subspace containing and a positive linear functional defined on . Then has a continuous, positive extension to .

The above theorem and proposition are crucial for the similarity between the proof of the Theorem of dual representation for coherent risk measures in case where the domain is the whole reflexive space and the cone interior is nonempty (like the ones studied in [7]). The results in [7] and other related articles cannot be proved in the restricted case, because the Separation Theorem for convex sets cannot be applied in this case, in the sense that we desire the separating functionals to lie in . Thus, we assure that every positive linear functional has a continuous extension all over , by the previous theorem and proposition.

For the application of the above theorem and the above proposition we need the following.

Lemma 6.3. If is a wedge of then is a subspace of .

Proof. It suffices to verify that the set is closed under the addition of vectors and the scalar multiplication. If , then , where . Then
and by the properties of as a wedge, . Also, if , and by the properties of as a wedge, . Finally if and by the properties of as a wedge, .

Lemma 6.4. The set is a cone of .

Proof. Obviously, is a wedge. In order to prove that it is a cone, it suffices to prove that Suppose that . Then . Hence, .

Corollary 6.5. The set is a subspace of .

We remark—as it is also mentioned in the Appendix—that denotes the natural embedding map from to the second dual space of .

Theorem 6.6. If is a closed wedge of a reflexive ordered normed space with nonempty cone interior containing the -interior point , is a -restricted coherent risk measure with -closed acceptance set , then for any , where . On the other hand, every defined through (6.3), is a -restricted coherent risk measure.

Proof. If we consider a -restricted coherent risk measure , then admits a representation like the one indicated in (6.3). To see this, we remark that for any and for any we have that
The singleton is a convex, weakly compact set and is by assumption a weakly closed set of which is also convex (since it is a wedge), since is an -restricted coherent risk measure. Since these two sets are disjoint, from the Strong Separation Theorem for convex sets in locally convex spaces there is some , an and a such that for any . Hence we take that
We have to notice that every depends on . The functional takes negative values on since if there is some such that , then for any , we take . Then if being a contradiction according to the previous separation argument. Then since we have that is a positive linear functional on the subspace . But from Theorem [3, Corollary  1.6.2] and Proposition [3, Proposition  3.1.8] since is an order unit of which belongs to , has a continuous, linear extension all over . For the sake of brevity, we denote again this functional by and we have that . If we suppose that , then we would take for any . This is true since , for appropriate , for any , because is an order-unit of . Hence . But since by assumption, this implies for any . Hence because . This implies that the initial assumption that is not true and holds. Hence we may suppose that holds, or else .
Since lies on , we have by the above separation inequalities and the fact that that
Hence if we denote by we have that for any . On the other hand, we have that for any , . This implies for any , since and since and this implies . implies for any . This implies . Finally we get that for any and any , This implies , for any .
For the opposite direction, it suffices to show that any , defined through (6.3), is an -restricted coherent risk measure. For this, we have to verify that satisfies the properties of a -restricted coherent risk measure as follows.(i)(-translation invariance): for any and for these for which .(ii)(Subadditivity): where and because is a wedge.(iii)(Positive homogeneity): where and for any , since is a wedge.(iv)(-Monotonicity): If where in terms of the partial ordering of induced by , then , hence for any since . Then by taking suprema over we have .

Remark 6.7. A difference between the representation Theorem [1, Theorem  2.12] and the previous dual characterization theorem for restricted coherent risk measures is that our result holds for closed wedges and for coherent risk measures defined on reflexive spaces, while [1, Theorem  2.12] holds for any convex, closed, and cash-invariant subset of some space with . Also, in [1, Theorem  2.12] an initial lower semicontinuous, unrestricted measure is mentioned, so that the equivalent restricted one is weakly (weak-star in the case of ) continuous. Such conditions are absent in our Theorem and the only condition is that the acceptance set of the restricted coherent measure is weakly closed.

#### 7. The Optimization Problem

The optimization problem that an insurance company faces at time-period is the following: where denotes the surplus of the insurance company at time-period . The problem (7.1) expresses the attempt of the insurance company to minimize the solvency capital given an initial economic capital constraint. The set is the subspace of the attainable surplus variables. A direct method of determination of optimal points of on , is via the min-max theorem reminded by Delbaen in [12, page  10]. Specifically, the statement of the previously mentioned min-max theorem is the following: Let be a compact, convex subset of a locally convex space . Let be a convex subset of an arbitrary vector space . Suppose that is a bilinear function . For each , we suppose that the partial (linear) function is continuous on . Then we have that

The case of the spaces mentioned are mainly the case where and is any EMM of the financial market shaped by the surplus asset of the insurance company and the bank-account asset whose interest rate process is constant and equal to . Under the above assumptions, we have the following.

Theorem 7.1. The problem (7.1) has a solution.

Proof. Suppose that is the space endowed with the weak topology. represents the convex set of the constraints. The compact set is the weakly compact base defined by the numeraire asset on the cone . The bilinear function we need in order to apply the above min-max Theorem is defined as follows: (), where The partial function is weakly continuous, since for each and each weakly convergent net , we have that , where is the net’s weak limit. Hence we have that , for any . This implies that weak continuity holds, since the images of through for any converge to the image of through , too. Hence, by Representation Theorem 6.6 and by weak compactness of the base we have that because the following result holds.

Proposition 7.2 (see [21, Propsition  3.1]). A function satisfying the min-max equality if and only if it has a saddle-point.

The existence of the last saddle-point implies the existence of a minimization point of over , hence a solution to (7.1).

Hence, the minimum solvency capital is in this case or else the part of the capital that will be surely subtracted, so that the risk which arises from the cumulative obligations of the company to be eliminated (with respect to ), if we suppose that the constraint is active.

#### 8. A Note for the Specification of the Optimal Solution

We proved the existence of solution through saddle-point theory and inf-compactness. If we would like to specify optimal strategies which lead to an optimal solution , we actually have to restate the optimization problem (7.1) as follows: while the evolution of is given through where is a 2-dimensional Brownian motion and the claims’ SDE is where the Brownian motion is the second component of the above 2-dimensional Brownian motion, which is a 1-dimensional Brownian motion itself. Finally, Equations (8.1)–(8.4) describe a stochastic control problem.

The interesting point is that if we completely restate the problem (8.1)–(8.4) according to the dual Representation Theorem 6.6, we take: while the evolution of is given through where is a 2-dimensional Brownian motion and the claims’ SDE is where the Brownian motion is the second component of the above 2-dimensional Brownian motion, which is a 1-dimensional Brownian motion itself. Finally,

In the statement indicated by (8.5)–(8.8), are normalized functionals and not probability measures, like in classical approaches such as [15] in the Chapter which refer to utility optimization; hence, there are crucial differences related to the model presented here.

#### Appendix

In this paragraph, we give some essential notions and results from the theory of partially ordered linear spaces which are used in the previous sections of this paper.

Let be a (normed) linear space. A set satisfying and for any is called wedge. A wedge for which is called cone. A pair where is a linear space and is a binary relation on satisfying the following properties:(i) for any (reflexive),(ii)If and then , where (transitive),(iii)If then for any and for any , where (compatible with the linear structure of ),is called partially ordered linear space.

The binary relation in this case is a partial ordering on . The set is called (positive) wedge of the partial ordering of . Given a wedge in , the equivalent partial ordering relation defined as follows: is a partial ordering on , called partial ordering induced by on . If the partial ordering of the space is antisymmetric, namely if and implies , where , then is a cone. If is partially ordered by , then any set of the form where is called order-interval of . is an order unit of , if , where is the order-interval of and we suppose that is ordered by , while . If is an interior point, then is an order unit of . If is an order unit and is a Banach space, then is an interior point of . denotes the linear space of all linear functionals of , while is the norm dual of , in case where is a normed linear space.

Suppose that is a wedge of . A functional is called positive functional of if for any . is a strictly positive functional of if for any . A linear functional where is a normed linear space, is called uniformly monotonic functional of if there is some real number such that for any . In case where a uniformly monotonic functional of exists, is a cone. for any is the dual wedge of in . Also, by we denote the subset of . It can be easily proved that if is a closed wedge of a reflexive space, then . If is a wedge of , then the set is the dual wedge of in , where denotes the natural embedding map from to the second dual space of . Note that if for two wedges of holds, then . If is a cone, then a set is called base of if for any there exists a unique such that . The set where is a strictly positive functional of is the base of defined by . is bounded if and only if is uniformly monotonic. If is a bounded base of such that then is called well based. If is well based, then a bounded base of defined by a exists.

The partially ordered vector space whose positive cone is is a vector lattice if for any , the supremum and the infimum of with respect to the partial ordering defined by the cone exist in . In this case and are denoted by and , respectively. If so, is the absolute value of and if is also a normed space such that for any , then is called normed lattice.

Finally, we remind that the usual partial ordering of an space, where is a probability space is the following: if and only if the set is a set lying in of -probability .

All the previously mentioned notions and related propositions concerning partially ordered linear spaces are contained in [3].

A subset of a convex set in is called extreme set or else face of , if whenever , where and implies . If is a singleton, is called extreme point of .

A subset of some Euclidean space of the form , where is called half-space and more specifically closed half-space of . The intersection of finitely many closed half-spaces is called polyhedral set.