# Restricted Coherent Risk Measures and Actuarial Solvency

Authors: Christos E. Kountzakis
Journal: Advances in Decision Sciences
 (2012)

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## 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.


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## Body

## 1. Overview of the Paper

The aim of this paper is to propose a min-max scheme for the specification of thesolvency capital for an insurance company. We actually suppose that the initial surplus of the company is equal to u. 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 T, the final surplus UT of the company is formulated by the corresponding stochastic process {Ut,t∈[0,T]}. In order to anticipate the probable risk which arises from the variability of UT 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 ρ:𝒰→ℝ∪{∞}, it is more rational to define the risk measure on the set 𝒰, being the set of financial positions which represent the final surplus variables UT for the company, than defining the risk measure on the entire space L (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 ρ:𝒰→ℝ. 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 ρ(UT)≤u holds, where u 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 𝒰 of random variables, as we prove in the following. The specification of the surplus position UT which minimizes a coherent risk measure ρ:𝒰→ℝ of this kind, under the constraint ρ(U)≤u,U∈𝒰 implies a threshold of solvency capital needed by this way for any feasible (in the sense that ρ(U)≤u holds) surplus position. Namely, the optimal value of the problem is the part of the initial capital u that the insurance company expends in order to anticipate the risk, no matter what will happen (and which will be the final surplus UT=U 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 U∈𝒰 with respect to ρ:𝒰→ℝ 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 ρ(U)≤u,U∈𝒰.


## 2. Previous Results and Essential Notions

First, we remind two general definitions in whichA is a wedge of the space L.Definition 2.1.
A real-valued functionρ:L→ℝ∪{+∞} which satisfies the properties(i)
ρ(x+ae)=ρ(x)-a  (Translation Invariance),(ii)
ρ(λx+(1-λ)x)≤λρ(x)+(1-λ)ρ(y)  for any  λ∈[0,1] (Convexity) and,(iii)
y≥Ax  implies  ρ(y)≤ρ(x) (A-Monotonicity),where x,y∈L is called (A,e)-convex risk measure.Definition 2.2.
A real-valued functionρ:L→ℝ∪{+∞} is a (A,e)-coherent risk measure if it is an (A,e)-convex risk measure and it satisfies the following property: ρ(λx)=λρ(x) for any x∈E and any λ∈ℝ+ (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 Lp,1≤p≤∞ are called restricted. We have to remind that in [1], the restricted convex risk measures are studied on cash-invariant subsets of Lp-spaces, where 1≤p≤∞. A dual representation theorem is proved there [1, Theorem  2.12] for (L+p,1)-convex risk measures. A primal reference for the study of restricted convex risk measures is [2], in which such risk measures are studied in L∞ 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 𝒫 which implies a partial ordering for a locally convex topological vector space L and a numeraire Π∈L∖{0}. The examples of [2] focus on normed linear spaces and moreover on Lp 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 p=∞, because 1 is an interior point of L+∞. In the present paper, we are going to define restricted risk measures on wedges of ordered normed linear spaces containing the numeraire asset E, which may be proper subspaces of reflexive spaces-like the spaces Lp,1<p<∞. We define the restricted measures on such spaces, because this arises from our financial model (Itô processes correspond to L2 spaces according to Itô Isometry). As it is well known the Lp(Ω,ℱ,μ)-spaces are associated to the existence of the moments 𝔼μ(Xp) for 1≤p<∞, where X:Ω→ℝ is a ℱ-measurable random variable. If X∈L∞ 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., [4–6]). 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 0 and T 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, ρ(UT) is indicated as the solvency capital under ρ 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 X represents the time-T liabilities of an insurance company and K(X) is the economic capital associated with these liabilities, while P(X) 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 ρ, the solvency capital for X is equal to ρ(X). The functionals K,P are connected to ρ by the identity ρ(X)=K(X)+P(X) for any X 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 CVaR. 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 L∞(Ω,ℱ,μ). In this case, L∞ is ordered by its usual partial ordering and the numeraire asset is 1, which is an interior point of L+∞. The important is that without the assumption of the existence of interior points in the positive cone L+ 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[0,T] for some constant T>0 representing the time-horizon in our model and an infinite set of states of the world Ω. We suppose that all the transaction decisions take place at time 0. The state ω∈Ω faced by the investors is contained in some event A∈ℱ, where ℱ represents some σ-algebra of subsets of Ω which provides information about the states being available at time T. We denoted by μ the objective probability measure on the measurable space (Ω,ℱ) and we consider the probability space (Ω,ℱ,μ). The information evolution along time-horizon [0,T] is represented by a filtration 𝔽=(ℱt)t∈[0,T] of (Ω,ℱ,μ).We consider a diffusion model for the capital of an insurance company. We assume an insurance company, whose𝔽-adapted capital process (Ct)t∈[0,T] evolves according to the following stochastic differential equation;
(3.1)dCt=(m(t,Ct)-ϕtrtBt)dt+θts(t,Ct)dWt(1),μ—a.e.,
where C0=c is some constant which denotes the initial capital of it and W(1) is one-dimensional Brownian motion with respect to 𝔽. The form of this stochastic differential equation is explained as follows: during an infinitesimal time, the company receives premiums equal to m(t,Ct). The company borrows money from a “bank account asset,” in order to invest to a risky asset, whose volatility process is s(t,Ct),t∈[0,T]. The process pt=∫0t‍m(s,Cs)ds may be interpreted as the total premium process, or else it indicates the total premium payments that the insurance company receives till time-period t. The process {m(t,Ct),t∈[0,T]} is the premium payment intensity and we may suppose that the pure diffusion term ∫0t‍θus(u,Cu)dWu(1) corresponds to the payoff of the investment of these received premia to some risky assets whose volatility-term is s(t,Ct),t∈[0,T] with respect to a one-dimensional Brownian motion W(1). The premia are invested to this asset as soon as they are received by the company. Bt,t∈[0,T] denotes the stochastic process which represents the evolution of the “bank account” asset. Hence, we have dBt=rtBtdt and B0=1. We may suppose that the interest-rate process (rt)t∈[0,T] of the asset which represents a bank account is uniformly bounded, or else that rt(ω)≤M for any (t,ω)∈[0,T]×Ω. It is usual in the recent literature, the capital process (Ct)t∈[0,T] to have a diffusion form (see, e.g., [13, 14]). The set of admissible trading strategies for the investment of the capital of the insurance company to the risky asset is denoted by Θ={(θt)t∈[0,T]}. These processes are adapted to the filtration 𝔽 of (Ω,ℱ,μ). Θ includes those processes for which θt(ω)=at,at∈ℝ,t∈[0,T] for any ω∈Ω. Φ is a set of admissible trading strategies for the bank account asset, while Θ and Φ have the same properties. The integrability properties that the investment strategies ϕ∈Φ,θ∈Θ have to satisfy are the following:
(3.2)∫0T|θtrt|dt<∞,μ—a.e.,∫0T|ϕtrt|dt<∞,μ—a.e.,∫0Tθt2s2(t,Ct)dt<∞,μ—a.e.These integrability conditions are implied from [15, Definition  2.1, page  7]. We also may suppose that for any t∈[0,T] the random variables θt,ϕt 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
(3.3)dZt=q(t,Zt)dt+σ(t,Zt)dWt(2),μ—a.e.,
where Z0=z is the initial total claim and W(2) is a one-dimensional Brownian motion with respect to the same filtration 𝔽, being independent from W(1). Hence, the stochastic process {Ut,t∈[0,T]} 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:
(3.4)dUt=(m(t,Ut+Zt)-ϕtBtrt-q(t,Zt))dt+[θts(t,Ut+Zt),-σ(t,Zt)]·[dWt(1)dWt(2)],μ—a.e.,
where U0=u=c-z. With notation of dWt=[dWt(1)dWt(2)] for the stochastic differential of the 2-dimensional Brownian motion W (W(1),W(2) are independent) and l(t,Ut,Zt,ϕt)=m(t,Ut+Zt)+ϕtBtrt-q(t,Zt), t∈[0,T], while v(t,Ut,Zt,θt)=[θts(t,Ut+Zt),-σ(t,Zt)], t∈[0,T], the stochastic differential equation for {Ut,t∈[0,T]} is obtained in the form
(3.5)dUt=l(t,Ut,Zt,ϕt)dt+v(t,Ut,Zt,θt)dWt,μ—a.e.,
where U0=u. 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 W=(W(1),W(2)) is a 2-dimensional Brownian motion with respect to the filtration 𝔽W generated by it, that means 𝔽W⊆𝔽. Hence, the surplus process can be considered as a financial asset whose drift is l(t,Ut,Zt,ϕt) and whose volatility process with respect to the 2-dimensional Brownian motion W is v(t,Ut,Zt,θt), t∈[0,T], with realizations as vectors of ℝ2. It is not restrictive to require
(3.6)vt(Ut(ω),Zt(ω),θt(ω))≠0,λ[0,T]⊗μ—a.e.,
where λ[0,T] denotes the Lebesgue measure on [0,T], for each portfolio strategy (θt)t∈[0,T]. It implies that
(3.7)rankvt(Ut(ω),Zt(ω),θt(ω))=1,λ[0,T]⊗μ—a.e.
This holds when for example
(3.8)σ(t,Zt(ω))≠0,λ[0,T]⊗μ—a.e.
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
(3.9)dBt=rtBtdt,B0=1,μ—a.e.,
where r 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 by𝒰. This set is actually
(3.10)𝒰={U∈L2(Ω,ℱ,μ)∣U=u+∫0T‍l(t,Ut,Zt,ϕt)dt+∫0T‍v(t,Ut,Zt,θt)dWt}.
The fact that 𝒰 is a subspace under the strategies is deduced as follows: if
(3.11)U1=u+∫0T‍l(t,Ut,Zt,ϕt1)dt+∫0T‍v(t,Ut,Zt,θt1)dWt,
for some corresponding strategies ϕ1,θ1 and also
(3.12)U2=u+∫0T‍l(t,Ut,Zt,ϕt2)dt+∫0T‍v(t,Ut,Zt,θt2)dWt,
for some corresponding strategies ϕ2,θ2, then
(3.13)λ1·U1+λ2·U2=u+∫0T‍(m(t,Ut+Zt)-(λ1·ϕt1+λ2ϕt2)Btrt-q(t,Zt))dt+∫0T‍[(λ1·θt1+λ2θt2)s(t,Ut+Zt),-σ(t,Zt)]·[dWt(1)dWt(2)]=u+∫0T‍l(t,Ut,Zt,λ1ϕt1+λ2ϕt2)dt+∫0T‍v(t,Ut,Zt,(λ1θt1+λ2θt2))dWt∈𝒰,
where λ1,λ2∈ℝ.The set of the attainable surplus variables𝒰 of the specific market is not equal to L=L2(Ω,ℱTW,Q) for any Equivalent Martingale Measure Q of it. Hence we consider as the space in which the positions U=UT lie in, any of the spaces L2(Ω,ℱTW,Q) for any Equivalent Martingale Measure Q of the market consisted by the surplus asset and the bank account asset. Every such Qθ,ϕ is defined as follows: The market prices of risk for the surplus-bond market are given by the following process equation for u:
(3.14)[θts(t,Ut+Zt),-σ(t,Zt)]·ut=m(t,Ut+Zt)+ϕtBtrt-q(t,Zt)-rt.
The equivalent Radon-Nikodym derivative of Qu with respect to μ is
(3.15)dQudμ=exp{-∫0T‍utdWt-12∫0T‍∥ut∥22dt}.The spacesL2(Ω,ℱTW,Qu) 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 L is an ordered linear space with nonempty cone interior. As in [7], the order structure of L is given by a closed wedge L+ which is not necessarily the positive cone L+p of the pointwise partial ordering if L=Lp. 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 L+ 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 L+ of L.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 L is the following:
(3.16)K(f,a)={x∈L∣〈x,f〉≥a∥x∥},f∈L*,∥f∥=1,a∈(0,1).
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 coneC in the normed linear space L, which has a base B, such that 0∉B+δB(0,1)¯. Let δ∈(0,1) be such that
(3.17)2δB(0,1)∩B=∅,
where B(0,1) denotes the closed unit ball in L. If
(3.18)Kn=cone¯(B+δjB(0,1)),j∈ℕ,
then C⊆Kj+1⊆Kj,j∈ℕ, Kj is a cone for any j≥2,C∖{0}⊆int(Kj),j≥1. About these cones, see for example [19, Lemma  2.1]. For example, a Bishop-Phelps cone C=K(f,a) in a reflexive space which is a well-based cone as the construction of the Kn,n∈ℕ requires, provides a set of interior points C∖{0} of the cone Kj,j≥1. If we consider the base Bf={x∈C∣〈x,f〉=1} defined by f, this base is a closed set where 0∉Bf. Hence there is a g≠0,g∈L* such that g(y)≥δ'>0 for any y∈Bf. g can be selected to be such that ∥g∥=1, hence ∥y∥≥g(y)≥δ'>0. By setting δ'=2δ, we may construct a sequence of approximating cones Kn,n∈ℕ, since we can set g=f, δ'=a∈(0,1). We remind that if D is a convex set, then the set cone(D)={x∈L∣x=λd,d∈D,λ∈ℝ+} is a wedge and by cone¯(D), we denote its norm (or weak) closure.
We remind that in general, theacceptance set𝒜ρ of a risk measure ρ:L→ℝ∪{+∞} (where the infinity value may be omitted) is the set 𝒜ρ={X∈L∣ρ(X)≤0}. If the risk measure is restricted, namely it is of the form ρ1:𝒰→ℝ∪{+∞}, where 𝒰⊆L, the acceptance set of ρ1 is 𝒜ρ1={X∈𝒰∣ρ1(X)≤0}.


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

The insurance company determines a numeraire assetE which corresponds to an insurable claim and belongs to an acceptance set 𝒜. This set 𝒜, namely demonstrates the exposure limits of risk for this insurance company. If U∈𝒜, then the company can anticipate the risk arising from the variability of U among the states of Ω. 𝒜 is a wedge of L, because it should satisfy the main properties of the acceptance sets of the coherent risk measures introduced in [18]:(i)
𝒜+𝒜⊆𝒜.(ii)
λ𝒜⊆𝒜,λ∈ℝ+.We remind that a setC⊆L satisfying C+C⊆C and λC⊆C for any λ∈ℝ+ is called wedge. A wedge for which C∩(-C)={0} is called cone. The coherence of the acceptance set 𝒜 is related to the property of Monotonicity. In [18, Ax. 2.1], a coherent acceptance set is supposed to contain the cone L+ of the space in which the financial positions lie in. In the case where the risk measure is defined in the whole subspace 𝒰, this property can be expressed in the following way:
(4.1)𝒰∩L+⊆𝒜,
which indicates that the cone of the induced partial ordering of the subspace 𝒰 of L is contained in the acceptance set. The acceptance set 𝒜 is not necessarily a subset of 𝒰 because the risk exposure limits are rather independent from the specific surplus process (Ut,t∈[0,T]). Also, if we talk about convex risk measures which are not coherent 𝒜 is an unbounded convex subset of L containing L+.IfD is a subspace of L, this subspace becomes a partially ordered linear space itself by using the order structure of (L,≥). In this case, we call D an ordered subspace of (L,≥) whose positive wedge is D∩L+. 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.Ifρ:𝒰→ℝ, then the value ρ(U),U∈𝒰 can be understood as the solvency capital for the financial position U under ρ or else as the minimum amount of shares of E that allows U jointly with these shares of E to belong to the acceptance set of ρ,
(4.2)𝒜ρ={U∈𝒰∣ρ(U)≤0}.On the other hand, thesolvency position that the insurance company needs in order to be insured in this case is ρ(U)E. Since the numeraire asset E may be normalized according to a price-functional π∈L*, namely a strictly positive functional of L+ such that 〈E,π〉=1, then the solvency capital of E with respect to ρ is equal to -1. Also note that if the space L is reflexive, then the numeraire asset E defines a base on the cone L+0, because L+00=(L+0)0=L+. This base is weakly compact due to the fact that E is an interior point of L+ and also due to [3, Theorem  3.8.6(i)] which indicates that if a wedgePof a locally convex spaceLhas an interior point, thenP0has aσ(L*,L)-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 L is an ordered normed linear space with positive wedge P,then a linear functional f of L is uniformly monotonic if and only if it is an interior point of P0 with respect to the norm topology of L*. 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 L+0={π∈L*∣〈X,π〉≥0,∀X∈L+}. 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 subsetN⊆L is E- translation invariant (cash translation invariant) if E∈N (1∈N) and for any U∈N and any λ∈ℝ,U+λE∈N(U+λ1∈N).Definition 5.2.
The subsetN⊆L is weakE- translation invariant (weak cash translation invariant) if there is some t∈ℝ such that for any U∈N,U+λE∈N(U+λ1∈N), where λ≥t.Lemma 5.3.
A wedgeN of L is weak E-translation invariant with respect to any element E of it.Proof.
IfU∈N, then for any t≥0, U+tE∈N. Hence N is weak E-translation invariant.The definition of translation invariance for a risk measure defined on a wedgeN takes a more general form.Definition 5.4.
LetN be a wedge of L and ρ be a function ρ:N→ℝ. ρ is E-translation invariant if
(5.1)ρ(U+aE)=ρ(U)-a,
for any U∈N and any a∈ℝ such that U+aE∈N.Definition 5.5.
LetN be a wedge of L, ρ be a function ρ:N→ℝ, and L+ be a cone of L. Then ρ is L+-monotone (or equivalently it satisfies the L+-monotonicity), if the inclusion
(5.2)N∩L+⊆𝒜ρ,
holds.Definition 5.6.
LetN be a wedge of L and the function ρ:N→ℝ satisfy the properties of (L+,E)-Translation Invariance, Subadditivity, Positive Homogeneity and L+-Monotonicity. Then we call it (L+,E)-restricted coherent.Definition 5.7.
A risk measureρ:N→ℝ satisfies the Convexity property if it is a convex function
(5.3)ρ(λU1+(1-λ)U2)≤λρ(U1)+(1-λ)ρ(U2),
for any λ∈[0,1] and U1,U2∈N.Definition 5.8.
IfN is a wedge of the ordered linear space L and a function ρ:N→ℝ satisfies the properties of (L+,E)-translation invariance, convexity, and L+-monotonicity, then it is called (L+,E)-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 thatN is a (closed) wedge of an ordered normed space L (which is reflexive if it is required and equal to some appropriately defined L2-space, too) such that N∩L+ is a nonempty wedge of L containing the numeraire E.


## 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]).
LetL be an ordered linear space and J be a linear subspace of it containing an order unit of L. Then a positive linear functional defined on J has a positive extension to L.The following Proposition assures the continuity of this extension.Proposition 6.2 ([3, Pr. 3.1.8]).
LetL be an ordered locally convex space with positive wedge P and suppose that E∈intP. Let J be a linear subspace containing E and f a positive linear functional defined on J. Then f has a continuous, positive extension to L.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 spaceL and the cone interior intL+ 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 L*. Thus, we assure that every positive linear functional has a continuous extension all over L, by the previous theorem and proposition.For the application of the above theorem and the above proposition we need the following.Lemma 6.3.
IfD is a wedge of L then D-D is a subspace of L.Proof.
It suffices to verify that the setD-D is closed under the addition of vectors and the scalar multiplication. If y1,y2∈D-D, then y1=x1-x2,y2=x3-x4, where x1,x2,x3,x4∈D. Then
(6.1)y1+y2=(x1+x3)-(x2+x4),
and by the properties ofD as a wedge, x1+x3∈D,x2+x4∈D. Also, if λ≥0, λy1=λx1-λx2 and by the properties of D as a wedge, λx1∈D,λx2∈D. Finally if λ<0,λy1=(-λ)x2-(-λ)x1 and by the properties of D as a wedge, (-λ)x1∈D,(-λ)x2∈D.Lemma 6.4.
The setD=N∩L+ is a cone of L.Proof.
Obviously,D=N∩L+ is a wedge. In order to prove that it is a cone, it suffices to prove that
(6.2)D∩(-D)={0}.
Suppose that n∈D∩(-D). Then n∈L+,n∈(-L+). Hence, n∈L+∩(-L+)={0}.Corollary 6.5.
The setN∩L+-N∩L+ is a subspace of L.We remark—as it is also mentioned in the Appendix—that^:L→L** denotes the natural embedding map from L to the second dual space L** of L.Theorem 6.6.
IfN is a closed wedge of a reflexive ordered normed space L with nonempty cone interior containing the L+-interior point E, ρ:N→ℝ is a (L+,E)-restricted coherent risk measure with σ(L,L*)-closed acceptance set 𝒜ρ, then
(6.3)ρ(U)=sup{〈-U,π〉∣π∈BE},
for any U∈N, where BE={y*∈L+0∣E^(y*)=1}. On the other hand, every ρ defined through (6.3), is a (L+,E)-restricted coherent risk measure.Proof.
If we consider a(L+,E)-restricted coherent risk measure ρ, then ρ admits a representation like the one indicated in (6.3). To see this, we remark that for any U∈N and for any ϵ>0 we have that
(6.4)U+(ρ(U)-ϵ)E∉𝒜ρ.
The singleton{U+(ρ(U)-ϵ)E} is a convex, weakly compact set and 𝒜ρ is by assumption a weakly closed set of L which is also convex (since it is a wedge), since ρ is an (L+,E)-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 ℓ∈L*,ℓ≠0, an α∈ℝ and a δ>0 such that
(6.5)〈U+(ρ(U)-ϵ)E,ℓ〉≥α+δ>α≥〈V,ℓ〉
for any V∈𝒜ρ. Hence we take that
(6.6)〈U+(ρ(U)-ϵ)E,ℓ〉>sup{〈V,ℓ〉∣V∈𝒜ρ}.
We have to notice that everyℓ depends on U. The functional ℓ takes negative values on N∩L+ since if there is some Z0∈N∩L+∖{0} such that 〈Z0,ℓ〉>0, then for any λ∈ℝ+, we take λZ0∈N∩L+. Then if λ→+∞(6.7)〈λ·Z0,ℓ〉>〈U+(ρ(U)-ϵ)E,ℓ〉,
being a contradiction according to the previous separation argument. Then since we have that -ℓ is a positive linear functional on the subspace J=N∩L+-N∩L+. But from Theorem [3, Corollary  1.6.2] and Proposition [3, Proposition  3.1.8] since E is an order unit of L which belongs to N∩L+, -ℓ has a continuous, linear extension all over L. For the sake of brevity, we denote again this functional by -ℓ and we have that -ℓ∈L+0. If we suppose that 〈E,-ℓ〉=0, then we would take 〈U,-ℓ〉=0 for any U∈L. This is true since U∈[-n0(U)E,n0E], for appropriate n0(U)∈ℕ, for any U∈L, because E is an order-unit of L. Hence n0(U)〈E,ℓ〉≤-〈U,ℓ〉≤-n0(U)〈E,ℓ〉. But since -〈E,ℓ〉=0 by assumption, this implies 〈U,-ℓ〉=0 for any U∈L. Hence -ℓ=0 because L=∪n=1∞[-nE,nE]. This implies that the initial assumption that 〈E,-ℓ〉=0 is not true and 〈E,-ℓ〉>0 holds. Hence we may suppose that
(6.8)〈E,-ℓ〉=1
holds, or else -ℓ∈BE.
Since-ℓ lies on BE, we have by the above separation inequalities and the fact that 0∈𝒜ρ that
(6.9)〈U,ℓ〉-ρ(U)+ϵ>0.
Hence if we denote-ℓ by π we have that ρ(U)<〈-U,π〉+ϵ≤sup{〈-U,π〉∣π∈BE}+ϵ for any ϵ>0. On the other hand, we have that for any U∈N, U+ρ(U)E∈𝒜ρ. This implies 〈U,π〉+ρ(U)≥0 for any π∈BE, since BE⊆L+0 and since L+⊆𝒜ρ and this implies 𝒜ρ0⊆L+0. π(U)+ρ(U)≥0 implies ρ(U)≥〈-U,π〉 for any π∈BE. This implies ρ(U)≥sup{〈-U,π〉∣π∈BE}. Finally we get that for any U∈N and any ϵ>0,
(6.10)sup{〈-U,π〉∣π∈BE}≤ρ(U)<sup{〈-U,π〉∣π∈BE}+ϵ.
This implies ρ(U)=sup{〈-U,π〉∣π∈BE}, for any U∈N.
For the opposite direction, it suffices to show that anyρ:N→ℝ, defined through (6.3), is an (L+,E)-restricted coherent risk measure. For this, we have to verify that ρ satisfies the properties of a (L+,E)-restricted coherent risk measure as follows.(i)
(E-translation invariance):
(6.11)ρ(U+aE)=sup{〈-U-aE,π〉∣π∈BE}=sup{〈-U,π〉-a∣π∈BE}=sup{〈-U,π〉∣π∈BE}-a=ρ(U)-a,
for any U∈N and for these a∈ℝ for which U+aE∈N.(ii)
(Subadditivity):(6.12)ρ(U1+U2)=sup{π(-U1-U2)∣π∈BE}≤sup{〈-U1,π〉∣π∈BE}+sup{〈-U2,π〉∣π∈BE}=ρ(U1)+ρ(U2),
where U1,U2∈N and U1+U2∈N because N is a wedge.(iii)
(Positive homogeneity):(6.13)ρ(λU)=sup{〈-λ·U,π〉∣π∈BE}=sup{λ·〈-U,pi〉∣π∈BE}=λ·sup{〈-U,π〉∣π∈BE}=λ·ρ(U),
where λ∈ℝ+ and λ·U∈N for any U∈N, since N is a wedge.(iv)
(L+-Monotonicity): If U′≥U where U,U′∈N in terms of the partial ordering of L induced by L+, then -U≥-U′, hence
(6.14)〈-U,π〉≥〈-U′,π〉
for any π∈BE since BE⊆L+0. Then by taking suprema over BE we have ρ(U)≥ρ(U′).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 Lp space with 1≤p≤∞. 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 L∞) continuous. Such conditions are absent in our Theorem and the only condition is that the acceptance set of the restricted coherent measure ρ is weakly closed.


## 7. The Optimization Problem

The optimization problem that an insurance company faces at time-period0 is the following:
(7.1)Minimizeρ(U)subjecttoU∈𝒰,ρ(U)≤u,
where u denotes the surplus of the insurance company at time-period 0. The problem (7.1) expresses the attempt of the insurance company to minimize the solvency capital given an initial economic capital constraint. The set 𝒰 is the subspace of the attainable surplus variables. A direct method of determination of optimal points of ρ on N1={U∈𝒰,ρ(U)≤u}, 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: LetKbe a compact, convex subset of a locally convex spaceY. LetVbe a convex subset of an arbitrary vector spaceM. Suppose thatFis a bilinear functionF:M×Y→ℝ. For eachl∈V, we suppose that the partial (linear) functionF(l,·)is continuous onY. Then we have that(7.2)infl∈Vsupk∈KF(l,k)=supk∈Kinfl∈VF(l,k).The case of the spaces mentioned are mainly the case whereL=L2(Ω,ℱ,Q) and Q 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 r. Under the above assumptions, we have the following.Theorem 7.1.
The problem (7.1) has a solution.Proof.
Suppose thatM is the space L=L2(Ω,ℱ,Q) endowed with the weak topology. V represents the convex set N1⊆L2(Ω,ℱ,Q) of the constraints. The compact set K is the weakly compact base BE defined by the numeraire asset on the cone L+0. The bilinear function we need in order to apply the above min-max Theorem is defined as follows: F:M×K→ℝ (F:L2×L2→ℝ), where
(7.3)F(x,π)=〈-x,π〉.
The partial function Fx:L2→ℝ is weakly continuous, since for each x∈L2 and each weakly convergent net (πa)a∈A⊆L2, we have that 〈x,πa〉→〈x,π〉, where π is the net’s weak limit. Hence we have that 〈-x,πa〉→〈-x,π〉, for any x∈L2=M. This implies that weak continuity holds, since the images of (πa)a∈A through Fx for any x converge to the image of π through Fx, too. Hence, by Representation Theorem 6.6 and by weak compactness of the base K=BE we have that
(7.4)infU∈N1ρ(U)=infU∈N1supπ∈BEF(U,π)=supπ∈BEinfU∈N1F(U,π)=F(U*,π*)=ρ(U*),
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(U*,π*) implies the existence of a minimization point U* of ρ over N1, hence a solution to (7.1).Hence, the minimum solvency capital isρ(U*) in this case or else the part of the capital u that will be surely subtracted, so that the risk which arises from the cumulative obligations of the company to be eliminated (with respect to ρ), if we suppose that the constraint ρ(U)≤u 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(θ*,ϕ*)∈Θ×Φ which lead to an optimal solution UT*(θ*,ϕ*), we actually have to restate the optimization problem (7.1) as follows:
(8.1)Minimizeρ(UT(θ,ϕ)),subjecttoU∈𝒰,ρ(U)≤u,
while the evolution of Ut,t∈[0,T] is given through
(8.2)dUt=l(t,Ut,Zt,ϕt)dt+v(t,Ut,Zt,θt)dWt,
where Wt=(Wt(1),Wt(2)),t∈[0,T] is a 2-dimensional Brownian motion and the claims’ SDE is
(8.3)dZt=q(t,Zt)dt+σ(t,Zt)dWt(2),
where the Brownian motion Wt(2),t∈[0,T] is the second component of the above 2-dimensional Brownian motion, which is a 1-dimensional Brownian motion itself. Finally,
(8.4)θ∈Θ,ϕ∈Φ.
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:
(8.5)Minimizesupπ∈BE〈-UT(θ,ϕ),π〉,subjecttoU∈𝒰,ρ(U)≤u,
while the evolution of Ut,t∈[0,T] is given through
(8.6)dUt=l(t,Ut,Zt,ϕt)dt+v(t,Ut,Zt,θt)dWt,
where Wt=(Wt(1),Wt(2)),t∈[0,T] is a 2-dimensional Brownian motion and the claims’ SDE is
(8.7)dZt=q(t,Zt)dt+σ(t,Zt)dWt(2),
where the Brownian motion Wt(2),t∈[0,T] is the second component of the above 2-dimensional Brownian motion, which is a 1-dimensional Brownian motion itself. Finally,
(8.8)θ∈Θ,ϕ∈Φ.In the statement indicated by (8.5)–(8.8), π 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.



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*Source: 350765-2012-12-19.xml*