Equity allocation and portfolio selection in insurance
Erik Taflin
AXA, 23, Avenue Matignon, 75008 Paris Cedex, France
Received 1 September 1999; accepted 1 December 1999
Abstract
A discrete time probabilistic model, for optimal equity allocation and portfolio selection, is formulated so as to apply to (at least) reinsurance. In the context of a company with several portfolios (or subsidiaries), representing both liabilities and assets, it is proved that the model has solutions respecting constraints on ROEs, ruin probabilities and market shares currently in practical use. Solutions define global and optimal risk management strategies of the company. Mathematical existence results and tools, such as the inversion of the linear part of the Euler–Lagrange equations, developed in a preceding paper in the context of a simplified model are essential for the mathematical and numerical construction of solutions of the model. © 2000 Elsevier Science B.V. All rights reserved.
JEL classification:C6; G11; G22; G32
MSC:90Axx; 49xx; 60Gxx
Keywords:Insurance; Equity allocation; Portfolio selection; Value at risk
1. Introduction
The context of this paper is a reinsurance companyH, with several portfolios (or subsidiaries), being described by a Cramèr–Lundberg like utility functionU, whose value, at timet, is simply the difference between the accumulated net incomes and the accumulated claims, in the time interval [0, t[ (cf. Embrechts et al., 1997). The corporate financial problem considered is:allocate equity to the different portfolios(or subsidiaries),and select the portfolios, such that the annual ROEs are satisfactory, such that the probability of ruin of H and the probability of non-solvency of each portfolio are acceptable and such that the expected final overall profitability is optimal. We here think of a portfolio as being a portfolio of insurance contracts or a portfolio of invested assets, which allows the inclusion of both liabilities and assets in the problem.
The purpose of the present paper is threefold: Firstly, to give a general probabilistic set-up of the above problem. This turns out to be possible in terms of an optimization problem with constraints and where the unknown variable is a stochastic process. We refer to this model as theNon-Linear Model. As we will see, mathematically, the cases with and without portfolios of invested assets are identical, modulo a change of names of variables. We therefore only consider the case where no invested assets are present. The value of the utility function U, at timet, is then the difference between the accumulated net premiums and the accumulated claims, in the time interval [0, t[.
Secondly, to develop a method which allows the construction of approximate solutionssatisfying the constraints of the Non-Linear Model. In fact, due to the non-solvency probabilities, the problem is highly non-linear and it seems difficult to solve it directly. This is done by considering a simplified portfolio selection model, withstronger constraintsthan in the original portfolio selection model and which is easier to solve. We prove that it is possible to choose the simplified problem so as to be the optimization of the final expected utility under constraints on the variance of the utility, the expected annual ROEs and other piecewise linear inequality constraints. We refer to this model as theQuadratic Model. For this model, the existence of solutions in a Hilbert space of adapted (to the claims processes) square integrable processes, is proved under mild hypotheses on the result processes of the portfolios. For their construction, a Lagrangian formalism with multipliers is introduced. Thirdly, to indicate that a basic arbitrage principle must be added to the model, in order to eliminate a degeneracy of the allocation problem. This is done by showing that the initial equity allocation and the future dividends generically are non-unique for optimal solutions, although the portfolios are unique. It is hinted that certain choices of limits on ruin and non-solvency probabilities re-establish uniqueness.
Before giving more detailed results, we shall go back to the motivation of the problem and also introduce some notation.
The reinsurance companyHis organized as a holding, with subsidiariesS(1), . . . , S(ℵ), whereℵ ≥1 is an integer. The companiesS(i)here correspond only to a division of the activities ofHinto parts, whose profitability, portfolio selection and certain other properties need to be considered individually. This allows localization of capital flows and results. The subsidiaryS(i)can cease its activities, which can be beneficial forH. The portfolioθ(i)ofS(i)has N(i)≥1 different types of contractsθj(i). It is decomposed intoθ(i)=ξ(i)+η(i), whereξ(i)is the run-off, at time t =0, concluded at a finite number of past timest <0 andη(i)is the (present and future) underwriting portfolio, to be concluded at a given finite number of times 0, . . . ,T ,¯ T¯ ≥1. If a subsidiaryS(i)ceases its activities at a time t =tc, (past, present or future), then it only continues to manage1 its run-off for timest≥tc. The companyHpays dividends2 D(t ), to the shareholders at timet. By conventionD(0)=0. Some rules determining the dividends
D(t ), in different situations, have been established.3 The initial equityK(0), ofHatt =0, is known. The problem is to determine att =0,
• (i) the equityK(i)(0)ofS(i)att =0 (1≤i≤ ℵ),
• (ii) the dividendD(i)(n), whichS(i)shall payHat timen≥1, where the determination of the dividend for time n, takes into account the observed claims during the periods precedingn,
• (iii) the portfolio selection forS(i), (i.e. underwriting targets for present and future periodsn≥ 0), where the determination of the underwriting targetη(i)j (n)for the time interval [n, n+1[ takes into account the observed claims during the periods precedingn,
such that the constraints are satisfied and the expected utility is optimal.
We postulate that the accessible informationFt at time t is given by the filtration generated by the claims processes.4 The output of the model is then given by the (certain) initial equitiesK(i)(0), by the present certain η(i)j (0)and future randomη(i)j (n), n≥1 underwriting levels of different types of contracts and by future random dividendsD(i)(n), n≥1 for each portfolio. By convention we setD(i)(0)=0, which is no restriction. The future random equitiesK(i)(n), n≥1 are then simply obtained by budget constraint equations. Points (i) and (ii) indicate that the dividends and the (target) underwriting levels form stochastic processes adapted to the filtration generated by the claims processes. The future random variables (subsequent underwriting levels and dividends) define a strategy of reactivity to the occurrence of future exterior random events. The uncertainty of these variables is reduced by
1The other possibility, which is not to keep the run-off, was considered in Taflin (1997).
2Here,D(t )can be positive or negative. By convention,D(t ) <0 corresponds to an increase in capital.
3For example,D(t )can simply be a function of the resultC(t )such thatD(t )=x(C(t )−c), ifC(t )≥c, andD(t )=0, ifC(t ) < c, where
0≤x≤1 andc≥0 are given real numbers.
the future increase of information. Thus underwriting levels and dividends, at a given future time, become certain when that given time is reached.
The probabilistic set-up of theNon-Linear Modelis given in Section 2.1 and is summarized by Eq. (2.9). It extends the stochastic model first considered in Taflin (1997). We justify, in Remark 2.2, that the models with and without invested assets are mathematically identical. TheQuadratic Model, which permits a constructive approach through a Lagrangian formalism, is given in Section 2.2 and is summarized by Eq. (2.16). The constraints of the Quadratic Model are stronger than those of the Non-Linear Model (Theorem 2.5). We establish (see Theorem 2.6), under certain mild conditions, (h1)–(h4), on the result processes for the unit contracts, that the optimization problem (2.9) has a solution. In Remark 2.7 some relations between solution of the Quadratic Model and approximate solutions of the original Non-Linear Model (2.9) are considered. Theorem 2.9 indicates that the solution, generically, is non-unique. Condition (h1) says that the final utility (sum of all results) of a unit contract, written at timek, is independent of events occurring beforekand that the intermediate utilities are not “too much” dependent. This is a starting point, since in practice, this is generally not exactly true, among other things because of feed-forward phenomena in the pricing. Condition (h2) is equivalent to the statement that no non-trivial linear combinations of final utilities, of contracts written at a given time, is a certain random variable. This can also be coined, in more financial terms: a new business portfolio (or underwriting portfolio)η(t ), constituted at timet, cannot be risk-free. Condition (h3) says that the final utility of unit contracts, written at different times are independent. Similarly, condition (h4) says that the final utility of unit contracts, written by different subsidiaries are independent. These conditions, which exclude interesting situations, like cyclic markets, have been chosen for simplicity. They can largely be weakened without altering the results of this paper. An important point is that no particular distributions (statistical laws) are required. The properties of these two models are mainly derived by considering the even simpler model of Taflin (1998), here called theBasic Model, the essential facts of which are summed up in Section 2.3. The portfolio in Taflin (1998) is an extension of Markowitz portfolio (Markowitz, 1952) to a multiperiod stochastic portfolio, as suggested by Harrison and Pliska (1979) (cf. see also Duffie, 1992; Dana and Jeanblanc-Picqué, 1994). One of the new features is that future results of contracts written at different times are distinguishable, which easily allows to consider contracts with different maturity times. The square root of the variance of the utility of a portfolio defines a norm, which is equivalent to the usualL2-norm of the portfolio (see Theorem 2.11). This is one of the major technical tools in the proof of the results of the present paper. There is existence and uniqueness of an optimal portfolio (see Theorem 2.12). Let us here also mention that a Lagrangian formalism is given in Taflin (1998), as well as essential steps in the construction of the optimal solution (Taflin, 1998, formula (2.17)). Namely, an effective method of calculating the inverse of the linear integral operator defined by the quadratic part of the Lagrangian, is established. The algorithm only invokes finite dimensional linear algebra and the conditional expectation operator. Moreover, the determination of Lagrange multipliers is also considered in Taflin (1998). The proofs of the results of the present paper are given in Section 3. For computer simulations, in the simplest cases see Dionysopoulos (1999).
2. The model and main results
2.1. Non-Linear Model
Mathematically, the probabilistic context of the model is given by a (separable perfect) probability space (Ω, P ,F)and a filtrationA= {Ft}t∈N, of sub-σ-algebras of theσ-algebraF, i.e.F0= {Ω,∅}andFs ⊂Ft ⊂F for 0≤s≤t. By conventionFt =F0fort <0.
To introduce the portfolios and utility functions of the subsidiaries,S(1), . . . , S(ℵ), let us consider the subsidiary S(j ). The portfolio ofS(j )is composed ofN(j )≥1 types of insurance contracts. By a unit contract, we denote an insurance contract whose total premium is one currency unit.5 The utilityu(j )i (t, t′), att′∈Nof the unit contract
i,1≤i ≤N(j ), concluded att ∈Z, is by definition the accumulated result in the time interval [0, t′[ ift <0, in the time interval [t, t′[ if 0≤t ≤t′andu(j )i (t, t′)=0 if 0≤t′≤t.6 We suppose thatui(j )(t, t′)isFt′-measurable and that(u(j )(t, t′))t′≥0 is an element of the space7 E(RN(j )), of processes, with finite moments of all orders. Since, for givent ∈Z, the process(u(j )i (t, t′))t′≥0isA-adapted, it follows that(u(j )(t, t′))t′≥0∈E(RN(j ),A). The final utility of the unit contracti, concluded att, which is given byu(j )i ∞(t )=u(j )i (t, s′)(=u(j )i (t,∞)), when the contract does not generate a flow after the times′,s′≥0, isFs′-measurable. We suppose that there exists a timeT (independent oft) such that the unit contracts concluded att ∈Z, do not generate a flow after the timet+T. Let the amount of the contract of typei, where 1≤i≤N(j ), concluded at timet ∈Z, beθi(j )(t ). In other words,θi(j )(t ) is the number of unit contracts of typei. Here, the run-offξ(j )(s) =θ(j )(s) ∈ RN(j ), s <0 is a certain vector (i.e.F0-measurable) andη(j )(s)=θ(j )(s), s ∈ {0, . . . ,T¯}is aFs-measurable random vector, taking its value in
RN(j ). We introduce as upplementary valueτf, which is a final state of the process(θ(j )(t ))t∈Z, reached when the
activity of the companyS(j )ceases. In the sequel the (certain) random variablesξ(j )(t )and the random variables η(j )(t )take values inRN(j )∪ {τf}.8 Moreover, it is supposed that the component of(η(j )(t ))t≥0, inRN
As already mentioned, the underwriting portfolioηshall satisfy constraints given by the market, the shareholders, etc. To introduce these constraints, letI be an index set andG= {gα|α∈I}be a set of functionsgα :N× ˜P →
L2(Ω,R)such that the valuegα(t, ξ, η), whereξ ∈ ˜Pr,η∈ ˜Pu, is independent ofη(t′), fort′> t. In this paper, we will say thatη7→gα(t, ξ, η)is a causal function (ofη). We define
C(ξ, G)= {η∈ ˜Pu|gα(t, ξ, η)≥0, t ≥0}, (2.1)
which is the set of all underwriting portfoliosηcompatibles with the run-offξ ∈ ˜Prand with the set of constraints G. We note that the process(gα(t, ξ, η))t≥0∈E2(R).
The utilityU(j )(t, θ(j )), at timet∈Zof a portfolioθ(j )∈ ˜P(j ), is defined by11
(U(j )(t, θ(j )))(ω)=X k≤t
(θ(j )(k))(ω)·(u(j )(k, t ))(ω)
6By result we here mean the net technical result including interest rates revenues from reserves. 7Let 1≤q <∞. Then(Xi) 9More precisely it is supposed that(p◦η(t ))t
≥0 ∈E2(RN,A), where the functionp :RN∪ {τf} →RNis defined byp(τf) =0 and
p(x)=x, forx∈RN.
10As usually a sequence(Xn)n
≥1inE˜2(RN∪ {τf},A)converges in distributions toX∈ ˜E2(RN∪ {τf},A)ifE(f (Xn))converges toE(f (X)),
for all real bounded continuous functionsf onRN∪ {τf}. This defines the topology of convergence in distributions (or the d-topology) on
˜
E2(RN∪ {τf},A). When a subsetF⊂ ˜E2(RN∪ {τ
f},A)is said to have a property of a topological vector space, such as being bounded, it is
meant that the setF′⊂E2(RN,A)×E2(R,A), given byF′ = {(p◦η, λ◦η)|η∈F}, whereλ(τ
f)=1 andλ(x)=0, ifx∈RN, has that
property. This convention will similarly be used for all spaces of functions with values inRN∪ {τf}.
11Here, the scalar product inRNisx·y=P
if(θ(j )(k))(ω)6=τffork≤tand by
(U(j )(t, θ(j )))(ω)= X k≤(tc)(ω)−1
(θ(j )(k))(ω)·(u(j )(k, t ))(ω)
if(tc)(ω) = inf{k ∈ Z|(θ(j )(k))(ω) = τf} ≤ t. The utilityU(j )so defined can be written as the following two forms:
U(j )(t, θ(j ))=X k≤t∗
θ(j )(k)·u(j )(k, t )=X k≤t
(p◦θ(j )(k))·u(j )(k, t )=U(j )(t, p◦θ(j )), (2.2)
wheret ∈ Z, (t∗)(ω) =min((tc)(ω)−1, t )andp : RN∪ {τf} → RN is defined byp(τf) =0 andp(x) =x, forx ∈RN(j ), (cf. footnote 9).U(j )(t, θ(j ))isFt-measurable. We have here chosen to keep the run-off for times, larger or equal totc, whenS(j )ceases its activities. Another possibility is not to keep the run-off (cf. footnote 1), in which case the utility is given by
U(j )(t, θ(j ))=U(j )(t∗, θ(j ))
fort ∈ Z. The stochastic process(U(j )(t, θ(j )))t≥0is an element of the spaceEp(R,A)for 1≤ p < 2, which follows directly from Hölder’s inequality. However, without further hypotheses, it does not in general have finite variance. The utility of an aggregate portfolioθ∈ ˜Pis defined by
U (t, θ )= X 1≤j≤ℵ
U(j )(t, θ(j )). (2.3)
The result of a portfolioθ(j )∈ ˜P(j ), for the time period [t, t+1[, t ∈Z, is defined by
(1U(j ))(t+1, θ(j ))=X k≤t∗
θ(j )(k)·(u(j )(k, t+1)−u(j )(k, t )), (2.4)
wheret∗is defined as in formula (2.3).(1U(j ))(t+1, θ(j ))isFt+1-measurable. Sinceu(j )(t+1, t+1) =0, formulas (2.2) and (2.4) give
(1U(j ))(t+1, θ(j ))=U(j )(t+1, θ(j ))−U (t, θ(j )) (2.5)
fort ≥0. The result of an aggregate portfolioθ∈ ˜Pis defined by
(1U )(t, θ )= X 1≤j≤ℵ
(1U(j ))(t, θ(j )). (2.6)
The companyS(j )has an initial equityK(j )(0)∈Ratt =0, and pays dividendsD(j )(t )at timet ≥0, D(j )(0)= 0. We suppose that(D(j )(t ))t≥0∈E2(R,A). The dividend can be negative, which as a matter of fact is an increase of equity. The expression (2.5), of the result for the period [t, t+1[, shows that the equityK(j )(t+1)at timet+1 is given by
K(j )(t+1)=K(j )(t )+(1U(j ))(t+1, θ(j ))−D(j )(t+1), (2.7)
wheret ≥0. We have that(K(j )(t ))t≥0∈Ep(R,A)for 1≤p <2. The dividendsDpaid to the shareholders by the companyH and the equity ofHare now given byD=P1≤j≤ℵD(j )andK=P1≤j≤ℵK(j ), respectively.
The companies H andS(1), . . . , S(ℵ) shall satisfy solvency conditions, which are expressed as lower limits on the equity. For a portfolio,θ(j ) ∈ ˜P(j ), let θ(j ) = ξ(j )+η(j ) be its decomposition into ξ(j ) ∈ ˜Pr(j ) and η(j ) ∈ ˜Pu(j ). Moreover, let(m(j )(t, θ(j )))t≥0 ∈ E2(R,A)be a process, called solvency margin, such thatη(j )7→
as in Eq. (2.2). We define the non-solvency probability, for the portfolioθ(j )ofS(j ), with respect to the solvency marginm(j )by
Ψ(j )(t, K(j ), θ(j ), m(j ))=P (inf{K(j )(n)−m(j )(n, θ(j ))|0≤n≤t}<0), (2.8) wheret ≥0. The most usual case ism(j )=0, i.e. positive equity, which gives the usual ruin probability. Similarly, we define the non-solvency probability, for the portfolioθ ∈ ˜P ofH, with respect to the solvency marginmby Ψ (t, K, θ, m)=P (inf{K(n)−m(n, θ )|0≤n≤t}<0).
We can now formulate the optimization problem. To precise the unknown processes (or variables), already mentioned in (i)–(ii) of Section 1, we introduce the Hilbert spaceP(j )
u,T¯ (resp. the complete metric space
(j ), with the underwriting horizonT¯ ∈N+1 fixed (independent ofj),
Before introducing the constraints, we recall that no flows are generated by contracts after a certain timeT¯+T. Therefore, equityK(j )(t )is constant fort ≥ ¯T +T. We suppose that the solvency margins are also chosen such that they are constant for sufficiently large times. We chooseT such that they are constant fort ≥ ¯T +T.
The constraints are: 0). To be general, we only suppose that there are real valued functions Fα, α ∈ I, an index set, such that
Fα(t, η,K(E 0),D)E ≤ Cα(t, K(0), ξ ), whereCα(t, K(0), ξ )are constants only depending on the initial equity
processes, which are causal functions of ηand which satisfy(( c(j )i (η))(t ))(ω) ∈ Rand((c¯i(j )(η))(t ))(ω) ∈ ]− ∞,∞] (market constraints),
• (c7) Ψ(j )(t, K(j ), ξ(j )+η(j ), m(j )) ≤ ǫ(j )(t ), where (m(j )(t, ξ(j )+η(j )))t≥0 ∈ E2(R,A) is a given sol-vency margin andǫ(j )(t ) ∈ [0,1] is a given acceptable non-solvency probability ofS(j ) for 1≤ j ≤ ℵand t ∈N,
• (c8)τfis a final state of the processη(j )such that ift > (tf(j ))(ω), then(η(j )(t ))(ω)=τf, where(tf(j ))(ω)is the smallest time inNsuch thatK(j )(tf(j )) < m(j )(tf(j ), ξ(j )+η(j )), and ift≤(tf(j ))(ω),(η(j )(t−1))(ω)∈RN(j ),
(K(j )(t−1))(ω) > (m(j )(t−1, ξ(j )+η(j )))(ω), then(η(j )(t ))(ω)∈RN(j )forω∈ Ω(a.e.), 1≤j ≤ ℵand t ≥1 (the activity ofS(j )ceases just after that the solvency margin is not satisfied).
LetCcbe the set of all(η,K(E 0),D)E , satisfying (v1)–(v3) and satisfying the constraints (c1)–(c8). Thus we sum up the constraints (c1)–(c8) on the form:
• (c)(η,K(E 0),D)E ∈Cc.
Among all the possible functions to optimize, we simply choose the expected value of the final utility,η 7→
E(U (∞, η+ξ )). The optimization problem is now: given the initial equityK(0), the dividend processD(ξ+η), as a function ofη, and the run-offξofH, satisfying (d1), (d2) and (d3), respectively, find the solutions(η,ˆ K(ˆE 0),D)ˆE ∈Cc of the equation
E(U (∞,ηˆ+ξ ))= sup (η,K(0),E D)E ∈Cc
E(U (∞, η+ξ )). (2.9)
Due to the constraints (c4) on the ruin probability and (c7) on the non-solvency probabilities, the resolution of this optimization problem leads to highly non-linear equations. This is true even in the case of practical applications, where the other constraints usually are piecewise linear.
Remark 2.1. The constraints(c1)and(c2)are just budget constraints. We have chosen the simplest form of the constraints(c3)on the ROE. Another possibility, is to strengthen it so that the ROE for the time interval[t, t+1[is satisfied conditionally to the information available at time t, i.e.E((1U )(t+1, ξ+η)|Ft)≥c(t )P1≤j≤ℵK(j )(t ). Of course, the expected value of the final utility will then in general be smaller for an optimal solution. (c4)is just a ruin constrain for H. (c5)is a very general constraint, which should cover most cases coming up in applications. (c6) says that, ifS(j )is in business at time t, then the underwriting targets for t,(η(j )i (t ))(ω)is in a given semi-bounded or bonded closed interval. Often the limits are proportional to(η(j )i (t−1))(ω)andη(j )i ≥0.In constraint(c8),
(tf(j ))(ω)is the smallest time such that the solvency margin is strictly negative forS(j )in the stateω.The constraint says thatS(j )ceases its activities for times larger than(tf(j ))(ω).Moreover, it says that, ifS(j )has not yet ceased its activities and if the solvency margin is strictly positive for a time t before(tf(j ))(ω),thenS(j )does not cease its activities att+1.So, it is only when the solvency margin is zero, there is a choice.
The constraints (c1)–(c5) and (c7) have a form as in formula (2.1). The constraints (c6) and (c8) can also be written on this form, which we give for later reference. Letλj :RN
(j )
∪ {τf} 7→ {0,1}be defined byλj(τf)=1 andλj(x)=0, x∈RN
(j )
. The constraint (c6) is then given by
(1−λj ◦η(j )(t ))(ηi(j )(t )−( c (j )
i (η))(t ))≥0, (2.10)
and
(1−λj ◦η(j )(t ))((c¯(j )i (η))(t ))−η (j )
i (t ))≥0, (2.11)
where 1 ≤ j ≤ ℵ, 1 ≤i ≤ N(j )and 0≤ t ≤ ¯T +T. In the case of (c8), lets(j )(t, x) =min0≤k≤t(K(j )(k)−
H (s)=0 ifs <0 andH (s)=1 if 0≤s. The constraint (c8) is then given by
(1−λj ◦η(j )(t ))s(j )(t−1, x)≥0, (2.12)
λj◦η(j )(t )s(j )(t−1, x)≤0, (2.13)
and
H (s(j )(t−1, x))(λj◦η(j )(t ))(K(j )(t−1)−m(j )(k, ξ(j )+η(j )))=0, (2.14) where 1≤j ≤ ℵ, 1≤i≤N(j )and 1≤t ≤ ¯T +T +1.
Remark 2.2. We shall illustrate that the mathematical formalism, in the cases with and without invested assets, are identical. Letη=(η′, η(ℵ))and letu(ℵ)(t, t′)=p(t+1)−p(t )+d(t+1),fort < t′,wherep, d∈E(RN(ℵ),A), d(0)=0,p(t )=p(T¯ +1)andd(t )=0,fort >T¯.Then
U (t, η)=U′(t, η′)+ X 0≤k≤t−1
η(ℵ)(k)·(p(k+1)−p(k)+d(k+1)), (2.15)
whereU′is the utility function given byU′(t, η′)=U (t, (η′,0)),i.e. for the portfolios1,2, . . . ,ℵ −1.The second term on the right-hand side of(2.15)is exactly the accumulated income, in the time interval[0, t[,from a portfolio
η(ℵ)of invested assets, with pricep(t )(ex dividend)at time t and paying dividendd(t )at a time just before t.This shows that the invested assets are taken into account by the model of this paper, as a particular case.
Remark 2.3. Formula(2.7)is related to the gain-process and to the self-financing condition used in finance. In fact it says that the equityK(j )of the companyS(j )is updated by the total gain minus the non-reinvested dividends. To have an explicit example in our context of a condition corresponding to a self-financing strategy in finance, let us consider the case of Remark 2.2withℵ = 1 (so U′ = 0)and with D = 0. The equity is then given by
K(t )=K(0)+U (t, η)(cf. formula(2.15))where U is given by the second term of the right-hand side of formula (2.15).If η(t )·p(t ) = K(t ),for0 ≤ t ≤ ¯T then, using that here U (t, η) = U (T , η)¯ for T < t¯ ≤ ¯T +T, it follows thatηis a self-financing strategy in the usual sense(cf. Duffie, 1992,Section6.K).Let us also admit that the company can invest positive amounts in a bond with positive price. Let p0(t ) > 0 for t ≥ 0, be the price of the bond. Ifη(t )·p(t ) ≤ K(t )then the rest is invested in a positive quantityη0(t )of bonds such that
K(t )=η(t )·p(t )+η0(t )p0(t )for0≤t ≤ ¯T +T.The strategyη¯=(η0, η1, . . . , ηN)is then self-financing. 2.2. Quadratic Model
The constraints (c4) on the ruin probability and (c7) on the non-solvency probabilities are replaced bystronger quadratic constraints, in this model. It will also be supposed that the non-ruin and non-solvency margins are satisfied in the mean. We introduce the constraints, whereV denotes the variance operator:
• (c′4)V (P1≤j≤ℵK(j )(t ))≤ǫ′(t )(δ(t )K(0))2andE(P1≤j≤ℵK(j )(t ))≥δ(t )K(0),t ∈N, whereǫ′(t )≥0 and δ(t ) >0,
• (c′7)V (K(j )(t )−m(j )(t, ξ(j )+η(j )))≤ǫ′(j )(t )(δ(j )(t )K(0))2andE(K(j )(t )−m(j )(t, ξ(j )+η(j )))≥δ(j )(t )K(0), for 1≤j ≤ ℵandt ∈N, wherem(j )’s are as in (c7),ǫ′(j )(t )≥0 andδ(j )(t ) >0.
For given initial equityK(0), dividend processD(ξ +η), as a function ofη, and run-offξ ofH, satisfying (d1), (d2) and (d3), respectively, letCc′ be the set of variables(η,K(E 0),D),E satisfying (v1), (v2) and (v3) and satisfying the constraints (c1)–(c3), (c4′), (c5), (c6), (c′7) and (c8). We sum up the constraints of the Quadratic Model on the form:
• (c′)(η,K(E 0),D)E ∈Cc′.
The optimization problem, in the case of the Quadratic Model, is now: given the initial equityK(0), the dividend processD(ξ +η), as a function ofη, and the run-offξ ofH, satisfying (d1), (d2) and (d3), respectively, find the solutions(η,ˆ K(ˆE 0),D)ˆE ∈Cc′of the equation
E(U (∞,ηˆ+ξ ))= sup (η,K(0),E D)E ∈C
c′
E(U (∞, η+ξ )). (2.16)
The constraints in the quadratic optimization problem (2.16) are stronger than those in the original problem (2.9).
Theorem 2.5. IfP0≤k≤tǫ′(k)≤ǫ(t )andP0≤k≤tǫ′(j )(k)≤ǫ(j )(t ),fort∈Nand1≤j ≤ ℵ,thenCc′ ⊂Cc. In order to give, in this paper, a mathematical analysis, which is as simple as possible, of optimization problem (2.16), we shall make certain (technical) hypotheses on the claims processes. The following hypotheses give a clear-cut mathematical context:
• (h1) independence with respect to the past:
1. u(p)∞(k)is independent ofFkfork∈Zand 1≤p≤ ℵ, 2. kE((u(p)(k, t ))2|Fk)kL∞ <∞fork < t,
• (h2) fork ∈ Zand 1 ≤p ≤ ℵ, theN ×N (positive) matrixc(p)(k)with elementscij(p)(k)=E((u (p)∞
i (k)−
E(u(p)i ∞(k))(u(p)j ∞(k)−E(u(p)j ∞(k)))is strictly positive, • (h3)u(p)i ∞(k)andu(p)j ∞(l)are independent fork6=l, • (h4)u(p)i ∞(k)andu
(r)∞
j (l)are independent forp6=r.
We note that the second point of (h1) is trivially satisfied ifu(p)(k, t )is independent ofFkfork < t.
The next theorem gives the existence of optimal solutions of problem (2.16). Approximations of these solutions can be constructed, using a Lagrangian formalism, as in the case of the Basic Model in Section 2.3. In order to state the theorem, we remind that ifξ is as in (2.16), then the functionsη(j ) 7→ m(j )(t, ξ(j )+η(j )),η 7→ D(ξ +η), η 7→ c(j )i (η),η 7→ ¯c(j )i (η)and(η,K(E 0),D)E 7→ Fα(t, η,K(E 0),D)E are defined forη(j ) ∈ ˜Pu,(j )T¯,η ∈ ˜Pu,T¯ and
(η,K(E 0),D)E ∈ ˜Pu,¯ T ×R
ℵ×E2(Rℵ,A). We also remind that formulas (2.5)–(2.7) give
K(j )(t )=K(j )(0)+U(j )(t, θ(j ))− X 1≤k≤t
D(j )(k), (2.17)
whereθ(j )∈ ˜P(j )¯
T , and the formula (2.3) gives
K(t )=K(0)+U (t, θ )− X 1≤k≤t
D(k, θ ), (2.18)
when constraints (c1) and (c2) are satisfied andθ ∈ ˜PT¯.
Theorem 2.6. Let the utilitiesu(p)(k, t )andu(p)∞(k),of unit contracts, satisfy(h1)–(h4).Let the functionsη(j )7→
m(j )(t, ξ(j )+η(j )),η 7→ D(ξ +η)and η 7→ c(j )i (η)toE2(R,A)map bounded sets into bounded sets. In the d-topology, letx = (η,K(E 0),D)E 7→ (x, U(j )(·, ξ(j )+η(j )), m(j )(·, ξ(j )+η(j )), D(ξ+η))be continuous, let
(η,K(E 0),D)E 7→ Fα(t, η,K(E 0),D)E ,α ∈ I,be lower semi-continuous and let η 7→ (c(j )i (η))(t, ω)and η 7→ −(c¯(j )i (η))(t, ω)be lower semi-continuous(a.e.).If
V
X
1≤k≤ ¯T+T
D(k, θ )
where0≤c <1,and ifCc′ is non-empty, then the optimization problem(2.16)has a solutionxˆ =(η,ˆ K(ˆE 0),D)ˆE ∈ Cc′.
Remark 2.7.
1. In applications it is easy to verify that the boundedness and continuity properties are satisfied. The variance condition simply translates that the final accumulated dividend are less volatile than the final accumulated result.
2. The solutionxˆ=(η,ˆ K(ˆE 0),D)ˆE of optimization problem(2.16)given by Theorem2.6is a sub-optimal solution of optimization problem(2.9),when the hypotheses of Theorem2.5are satisfied, i.e.xˆ ∈CcandE(U (∞,ηˆ+
ξ ))−sup(η,K(0),E D)E ∈CcE(U (∞, η+ξ )) ≤ 0.If the difference isδ,then an inspection of the proofs of this
paper shows thatδ→0whenǫ(j )(t )andǫ(t )go to zero, so the approximation is good for small non-solvency probabilities. The situation can be very different for large non-solvency probabilities.
Remark 2.8. The optimization problem(2.16)can be formulated as the optimization of a Lagrangian with multi-pliers. To show this letλjbe as in(2.10),letpj :RN
(j )
∪ {τf} →RN (j )
be defined bypj(τf)=0andpj(x)=x,for
x ∈RN(j ),letp=(p1, . . . , pℵ)andλ=(λ1, . . . , λℵ).LetET2¯(RN,A)be the subset of elementsη∈E2(RN,A),
such thatη(t )=0 fort >T¯.By definition, ifη∈ ˜Pu,T¯ thenp◦η∈E2¯ T(R
N,A),whereN =P
1≤j≤ℵN(j )and
λ◦η∈E2¯ T(R
ℵ,A).We introduce the Hilbert spacesH=E2
¯
T(R
N,A),H
0=H⊕ET2¯(Rℵ,A)andH1of elements
(K(E 0),D)E ∈ Rℵ⊕E2¯ T(R
ℵ,A),such thatD(E 0)= 0.LetH =H
0⊕H1.The optimization problem can now be formulated using the variablex =(α, β,K(E 0),D)E inH,where(α, β)∈H0and where for solutionsα=p◦ηand
β =λ◦η.The constraints(1−β(j ))β(j )=0andα(j )β(j )=0shall then be satisfied. The constraints(c1)–(c3), (c′4)and(c7′)are easily expressed in the new variables. The constraint (c6)is reformulated by using(2.10) and (2.11)and constraint(c8)by(2.12)–(2.14).This gives a Lagrangian with multipliers. We note that the function on the left-hand side of(2.14)is not differentiable, which leads to singularities in the Euler–Lagrange equation. Approximation schemes can be based on the inversion methods developed inTaflin (1998),for the linear part of the Euler–Lagrange equation. A detailed solution of this problem is the subject of future studies.
In general the solutionxˆ∈Cc′, of Theorem 2.6, is not unique. This fact can be traced back to a simplified case, namely where only the constraints (c1)–(c4) are considered and where allηi(j ) ≥ 0. To state the result letCc′′ be the set of all(η,K(E 0),D)E ∈ ˜Pu,T¯ ×Rℵ×E2(Rℵ,A), satisfying (v1)–(v3), (c1), (c2) and (c′4). We consider the following optimization problem: givenK(0)≥0,D=0 andξ =0, find the solutionsxˆ∈Cc′′of the equation
E(U (∞,η))ˆ = sup (η,K(0),E D)E∈C
c′′
E(U (∞, η)). (2.19)
Theorem 2.9. IfCc′′ is non-empty, then the optimization problem(2.19)has a solutionxˆ ∈ Cc′′.Moreover,ηˆis unique and(K(ˆE 0),D)ˆE only has to satisfyK(0)=P1≤j≤ℵKd(j )(0)and0=P
1≤j≤ℵDd(j ).
In the situation of the theorem, there is a whole hyperplane (Rℵ−1×E2(Rℵ−1,A)translated) of solutions in the variable(K(ˆE 0),D)ˆE . The generic case seems to be close to this case. To avoid a heavy “book-keeping” of solutions we only illustrate this by an informal remark instead of stating a formal theorem.
supplementary economic principle seems to be needed in order to guarantee uniqueness. If not guaranteed, then it does not always matter what we do with the equity!
To give an idea, without anticipating future work, how the problems discussed in Remarks 2.8 and 2.10 can be solved, we shall end this paragraph by a closer study of certain terms of the LagrangianLconstructed according to Remark 2.8. Let the LagrangianL0be the sum ofE(U (∞, η+ξ ))and the terms with multipliers corresponding to constraints (c1)–(c3), (c5) and (c6). To obtain explicit expressions of the termsL1andL2ofLcorresponding to the linear constraints and variance constraints respectively in (c4′) and (c7′) let
L(0)
There are Kuhn–Tucker conditions, which we do not give explicitly, corresponding to the Lagrange multipliersµ(j )i,t , where 0≤j ≤ ℵ,i=1,2 and 0≤t ≤ ¯T +T. LetL′=L0+L1+L2.
A possible algorithm to construct approximate solutions consists of determining the sequence(xn)n≥0, where
xn =(αn, βn,KEn(0),DEn)is defined likex =(α, β,K(E 0),D)E in Remark 2.8. Letβ0=0, and for givenβn, letxn be a solution of the optimization problem given by the Lagrangian(αn,KEn(0),DEn)7→L′(xn)with multipliers and let aβn+1be determined by constraint (c8). The complexity of this problem, for realistic choices (like piecewise linear) of the functions Fα, m(j ), D,c(j )i andc¯
(j )
A study of the LagrangianL′also gives a hint in what direction to search for a principle that breaks the degeneracy of the optimization problem (2.16). Under certain simple hypotheses on the given functions in Theorem 2.6, the derivative ofL′ with respect to the variable(K(E 0), E(D))E leads to an Euler–Lagrange equation, being a closed system in the variable(K(E 0), E(D))E . Its solution is degenerate except in the case when all constraints are saturated. This means that, from the point of view of degeneracy, certain choices of limits on ruin and non-solvency probabilities are singled out.
2.3. Basic Model
We shall here sum up certain results obtained in Taflin (1998) concerning a particularly simple model, which is an essential building block of the models already considered in Section 2.1 and 2.2. In that model it is supposed that the number of subsidiariesℵ =1, the run-offξ =0, the dividendsD=0 and it is supposed that there are no market limitations on the subscription levels, except that they are positive. It is also imposed that the portfolioηis an element of the Hilbert spaceH=Pu,T¯ (soη(ω)6=τf onΩ). We remind that, in this situation, the equity
K(t )=K(0)+U (t, η), (2.25)
whereK(0)≥0 is the initial equity att=0.
In the sequel of this paragraph, we closely follow Taflin (1998). Constraints on the variableηare introduced: • (C3)E((1U )(t+1, η))≥c(t )E(K(t )),c(t )∈R+is given (constraint on profitability),
• (C4)E((U (∞, η)−E(U (∞, η)))2)≤σ2, whereσ2>0 is given (acceptable level of the variance of the final utility),
• (C6) 0≤ηi(t ), where 1≤i≤N(only positive subscription levels),
Let (C0be the set of portfoliosη∈Hsuch that constraints (C3), (C4) and (C6) are satisfied. This is well-defined. In fact the quadratic form
η7→a(η)=E((U (∞, η))2) (2.26)
inH, has a maximal domainD(a), since for eachη∈H, the stochastic process(U (t, θ ))t≥0is an element of the spaceEp(R,A)for 1 ≤ p < 2 (which follows directly from Hölder’s inequality). The optimization problem is now, to find allηˆ∈C0, such that
E(U (∞,η))ˆ = sup η∈C0
E(U (∞, η)). (2.27)
The solution of this optimization problem is largely based on the study of the quadratic form
η7→b(η)=E((U (∞, η)−E(U (∞, η)))2) (2.28)
inH,with (maximal) domainD(b)=D(a).
We make certain (technical) hypotheses on the claims processes: • (H1)u∞(k)is independent ofFkfork∈N,
• (H2) fork ∈ NtheN×N (positive) matrixc(k)with elementscij(k) = E((u∞i (k)−E(u∞i (k)))(u∞j (k)−
E(u∞j (k))))is strictly positive, • (H3)u∞i (k)andu
∞
j (l)are independent fork6=l.
The next crucial result (Taflin, 1998, Lemma 2.2 and Theorem 2.3) shows that (the square root of) each one of the quadratic formsbandais equivalent to the norm inH.
Theorem 2.11. If the hypotheses(H1)–(H3)are satisfied, then the quadratic formsbandaare bounded from below and from above, by strictly positive numbers c and C respectively, where0< c≤C,i.e.
ckηk2H≤b(η)≤a(η)≤Ckηk2H (2.29)
The operatorsB(resp.A) inH, associated withb(resp.a), (by the representation theorem), i.e.
b(ξ, η)=(ξ, Bη)H (resp.,a(ξ, η)=(ξ, Aη)H) (2.30)
forξ ∈Handη ∈H, are strictly positive, bounded, self-adjoint operators ontoHwith bounded inverses. There existc ∈R, such that 0<cI≤ B ≤ A, whereI is the identity operator. It follows from formula (2.30), that an explicit expression ofAis given by
(Aη)(k)=E(U (∞, η)u∞(k)|Fk), (2.31)
and that an explicit expression ofBis given by
(Bη)(k)=E((U (∞, η)−E(U (∞, η)))u∞(k)|Fk) (2.32)
forη∈H, where 0≤k≤ ¯T.
The next result (Taflin, 1998, Corollary 2.6) solves the optimization problem of this paragraph.
Theorem 2.12. Let hypotheses(H1)– (H3)be satisfied. IfC0is non-empty, then optimization problem(2.27)has unique solutionηˆ ∈C0.
The solutionηˆis given by a constructive approach in Taflin (1998). In fact, in that reference a Lagrangian formalism, an algorithm to invert the operatorsAandBand approximation methods for determining the multipliers are given.
3. Proofs
Chebyshev’s inequality gives d-topology, can be given a compatible structure of a complete separable metric space (cf. Itô, 1984, Sections 2.7 and 2.8). So in the d-topology a set is closed (resp. compact) if and only if it is sequentially closed (resp. compact) (cf. Rudin, 1973, Theorem A4). The first term on the right-hand side is uniformly bounded for η ∈ A′, according to constraint (c4′). Since V (ξi(j )(k)u(j )i ∞(k))= |(ξi(j )(k)|2V (u(j )∞
i (k)) <∞, it follows that the second term also is uniformly bounded on
We have, according to Eq. (2.17)
V X
1≤k≤t
D(j )(k)
1/2
≤(V (K(j )(t )−m(j )(t, θ(j ))))1/2+(V (U(j )(t, θ(j ))))1/2+(V (m(j )(t, θ(j ))))1/2.
The first term on the right-hand side is uniformly bounded inη(j )forη∈B′(0, R1),according to constraint (c7′), the second according to Lemma 3.1 and the third according to the hypotheses of the theorem. This shows that there isR′<∞such that
V (D(j )(t )) < R′ (3.2)
for allD(j )(t )satisfying conditions (c′4) and (c′7).
According to constraint (c′7), we have thatE(K(j )(t ))≥δ(j )(t )K(0)+E(m(j )(t, ξ(j )+η(j ))). Sincem(j )(t, ξ(j )+
η(j ))is uniformly bounded (hypotheses of the theorem) inE2(R,A)forη∈B′(0, R
1), it follows thatE(K(j )(t )) is bounded from below onA′. In particular, constraint (c1) then gives that|K(j )(0)|is bounded onA′. Formula (2.17) and the fact thatE(U(j )(t, ξ(j )+η(j )))is uniformly bounded forη ∈ B′(0, R1), (cf. Lemma 3.1) show thatE(D(j )(t )), 1≤tare bounded from above.|E(D(t, ξ +η))|is uniformly bounded forη∈B′(0, R1), since, by the hypotheses of the theorem,D(t, ξ+η)is uniformly bounded inE2(R,A), forη∈ B′(0, R1). SinceD =
P
1≤j≤ℵD(j ), according to (c2), it follows thatE(D(j )(t, ξ(j )+η(j )))is bounded onA′. According to inequality (3.2) it follows then that there existsR2<∞such that
kD(j )(t )k + |K(j )(0)|< R2 (3.3) for all(η,0,K(E 0),D)E ∈A′. This inequality and the fact thatη∈B′(0, R1), show thatA′is a bounded subset ofH. Let(xn)n≥1be a sequence inA′, which converges in the d-topology ofHtox′ =(η′,0,KE′(0),DE′)∈H. The limit element then satisfies the constraints (c1) and (c2). In fact, the case of (c1) is trivial. By the hypotheses of the theorem, it follows thatx7→δ(x)=D(t, ξ+η)−P1≤j≤ℵD(j )(t ), is d-continuous onA′, sox′satisfies (c2).
By the hypotheses of the theorem and by formula (2.17), the function
x7→(U(j )(·, ξ(j )+η(j )), K(j )(·), K(j )(·)−m(j )(·, ξ(j )+η(j ))) (3.4)
is d-continuous on A′. The functions x 7→ E(U(j )(t, ξ(j )+η(j ))), x 7→ E(m(j )(t, ξ(j )+η(j ))) andx 7→
E(K(j )(t )−m(j )(t, ξ(j )+η(j )))are also d-continuous onA′. In factA′ is a bounded subset of H, so accord-ing to Lemma 3.1, the image ofA′under the first component of the function (3.4) is bounded inE2. According to formula (2.17), this is then also the case for the second component in (3.4). The boundedness inE2of the image ofA′under the functionx 7→K(j )(·)−m(j )(·, ξ(j )+η(j )), now follows by the hypotheses of the theorem. The d-continuity of the function (3.4) then give the d-continuity of the above expected values, by uniform integrability (cf. Billingsley, 1986, Theorem 25.12). The functionx 7→f (x)=K(j )(t )−m(j )(t, ξ(j )+η(j ))is d-continuous. Since the sequence(f (xn))n≥1converges in the d-topology, it follows thatE(|f (x′)|2)≤lim infn→∞E(|f (xn)|2) (cf. Billingsley, 1986, Theorem 25.11). Because of the already proved d-continuity onA′ of x 7→ E(f (x)), it follows that
V (f (x′))=E((f (x′))2)−(E(f (x′)))2≤lim inf
n→∞ V (f (xn)).
Constraint (c′7) then gives thatV (f (x′))≤ǫ′(j )(t )(δj(t )K(0))2, which proves that
V (K′(j )(t )−m(j )(t, ξ(j )+η′(j )))≤ǫ′(j )(t )(δj(t )K(0))2. (3.5) Once more, by the already proved proved d-continuity, it follows from (c′7) that
Inequalities (3.5) and (3.6) show that the limit pointx′satisfies constraint (c7′). Similarly, it is proved thatx′also satisfies constraint (c4′). This proves thatA′is d-closed.
Let(xn)n≥1be a sequence inA′. Then(xn)n≥1is tight, sinceA′is a bounded subset ofH. In fact, if the bound isR and iff (a)=infX∈A′P (|X| ≤a), thenf (a)=infX∈A′(1−P (|X|> a))≥1−a−2supX∈A′E(|X|2)≥1−(R/a)2. Hence lima→∞f (a)=1. This proves thatA′is d-compact, sinceA′is closed (cf. Itô, 1984, Theorem 2.5.3).
Let(yn)n≥1,yn=(p◦ηn, λ◦ηn,KEn(0),DEn), be a sequence inAand letxn= ¯p(yn). Extracting a subsequence we can suppose that(xn)n≥1converges in the d-topology tox∈A′, sinceA′is d-compact. Letgn=λ◦ηn. Then
gn:Ω → {0,1}ℵdefines a bounded sequence inE2(Rℵ,A). Extracting a subsequence we can suppose that(gn)n≥1 converges in the d-topology tog∈E2(Rℵ,A). It follows thatg:Ω → {0,1}ℵ(a.e.) (sinceg(j )
n is a d-convergent sequence of characteristic functions;g(j )n =(gn(j ))2). This proves that a subsequence of(yn)n≥1converges toy∈A, soAis d-compact.
We shall next prove that the subsetB ⊂A,of all elements inA, which satisfies conditions (c3), (c5), (c6) and (c8), is d-compact. As earlier in this proof letx =(p◦η, λ◦η,K(E 0),D)E ∈Aand let(yn)n≥1be a d-convergent sequence inAwith limity=(p◦η′, λ◦η′,KE′(0),DE′).y∈A, sinceAis d-compact. Suppose thatBis non-empty, otherwise there is nothing to prove.
Constraint(c3): Letyn satisfy (c3) forn ≥ 1. Due to formula (2.2), it is enough to consider the value of the constraint atxn= ¯p(yn). It is proved, as in the case of the d-continuity of functions in and following (3.4) that both sides of the inequality in (c3) are d-continuous functions onA′. This proves that (c3) is satisfied byp(y)¯ , since it is satisfied byp(y¯ n),n≥1. Therefore, it is satisfied byy.
Constraint(c5): Letyn satisfy (c5) forn ≥ 1. According to the hypotheses of the theorem, the functionx 7→
fα(x) = Fα(t, η,K(E 0),D)E is sequentially lower semi-continuous in the d-topology. This gives that fα(y) ≤ lim infn→∞fα(yn). It follows from (c5) thatfα(y)≤Cα(t, K(0), ξ ). This proves that (c5) is satisfied byy.
Constraint(c6): Letynsatisfy (c6) forn≥1. The functions,η(j ) 7→pj ◦η(j )fromP˜u(j )toPu(j ), andη(j )7→
λj ◦η(j )fromP˜u(j )toE2(R,A), are d-continuous. According to the hypotheses of the theorem,η 7→ (c(j )i ((p◦
η))(t, ω)is then lower semi-continuous (a.e.). This is then also the case for the functionη 7→ (b(j )i (η))(t, ω) =
(1−λj◦η(j ))(c(j )i ((p◦η))(t, ω). The first condition of (c6) is equivalent to(b(j )i (η))(t, ω)≤((pj◦η(j ))(t ))(ω)(a.e.). Since the left-hand side is lower semi-d-continuous, it follows that(b(j )i (η′))(t, ω)≤lim infn→∞(b(j )i (ηn))(t, ω). Since the member in the middle is d-continuous (a.e.), it then follows that(bi(j )(η′))(t, ω)≤ η′(j ))(t ))(ω)(a.e.). This inequality and a similar result in the case ofc¯i(j )show that (c6) is satisfied byy.
Constraint(c8): Letynsatisfy (c8) forn≥1. Let
s(j )(t, x)= min 0≤k≤t(K
(j )(k)−m(j )(k, ξ(j )+η(j ))),
whereK(j )(k)is evaluated atx ∈A. According to the hypotheses of the theoremx7→(x, s(j )(·, x))is d-continuous fromAtoA×E2(R,A). We shall prove by contradiction that (c8) is satisfied byy. We remind that, forx ∈ A,
(η(j )(t ))(ω)∈RN(j )or(η(j )(t ))(ω)=τfand we note that (a.e.) if(η
′(j )
(t ))(ω)=τfthen(η
′(j )
(t+s))(ω)=τf,s≥ 0. Suppose that for somet′≥0,(s(j )(t′, y))(ω) <0 for allω∈Q′, whereQ′is aFt′-measurable set, withP (Q′) > 0. Suppose also that(η′(j )(t′+1))(ω)6=τf, for allω∈Q′′, whereQ′′⊂Q′is aFt′-measurable set andP (Q′′) >0. Leth:R→Rbe given byh(a)= −1 ifa≤ −1,h(a)=aif−1< a≤0 andh(a)=0 if 0< a. The function (a, b)7→h(a)(1−b)fromR× {0,1}toRis bounded and continuous. According to the hypotheses onQ′andQ′′it follows thatE(h(s(j )(t′, y))(1−(λj◦η
′(j )
)(t′)))≤ −r,for somer >0. However sinceynsatisfies (c8) we have that
E(h(s(j )(t′, yn))(1−(λj◦η
′(j )
n )(t′)))=0 (3.7)
That y satisfies the second part of the constraint (c8) (i.e. (η′(j )(t ))(ω) ∈ RN(j ), ift ≤ (tf(j ))(ω)and(K′(j ) (t−1)−m(j )(t−1, ξ(j )+η′(j )))(ω) >0) is proved similarly. This ends the proof of constraint (c8).
We have now proved thaty ∈ B, soB is a d-closed subset ofA. ThusBis d-compact, sinceB ⊂AandAis compact.
Similarly, as after formulas (3.4), it follows thatη 7→ E(U (t, ξ +η))is d-continuous on bounded subsets of ˜
Pu,¯
T. The mapx 7→f (x)=E(U (t, ξ+η)), fromA, is then d-continuous. SinceB ⊂Ais d-compact, it follows that the mapB ∋ x 7→ f (x)= E(U (t, ξ+η)), attains its supremum,f (x)ˆ , at a pointxˆ ∈ B. This proves the
theorem.
Proof of Theorem 2.9. This is a direct consequence of Theorem 2.12.
Acknowledgements
The author would like to thank Jean-Marie Nessi, CEO of AXA-Ré, and his collaborators for many interesting discussions, which were the starting point of this work, the anonymous referee for several constructive suggestions and Gheorghe Postelnicu for finding a serious error in a preliminary version of this paper.
References
Billingsley, P., 1986. Probability and Measure, 2nd Edition. Wiley, New York.
Dana, R.A., Jeanblanc-Picqué, M., 1994. Marchés financiers en temps continue: valorisation et équilibre. Economica.
Dionysopoulos, T., 1999. Stochastic optimization of a simplified reinsurance portfolio. Memoir de DEA 1999, Probabilités et Applications. Université Pierre et Marie Curie, Paris.
Duffie, D., 1992. Dynamic Asset Pricing Theory. Princeton University Press, Princeton, NJ.
Embrechts, P., Klüppelberg, C., Mikosh, T., 1997. Modelling Extremal Events, Applications of Mathematics, Vol. 33. Springer, Berlin. Harrison, M., Pliska, S., 1979. Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20, 381–408. Itô, K., 1984. Introduction to Probability Theory, Cambridge University Press, Cambridge.
Markowitz, H., 1952. Portfolio selection. Journal of Finance 7, 77–91. Rudin, W., 1973. Functional Analysis. McGraw-Hill, New York.
Taflin, E., 1997. Allocation de fonds propres et pilotage de portefeuille. Rapport AXA-Ré et UAP Direction Scientifique.
