The model and main results

E. Taflin Insurance: Mathematics and Economics 27 2000 65–81 67 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 the Non-Linear Model is 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. The Quadratic 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, h 1 –h 4 , 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 h 1 says that the final utility sum of all results of a unit contract, written at time k, is independent of events occurring before k and 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 h 2 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 ηt, constituted at time t, cannot be risk-free. Condition h 3 says that the final utility of unit contracts, written at different times are independent. Similarly, condition h 4 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 the Basic 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 usual L 2 -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 filtration A = {F t } t ∈N , of sub-σ -algebras of the σ -algebra F , i.e. F = {Ω, ∅} and F s ⊂ F t ⊂ F for 0 ≤ s ≤ t. By convention F t = F for t 0. To introduce the portfolios and utility functions of the subsidiaries, S 1 , . . . , S ℵ , let us consider the subsidiary S j . The portfolio of S j is composed of N j ≥ 1 types of insurance contracts. By a unit contract, we denote an insurance contract whose total premium is one currency unit. 5 The utility u j i t, t ′ , at t ′ ∈ N of the unit contract 5 All flows are supposed actualized. 68 E. Taflin Insurance: Mathematics and Economics 27 2000 65–81 i, 1 ≤ i ≤ N j , concluded at t ∈ Z, is by definition the accumulated result in the time interval [0, t ′ [ if t 0, in the time interval [t, t ′ [ if 0 ≤ t ≤ t ′ and u j i t, t ′ = 0 if 0 ≤ t ′ ≤ t. 6 We suppose that u j i t, t ′ is F t ′ -measurable and that u j t, t ′ t ′ ≥0 is an element of the space 7 E R N j , of processes, with finite moments of all orders. Since, for given t ∈ Z, the process u j i t, t ′ t ′ ≥0 is A-adapted, it follows that u j t, t ′ t ′ ≥0 ∈ ER N j , A . The final utility of the unit contract i, concluded at t, which is given by u j ∞ i t = u j i t, s ′ = u j i t, ∞ , when the contract does not generate a flow after the time s ′ , s ′ ≥ 0, is F s ′ -measurable. We suppose that there exists a time T independent of t such that the unit contracts concluded at t ∈ Z, do not generate a flow after the time t + T . Let the amount of the contract of type i, where 1 ≤ i ≤ N j , concluded at time t ∈ Z, be θ j i t . In other words, θ j i t is the number of unit contracts of type i. Here, the run-off ξ j s = θ j s ∈ R N j , s 0 is a certain vector i.e. F -measurable and η j s = θ j s, s ∈ { 0, . . . , ¯ T } is a F s -measurable random vector, taking its value in R N j . We introduce as upplementary value τ f , which is a final state of the process θ j t t ∈Z , reached when the activity of the company S j ceases. In the sequel the certain random variables ξ j t and the random variables η j t take values in R N j ∪ {τ f }. 8 Moreover, it is supposed that the component of η j t t ≥ , in R N j , has finite variance, 9 i.e. η j t t ≥ ∈ ˜ E 2 R N j ∪ {τ f }, A, the space of processes with values in R N j ∪ {τ f } and whose component in R N j is an element of E 2 R N j , A . 10 We introduce the underwriting portfolio set ˜ P j u , of elements η j ∈ ˜ E 2 R N j ∪ {τ f }, A, such that only a finite number of η j t , t ∈ N are different from zero. Let P j u be the intersection of ˜ P j u and E 2 R N j , A . The spaces of run-off portfolios are defined by ˜ P j r = R N j ∪ {τ f } ∞ and P j r = R N j ∞ . If ξ j ∈ ˜ P j r and if η j ∈ ˜ P j u , then θ j = ξ j + η j ∈ ˜ P j = ˜ P j r × ˜ P j u , the portfolio set of S j . The aggregate portfolio θ = θ 1 , . . . , θ ℵ , of the company H , is an element of the portfolio set ˜ P = × 1≤j ≤ℵ ˜ P j . Let ˜ P r = × 1≤j ≤ℵ ˜ P j r and ˜ P u = × 1≤j ≤ℵ ˜ P j u . As already mentioned, the underwriting portfolio η shall satisfy constraints given by the market, the shareholders, etc. To introduce these constraints, let I be an index set and G = {g α |α ∈ I } be a set of functions g α : N × ˜ P → L 2 Ω, R such that the value g α t, ξ, η , where ξ ∈ ˜ P r , η ∈ ˜ P u , is independent of ηt ′ , for t ′ t . In this paper, we will say that η 7→ g α t, ξ, η is a causal function of η. We define Cξ, G = {η ∈ ˜ P u |g α t, ξ, η ≥ 0, t ≥ 0}, 2.1 which is the set of all underwriting portfolios η compatibles with the run-off ξ ∈ ˜ P r and with the set of constraints G . We note that the process g α t, ξ, η t ≥ ∈ E 2 R . The utility U j t, θ j , at time t ∈ Z of a portfolio θ j ∈ ˜ P j , is defined by 11 U j t, θ j ω = X k≤t θ j kω · u j k, t ω 6 By result we here mean the net technical result including interest rates revenues from reserves. 7 Let 1 ≤ q ∞. Then X i 0≤i ∈ E q R N if and only if X i : Ω → R N is F -measurable and kX i k L q Ω, R N = R Ω |X i ω| q R N dP ω 1q ∞ for i ≥ 0, where | | R N is the norm in R N . Let E q R N , A the sub-space of A-adapted processes in E q R N . We define ER N = ∩ q≥ 1 E q R N and ER N , A = ∩ q≥ 1 E q R N , A . 8 R N ∪ {τ f } is not a vector space. Let X l : Ω → R N ∪ {τ f }, l ∈ {1, 2} be two random variables and let a ∈ R. We define X 1 + X 2 ω = X 1 ω + X 2 ω if X l ω 6= τ f for l ∈ {1, 2} and X 1 + X 2 ω = τ f , if X l ω = τ f for l = 1 or l = 2. We also define aX 1 ω = aX 1 ω if X 1 ω 6= τ f and aX 1 ω = τ f if X 1 ω = τ f . This permits to continue to use the linear structure on the subspace R N . 9 More precisely it is supposed that p ◦ ηt t ≥ ∈ E 2 R N , A , where the function p : R N ∪ {τ f } → R N is defined by pτ f = 0 and px = x , for x ∈ R N . 10 As usually a sequence X n n≥ 1 in ˜ E 2 R N ∪ {τ f }, A converges in distributions to X ∈ ˜ E 2 R N ∪ {τ f }, A if Ef X n converges to Ef X, for all real bounded continuous functions f on R N ∪ {τ f }. This defines the topology of convergence in distributions or the d-topology on ˜ E 2 R N ∪ {τ f }, A. When a subset F ⊂ ˜ E 2 R N ∪ {τ f }, A is said to have a property of a topological vector space, such as being bounded, it is meant that the set F ′ ⊂ E 2 R N , A × E 2 R, A , given by F ′ = {p ◦ η, λ ◦ η|η ∈ F }, where λτ f = 1 and λx = 0, if x ∈ R N , has that property. This convention will similarly be used for all spaces of functions with values in R N ∪ {τ f }. 11 Here, the scalar product in R N is x · y = P 1≤i≤N x i y i . E. Taflin Insurance: Mathematics and Economics 27 2000 65–81 69 if θ j kω 6= τ f for k ≤ t and by U j t, θ j ω = X k≤t c ω− 1 θ j kω · u j k, t ω if t c ω = inf{k ∈ Z|θ j kω = τ f } ≤ t. The utility U 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 where t ∈ Z, t ∗ ω = mint c ω − 1, t and p : R N ∪ {τ f } → R N is defined by pτ f = 0 and px = x, for x ∈ R N j , cf. footnote 9. U j t, θ j is F t -measurable. We have here chosen to keep the run-off for times, larger or equal to t c , when S 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 for t ∈ Z. The stochastic process U j t, θ j t ≥ is an element of the space E p 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 portfolio θ ∈ ˜ P is defined by U t, θ = X 1≤j ≤ℵ U j t, θ j . 2.3 The result of a portfolio θ 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 where t ∗ is defined as in formula 2.3. 1U j t + 1, θ j is F t + 1 -measurable. Since u 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 for t ≥ 0. The result of an aggregate portfolio θ ∈ ˜ P is defined by 1U t, θ = X 1≤j ≤ℵ 1U j t, θ j . 2.6 The company S j has an initial equity K j 0 ∈ R at t = 0, and pays dividends D j t at time t ≥ 0, D j 0 = 0. We suppose that D j t t ≥ ∈ E 2 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 equity K j t + 1 at time t + 1 is given by K j t + 1 = K j t + 1U j t + 1, θ j − D j t + 1, 2.7 where t ≥ 0. We have that K j t t ≥ ∈ E p R, A for 1 ≤ p 2. The dividends D paid to the shareholders by the company H and the equity of H are now given by D = P 1≤j ≤ℵ D j and K = P 1≤j ≤ℵ K j , respectively. The companies H and S 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 ∈ ˜ P j r and η j ∈ ˜ P j u . Moreover, let m j t, θ j t ≥ ∈ E 2 R, A be a process, called solvency margin, such that η j 7→ m j t, ξ j + η j is a casual function and such that m j t, θ j = m j t, p ◦ θ j , where p is the projection 70 E. Taflin Insurance: Mathematics and Economics 27 2000 65–81 as in Eq. 2.2. We define the non-solvency probability, for the portfolio θ j of S j , with respect to the solvency margin m 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 where t ≥ 0. The most usual case is m j = 0, i.e. positive equity, which gives the usual ruin probability. Similarly, we define the non-solvency probability, for the portfolio θ ∈ ˜ P of H , with respect to the solvency margin m by Ψ t, K, θ, m = P inf{Kn − mn, 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 space P j u, ¯ T resp. the complete metric space 12 ˜ P j u, ¯ T of elements η j ∈ P j u , resp. η j ∈ ˜ P j u such that η j t = 0, resp. η j t ω = 0 or η j t ω = τ f , ω ∈ Ω for t ¯ T . We also introduce the spaces P j ¯ T = P j r × P j u, ¯ T , ˜ P j ¯ T = ˜ P j r × ˜ P j u, ¯ T , P u, ¯ T = × 1≤j ≤ℵ P j u, ¯ T , ˜ P u, ¯ T = × 1≤j ≤ℵ ˜ P j u, ¯ T , P ¯ T = P r × P u, ¯ T and ˜ P ¯ T = ˜ P r × ˜ P u, ¯ T . The unknown variables are: • v 1 the equity K j 0 ∈ R of S j at t = 0, • v 2 the dividend process D j = 0, D j 1, D j 2, . . . ∈ E 2 R, A paid by S j to H , • v 3 the underwriting portfolio η j ∈ ˜ P j u, ¯ T , of S j , with the underwriting horizon ¯ T ∈ N + 1 fixed independent of j , where 1 ≤ j ≤ ℵ. Thus the unknown variables of the subsidiary S j are the components of the variable Z j = K j 0, D j , η j 0, . . . , η i ¯ T , η j ¯ T + 1, . . . , where η j ¯ T +l can only take the values 0 and τ f for l 0. The optimization is done with respect to the variable Z = Z 1 , . . . , Z ℵ , satisfying v 1 –v 3 . It will be convenient to use a variable, which is obtained by a permutation of the coordinates of Z. Let E K 0 = K 1 0, . . . , K ℵ and E D = D 1 , . . . , D ℵ . We use the variable η, E K 0, E D instead of Z. Certain data are given as • d 1 initial equity K0 ∈ R + of H , • d 2 the dividend process Dθ ∈ E 2 R, A , with Dθ 0 = 0, which H pays the share holders, only depends of, past and present, aggregate results i.e. Dθ t = f t 1U 1, θ , . . . , 1U t, θ , t ≥ 1, where f t is a F t -measurable function, cf. footnote 3. • d 3 the run-off ξ ∈ ˜ P r of the subsidiaries. Before introducing the constraints, we recall that no flows are generated by contracts after a certain time ¯ T + T . Therefore, equity K j t is constant for t ≥ ¯ T + T . We suppose that the solvency margins are also chosen such that they are constant for sufficiently large times. We choose T such that they are constant for t ≥ ¯ T + T . The constraints are: • c 1 K0 = P 1≤j ≤ℵ K j 0, • c 2 D = P 1≤j ≤ℵ D j , • c 3 E1U t + 1, ξ + η ≥ ctE P 1≤j ≤ℵ K j t , where ct ∈ R + is given for t ∈ N constraint on ROE, • c 4 Ψ t, P 1≤j ≤ℵ K j , ξ + η, 0 ≤ ǫt, where ǫt ∈ [0, 1] is a given acceptable ruin probability of H for t ∈ N , • c 5 supplementary constraints, to be specified, on D j e.g. to increase the equity of S l , one can set D l = 0. To be general, we only suppose that there are real valued functions F α , α ∈ I , an index set, such that F α t, η, E K 0, E D ≤ C α t, K 0, ξ , where C α t, K 0, ξ are constants only depending on the initial equity and the run-off, • c 6 if η j t ω 6= τ f , then c j i ηt ω ≤ η j i t ω and η j i t ω ≤ ¯ c j i ηt ω , for ω ∈ Ω a.e., 1 ≤ j ≤ ℵ, 1 ≤ i ≤ N j and 0 ≤ t ≤ ¯ T . Here, c j i η ∈ E 2 R, A and ¯c j i η are given A-adapted 12 The metric ρ in ˜ P j u, ¯ T is defined by ρX, Y = P 1≤i≤ ¯ T kp ◦ X i − p ◦ Y i k 2 + P X −1 i {τ f }1Y −1 i {τ f } 2 12 , where 1 denotes the symmetric difference of sets and k k is the norm in L 2 R N j . The topology defined by the metric ρ, is referred to as the strong topology. E. Taflin Insurance: Mathematics and Economics 27 2000 65–81 71 processes, which are causal functions of η and which satisfy c j i ηt ω ∈ R and ¯c j i ηt ω ∈ ] − ∞, ∞] market constraints, • c 7 Ψ j t, K j , ξ j + η j , m j ≤ ǫ j t , where m j t, ξ j + η j t ≥ ∈ E 2 R, A is a given sol- vency margin and ǫ j t ∈ [0, 1] is a given acceptable non-solvency probability of S j for 1 ≤ j ≤ ℵ and t ∈ N , • c 8 τ f is a final state of the process η j such that if t t j f ω , then η j t ω = τ f , where t j f ω is the smallest time in N such that K j t j f m j t j f , ξ j + η j , and if t ≤ t j f ω , η j t − 1ω ∈ R N j , K j t − 1ω m j t − 1, ξ j + η j ω , then η j t ω ∈ R N j for ω ∈ Ω a.e., 1 ≤ j ≤ ℵ and t ≥ 1 the activity of S j ceases just after that the solvency margin is not satisfied. Let C c be the set of all η, E K 0, E D , satisfying v 1 –v 3 and satisfying the constraints c 1 –c 8 . Thus we sum up the constraints c 1 –c 8 on the form: • c η, E K 0, E D ∈ C c . Among all the possible functions to optimize, we simply choose the expected value of the final utility, η 7→ EU ∞, η + ξ . The optimization problem is now: given the initial equity K0, the dividend process Dξ + η, as a function of η, and the run-off ξ of H , satisfying d 1 , d 2 and d 3 , respectively, find the solutions ˆ η, ˆE K 0, ˆE D ∈ C c of the equation EU ∞, ˆ η + ξ = sup η, E K 0, E D∈C c EU ∞, η + ξ . 2.9 Due to the constraints c 4 on the ruin probability and c 7 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 c 1 and c 2 are just budget constraints. We have chosen the simplest form of the constraints c 3 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. E1U t + 1, ξ + η|F t ≥ ct P 1≤j ≤ℵ K j t . Of course, the expected value of the final utility will then in general be smaller for an optimal solution . c 4 is just a ruin constrain for H . c 5 is a very general constraint, which should cover most cases coming up in applications. c 6 says that, if S 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 − 1ω and η j i ≥ 0. In constraint c 8 , t j f ω is the smallest time such that the solvency margin is strictly negative for S j in the state ω . The constraint says that S j ceases its activities for times larger than t j f ω . Moreover, it says that, if S j has not yet ceased its activities and if the solvency margin is strictly positive for a time t before t j f ω , then S j does not cease its activities at t + 1. So, it is only when the solvency margin is zero, there is a choice. The constraints c 1 –c 5 and c 7 have a form as in formula 2.1. The constraints c 6 and c 8 can also be written on this form, which we give for later reference. Let λ j : R N j ∪ {τ f } 7→ {0, 1} be defined by λ j τ f = 1 and λ j x = 0, x ∈ R N j . The constraint c 6 is then given by 1 − λ j ◦ η j t η j i t − c j i ηt ≥ 0, 2.10 and 1 − λ j ◦ η j t ¯ c j i ηt − η j i t ≥ 0, 2.11 where 1 ≤ j ≤ ℵ, 1 ≤ i ≤ N j and 0 ≤ t ≤ ¯ T + T . In the case of c 8 , let s j t, x = min 0≤k≤t K j k − m j k, ξ j + η j , where K j k is evaluated at x = η, E K 0, E D . Let the step function H be defined by 72 E. Taflin Insurance: Mathematics and Economics 27 2000 65–81 H s = 0 if s 0 and H s = 1 if 0 ≤ s. The constraint c 8 is then given by 1 − λ j ◦ η j t s j t − 1, x ≥ 0, 2.12 λ j ◦ η j t s j t − 1, x ≤ 0, 2.13 and H s j t − 1, xλ j ◦ 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. Let η = η ′ , η ℵ and let u ℵ t, t ′ = pt + 1 − pt + dt + 1, for t t ′ , where p, d ∈ ER N ℵ , A , d 0 = 0, pt = p ¯ T + 1 and dt = 0, for t ¯ T . Then U t, η = U ′ t, η ′ + X 0≤k≤t−1 η ℵ k · pk + 1 − pk + dk + 1, 2.15 where U ′ is the utility function given by U ′ t, η ′ = U t, η ′ , 0, i.e. for the portfolios 1, 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 price pt ex dividend at time t and paying dividend dt 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 equity K j of the company S 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 2.2 with ℵ = 1 so U ′ = 0 and with D = 0. The equity is then given by Kt = 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 ηt · pt = Kt, for 0 ≤ t ≤ ¯ T then, using that here U t, η = U ¯ T , η for ¯ T t ≤ ¯ T + T , it follows that η is a self-financing strategy in the usual sense cf. Duffie, 1992, Section 6.K. Let us also admit that the company can invest positive amounts in a bond with positive price. Let p t 0 for t ≥ 0, be the price of the bond. If ηt · pt ≤ Kt then the rest is invested in a positive quantity η t of bonds such that Kt = ηt · pt + η t p t for 0 ≤ t ≤ ¯ T + T . The strategy ¯ η = η , η 1 , . . . , η N is then self-financing . 2.2. Quadratic Model The constraints c 4 on the ruin probability and c 7 on the non-solvency probabilities are replaced by stronger 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, where V denotes the variance operator: • c ′ 4 V P 1≤j ≤ℵ K j t ≤ ǫ ′ t δt K 2 and E P 1≤j ≤ℵ K j t ≥ δt K 0, t ∈ N, where ǫ ′ t ≥ 0 and δt 0, • c ′ 7 V K j t −m j t, ξ j +η j ≤ ǫ ′j t δ j t K 2 and EK j t −m j t, ξ j +η j ≥ δ j t K 0, for 1 ≤ j ≤ ℵ and t ∈ N, where m j ’s are as in c 7 , ǫ ′j t ≥ 0 and δ j t 0. For given initial equity K0, dividend process Dξ + η, as a function of η, and run-off ξ of H , satisfying d 1 , d 2 and d 3 , respectively, let C c ′ be the set of variables η, E K 0, E D, satisfying v 1 , v 2 and v 3 and satisfying the constraints c 1 –c 3 , c ′ 4 , c 5 , c 6 , c ′ 7 and c 8 . We sum up the constraints of the Quadratic Model on the form: • c ′ η, E K 0, E D ∈ C c ′ . Remark 2.4. Of course this model is not quadratic in η for general m , F α , c and ¯c. However, for common choices of these functions it is piecewise quadratic, which is the reason to keep the name quadratic . E. Taflin Insurance: Mathematics and Economics 27 2000 65–81 73 The optimization problem, in the case of the Quadratic Model, is now: given the initial equity K0, the dividend process Dξ + η, as a function of η, and the run-off ξ of H , satisfying d 1 , d 2 and d 3 , respectively, find the solutions ˆ η, ˆE K 0, ˆE D ∈ C c ′ of the equation EU ∞, ˆ η + ξ = sup η, E K 0, E D∈C c′ EU ∞, 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. If P 0≤k≤t ǫ ′ k ≤ ǫt and P 0≤k≤t ǫ ′j k ≤ ǫ j t , for t ∈ N and 1 ≤ j ≤ ℵ, then C c ′ ⊂ C c . 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: • h 1 independence with respect to the past: 1. u p∞ k is independent of F k for k ∈ Z and 1 ≤ p ≤ ℵ, 2. kEu p k, t 2 |F k k L ∞ ∞ for k t, • h 2 for k ∈ Z and 1 ≤ p ≤ ℵ, the N × N positive matrix c p k with elements c p ij k = Eu p∞ i k − Eu p∞ i ku p∞ j k − Eu p∞ j k is strictly positive, • h 3 u p∞ i k and u p∞ j l are independent for k 6= l, • h 4 u p∞ i k and u r∞ j l are independent for p 6= r. We note that the second point of h 1 is trivially satisfied if u p k, t is independent of F k for k 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 if ξ is as in 2.16, then the functions η j 7→ m j t, ξ j + η j , η 7→ Dξ + η, η 7→ c j i η , η 7→ ¯c j i η and η, E K 0, E D 7→ F α t, η, E K 0, E D are defined for η j ∈ ˜ P j u, ¯ T , η ∈ ˜ P u, ¯ T and η, E K 0, E D ∈ ˜ P u, ¯ T × R ℵ × E 2 R ℵ , 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 where θ j ∈ ˜ P j ¯ T , and the formula 2.3 gives Kt = K 0 + U t, θ − X 1≤k≤t Dk, θ , 2.18 when constraints c 1 and c 2 are satisfied and θ ∈ ˜ P ¯ T . Theorem 2.6. Let the utilities u p k, t and u p∞ k , of unit contracts, satisfy h 1 –h 4 . Let the functions η j 7→ m j t, ξ j + η j , η 7→ Dξ + η and η 7→ c j i η to E 2 R, A map bounded sets into bounded sets. In the d-topology, let x = η, E K 0, E D 7→ x, U j ·, ξ j + η j , m j ·, ξ j + η j , Dξ + η be continuous, let η, E K 0, E D 7→ F α t, η, E K 0, E D , α ∈ I , be lower semi-continuous and let η 7→ c j i ηt, ω and η 7→ − ¯c j i ηt, ω be lower semi-continuous a.e.. If V   X 1≤k≤ ¯ T +T Dk, θ   ≤ c 2 V U ∞, θ , 74 E. Taflin Insurance: Mathematics and Economics 27 2000 65–81 where 0 ≤ c 1, and if C c ′ is non-empty, then the optimization problem 2.16 has a solution ˆ x = ˆ η, ˆE K 0, ˆE D ∈ C c ′ . 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 solution ˆ x = ˆ η, ˆE K 0, ˆE D of optimization problem 2.16 given by Theorem 2.6 is a sub-optimal solution of optimization problem 2.9, when the hypotheses of Theorem 2.5 are satisfied, i.e. ˆ x ∈ C c and EU ∞, ˆ η + ξ − sup η, E K 0, E D∈C c EU ∞, η + ξ ≤ 0. If the difference is δ, then an inspection of the proofs of this paper shows that δ → 0 when ǫ j t and ǫt 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 let λ j be as in 2.10, let p j : R N j ∪ {τ f } → R N j be defined by p j τ f = 0 and p j x = x , for x ∈ R N j , let p = p 1 , . . . , p ℵ and λ = λ 1 , . . . , λ ℵ . Let E 2 ¯ T R N , A be the subset of elements η ∈ E 2 R N , A , such that ηt = 0 for t ¯ T . By definition, if η ∈ ˜ P u, ¯ T then p ◦ η ∈ E 2 ¯ T R N , A , where N = P 1≤j ≤ℵ N j and λ ◦ η ∈ E 2 ¯ T R ℵ , A . We introduce the Hilbert spaces H = E 2 ¯ T R N , A , H = H ⊕ E 2 ¯ T R ℵ , A and H 1 of elements E K 0, E D ∈ R ℵ ⊕ E 2 ¯ T R ℵ , A , such that E D 0 = 0. Let H = H ⊕ H 1 . The optimization problem can now be formulated using the variable x = α, β, E K 0, E D in H , where α, β ∈ H and where for solutions α = p ◦ η and β = λ ◦ η . The constraints 1 − β j β j = 0 and α j β j = 0 shall then be satisfied. The constraints c 1 –c 3 , c ′ 4 and c ′ 7 are easily expressed in the new variables. The constraint c 6 is reformulated by using 2.10 and 2.11 and constraint c 8 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 in Taflin 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 solution ˆ x ∈ C c ′ , of Theorem 2.6, is not unique. This fact can be traced back to a simplified case, namely where only the constraints c 1 –c 4 are considered and where all η j i ≥ 0. To state the result let C c ′′ be the set of all η, E K 0, E D ∈ ˜ P u, ¯ T × R ℵ × E 2 R ℵ , A , satisfying v 1 –v 3 , c 1 , c 2 and c ′ 4 . We consider the following optimization problem: given K0 ≥ 0, D = 0 and ξ = 0, find the solutions ˆ x ∈ C c ′′ of the equation EU ∞, ˆ η = sup η, E K 0, E D∈C c′′ EU ∞, η. 2.19 Theorem 2.9. If C c ′′ is non-empty, then the optimization problem 2.19 has a solution ˆ x ∈ C c ′′ . Moreover, ˆ η is unique and ˆE K 0, ˆE D only has to satisfy K 0 = P 1≤j ≤ℵ d K j 0 and 0 = P 1≤j ≤ℵ d D j . In the situation of the theorem, there is a whole hyperplane R ℵ−1 × E 2 R ℵ−1 , A translated of solutions in the variable ˆE K 0, ˆE D . 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. Remark 2.10. Suppose that a solution ˆ x of Theorem 2.9 satisfies condition c ′ 7 for given m j and ǫ j , 1 ≤ j ≤ ℵ. Apart from exceptional cases, there will be a whole neighbourhood of elements E K ′ 0, E D ′ in a submanifold homeomorphic to R ℵ−1 × E 2 R ℵ−1 , A , which also give solutions y = ˆ η, E K ′ 0, E D ′ . In the general situation of Theorem 2.5, this degeneracy can be partially reduced because some of the constraints will saturate. However, a E. Taflin Insurance: Mathematics and Economics 27 2000 65–81 75 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 Lagrangian L constructed according to Remark 2.8. Let the Lagrangian L be the sum of EU ∞, η + ξ and the terms with multipliers corresponding to constraints c 1 –c 3 , c 5 and c 6 . To obtain explicit expressions of the terms L 1 and L 2 of L corresponding to the linear constraints and variance constraints respectively in c ′ 4 and c ′ 7 let L 1 = − X 0≤t≤ ¯ T +T µ 1,t  K0 + E  Ut, θ − X 1≤k≤t Dk, θ   − δtK0   , 2.20 L j 1 =− X 0≤t≤ ¯ T +T µ j 1,t  K j 0+E  U j t, θ j − X 1≤k≤t D j k − m j t, ξ j +η j   − δ j t K   , 2.21 L 2 = X 0≤t≤ ¯ T +T µ 2,t  V  Ut, θ − X 1≤k≤t Dk, θ   − ǫ ′ t δt K 2   , 2.22 and let L j 2 = X 0≤t≤ ¯ T +T µ j 2,t  V  U j t, θ j − X 1≤k≤t D j k − m j t, ξ j + η j   −ǫ ′j t δ j t K 2   , 2.23 where 1 ≤ j ≤ ℵ. Using formulas 2.17 and 2.18, L i can be written as L i = X 0≤j ≤ℵ L j i , i = 1, 2. 2.24 There are Kuhn–Tucker conditions, which we do not give explicitly, corresponding to the Lagrange multipliers µ j i,t , where 0 ≤ j ≤ ℵ, i = 1, 2 and 0 ≤ t ≤ ¯ T + T . Let L ′ = L + L 1 + L 2 . A possible algorithm to construct approximate solutions consists of determining the sequence x n n≥ , where x n = α n , β n , E K n 0, E D n is defined like x = α, β, E K 0, E D in Remark 2.8. Let β = 0, and for given β n , let x n be a solution of the optimization problem given by the Lagrangian α n , E K n 0, E D n 7→ L ′ x n with multipliers and let a β n+ 1 be determined by constraint c 8 . The complexity of this problem, for realistic choices like piecewise linear of the functions F α , m j , D , c j i and ¯c j i , is similar to the complexity of the Basic Model of Section 2.3, with N = P 1≤j ≤ℵ N j . In fact the derivative of the function α 7→ L ′ x , giving the “complex part” of the Euler–Lagrange equations, is a piecewise linear affine map in H, whose linear part has properties similar to those of the operator B in formula 2.32 for details, see Taflin, 1998. We recall that the difficult parts, in this context, for the Basic Model was the inversion of B, which was explicitly done in Taflin 1998, Appendix A and the determination of multipliers ν ∈ H corresponding to constraints like η j i ≥ 0, for which an approximation algorithm was proposed in Taflin 1998, Appendix B. Numerical simulations for the Basic Model with ¯ T = 2 were done in Dionysopoulos 1999. A non-trivial random vector ν is present in this case. The resolution took some minutes in a low-end 1998 C-programming environment. 76 E. Taflin Insurance: Mathematics and Economics 27 2000 65–81 A study of the Lagrangian L ′ 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 of L ′ with respect to the variable E K 0, E E D leads to an Euler–Lagrange equation, being a closed system in the variable E K 0, E E D . 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 subsidiaries ℵ = 1, the run-off ξ = 0, the dividends D = 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 portfolio η is an element of the Hilbert space H = P u, ¯ T so ηω 6= τ f on Ω. We remind that, in this situation, the equity Kt = K 0 + U t, η, 2.25 where K0 ≥ 0 is the initial equity at t = 0. In the sequel of this paragraph, we closely follow Taflin 1998. Constraints on the variable η are introduced: • C 3 E1U t + 1, η ≥ ctEKt, ct ∈ R + is given constraint on profitability, • C 4 EU ∞, η − EU ∞, η 2 ≤ σ 2 , where σ 2 0 is given acceptable level of the variance of the final utility, • C 6 0 ≤ η i t , where 1 ≤ i ≤ N only positive subscription levels, Let C be the set of portfolios η ∈ H such that constraints C 3 , C 4 and C 6 are satisfied. This is well-defined. In fact the quadratic form η 7→ aη = EU ∞, η 2 2.26 in H, has a maximal domain Da, since for each η ∈ H, the stochastic process U t, θ t ≥ is an element of the space E p R, A for 1 ≤ p 2 which follows directly from Hölder’s inequality. The optimization problem is now, to find all ˆ η ∈ C , such that EU ∞, ˆ η = sup η∈C EU ∞, η. 2.27 The solution of this optimization problem is largely based on the study of the quadratic form η 7→ bη = EU ∞, η − EU ∞, η 2 2.28 in H, with maximal domain Db = Da. We make certain technical hypotheses on the claims processes: • H 1 u ∞ k is independent of F k for k ∈ N, • H 2 for k ∈ N the N × N positive matrix ck with elements c ij k = Eu ∞ i k − Eu ∞ i ku ∞ j k − Eu ∞ j k is strictly positive, • H 3 u ∞ i k and u ∞ j l are independent for k 6= 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 forms b and a is equivalent to the norm in H. Theorem 2.11. If the hypotheses H 1 –H 3 are satisfied, then the quadratic forms b and a are bounded from below and from above, by strictly positive numbers c and C respectively, where 0 c ≤ C, i.e. ckηk 2 H ≤ bη ≤ aη ≤ Ckηk 2 H 2.29 for η ∈ H . E. Taflin Insurance: Mathematics and Economics 27 2000 65–81 77 The operators B resp. A in H, associated with b resp. a, by the representation theorem, i.e. bξ, η = ξ, Bη H resp., aξ, η = ξ, Aη H 2.30 for ξ ∈ H and η ∈ H, are strictly positive, bounded, self-adjoint operators onto H with bounded inverses. There exist c ∈ R, such that 0 cI ≤ B ≤ A, where I is the identity operator. It follows from formula 2.30, that an explicit expression of A is given by Aηk = EU ∞, ηu ∞ k|F k , 2.31 and that an explicit expression of B is given by Bηk = EU ∞, η − EU ∞, ηu ∞ k|F k 2.32 for η ∈ 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 H 1 – H 3 be satisfied. If C is non-empty, then optimization problem 2.27 has unique solution ˆ η ∈ C . The solution ˆ η is given by a constructive approach in Taflin 1998. In fact, in that reference a Lagrangian formalism, an algorithm to invert the operators A and B and approximation methods for determining the multipliers are given.

3. Proofs