Verification theorem Directory UMM :Data Elmu:jurnal:I:Insurance Mathematics And Economics:Vol27.Issue2.2000:

C. Hipp, M. Plum Insurance: Mathematics and Economics 27 2000 215–228 217 If for surplus s = 0 there is a positive investment in the asset, then the risk in the asset will produce an immediate ruin and so 1 = ψ0 ψ 0, which cannot be optimal. So in the compound Poisson risk process case, the form of the optimal investment strategy cannot be constant. Consumers or supervisory authorities will use ruin probability as objective function. Shareholders would prefer other objective functions such as expected discounted dividends. Other optimization problems would be: optimiza- tion of reinsurance programs see Hoejgaard and Taksar, 1998; Schmidli, 1999 or of new business see Hipp and Taksar, 2000. 1.1. Bellman equation The computation of the optimal investment strategy is based on the Bellman equation see Fleming and Soner, 1993, p. 12, Eq. 5.3 ′ . This integro-differential equation for δs = 1 − ψs is derived as follows: consider a short interval [0, dt ] of length dt in which θ shares are held. Then there will be a claim of size Y ∼ Q with probability λ dt , and if Y s, then the risk process continues not leading to ruin with probability δs − Y . If no claim occurs in the time interval, then there will be no ruin in the future with probability δs + c dt + θxa dt + θxb dW t, and this will happen with probability 1 − λ dt. Taking expectations, we obtain with Itô’s lemma δs = δs + {λE[δs − Y − δs] + c + aθxδ ′ s + 1 2 b 2 θ 2 x 2 δ ′′ s } dt, s 0. Introducing A = θx and maximizing, we obtain the Bellman equation for our problem: sup A {λE[δs − Y − δs] + c + aAδ ′ s + 1 2 b 2 A 2 δ ′′ s } = 0, s 0. 1.3 The supremum exists whenever δ ′′ s 0, and in this case A = As = − a b 2 δ ′ s δ ′′ s . 1.4 The function δs will satisfy 1.3 only if δs is strictly concave, strictly increasing, twice continuously differen- tiable, and satisfies δs → 1 for s → ∞. In this case, the optimal investment strategy will be θ t = AT t − Xt − , where As is the optimizer 1.4. In this argument, we assume that δs is a smooth function. As this is not an essential assumption, we shall start from a smooth solution of 1.3 and use the following verification theorem.

2. Verification theorem

Here, we shall show that from Bellman’s equation, we obtain a strategy which dominates all admissible strategies. Theorem 2.1. Assume that there exists a solution δ ∗ s of the Bellman equation 1.3 with maximizing function A ∗ s with the following properties: δ ∗ 0, δ ′ ∗ 0 0, δ ∗ s = 0 for s 0, lim s →∞ δ ∗ s = 1, and δ ∗ s is twice continuously differentiable on {s 0}. Then δ ′ ∗ s 0 for s 0, and if θ t is an arbitrary admissible 218 C. Hipp, M. Plum Insurance: Mathematics and Economics 27 2000 215–228 investment strategy for which the reserve process T s, θ, t is defined on 0 ≤ t ∞, then the corresponding survival probability δs satisfies δs ≤ δ ∗ s, s ≥ 0, with equality for θ ∗ t = A ∗ T t −Xt. Proof. Assume that δ ′ ∗ s = 0, and s is the smallest positive value with this property. Then = sup A {λE[δ ∗ s − Y − δ ∗ s ] + 1 2 A 2 δ ′′ ∗ s }, which implies δ ′′ ∗ s ≤ 0, and thus E[δ ∗ s − Y − δ ∗ s ] ≥ 0; but this contradicts δ ∗ y δ ∗ s for 0 y s . Let now θ t be an arbitrary admissible strategy. We write T s, θ, t for the process with this investment strategy θ t and initial surplus s. Fix a positive ε to be chosen arbitrarily small later. Let τ ∗ and τ be the times of first ruin for T s + ε, θ ∗ , t and T s, θ, t , respectively, where θ ∗ t = A ∗ T s, θ ∗ , t −. Let θ 1 t be the strategy θ 1 t = θt + ε 2 Xt . Write τ 1 and δ 1 s for the corresponding ruin time and the survival probability using strategy θ 1 t when starting at s + ε, respectively. We have T s + ε, θ 1 , t = T s, θ, t + ε + aε 2 t + bε 2 W t , P {τ 1 τ } ≤ P {ε + aε 2 t + bε 2 W t 0 for some t } = exp − 2a 2 b 2 ε . Furthermore, T s + ε, θ 1 , t → ∞ on the set {τ = ∞}. 2.1 We observe that δ ∗ T s + ε, θ 1 , t ∧ τ 1 2.2 is a local supermartingale, while δ ∗ T s + ε, θ ∗ , t ∧ τ ∗ 2.3 is a local martingale. Since δ ∗ s is bounded and convergent for s → ∞, the process 2.2 is a supermartingale, and 2.3 is a martingale. Therefore, for all t 0 δ ∗ s + ε ≥ Eδ ∗ T s + ε, θ 1 , t ∧ τ 1 . Now let t → ∞. Since δ ∗ u → 1 for u → ∞, we obtain with 2.1 δ ∗ s + ε ≥ lim t →∞ Eδ ∗ T s + ε, θ 1 , t ∧ τ 1 1 τ =∞} = P{τ = ∞ and τ 1 = ∞} ≥ δs − P {τ 1 τ } ≥ δs − exp − 2a 2 b 2 ε . C. Hipp, M. Plum Insurance: Mathematics and Economics 27 2000 215–228 219 Now ε → 0 yields δ ∗ s ≥ δs. On the other hand, δ ∗ s + ε = Eδ ∗ T s + ε, θ ∗ , t ∧ τ ∗ which, with t tending to infinity, implies δ ∗ s + ε ≤ P {τ ∗ = ∞}.

3. Existence of a solution