Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model
Abstract
:1. Introduction
- x is the initial surplus;
- q is a discount factor;
- are the cumulative dividends;
- are the cumulative forced capital injections each time the process attempts to cross into , and is a proportional cost for injecting capital;
- is the process modified by capital injections and dividends, and .
- ; the optimal policy is (RP), i.e., pay dividends at an upper barrier and always inject capital at 0, implying that bankruptcy never occurs;
- ; the optimal policy is (AP), i.e., pay dividends in order to reflect the surplus process at some upper barrier , and never inject capital.
- we prove the optimality of policies such as those described at the very beginning for which, in general, (i.e., de Finetti and Shreve, Lehoczky and Gaver policies fail to be optimal);
- the resulting value function is not of class at 0 (in the sense that its derivative does not exist at and, more, the right-hand derivative at 0 is not k as assumed, for Brownian claims, in [14], 5.2). This can be found in Remark 14 and it implies, in particular, that the verification Theorem 4 has to be given for absolutely-continuous functions. Working with super-solutions also explains the particular form of the Equation (8) on .
- the optimal parameters and are completely characterized by non-trivial equations and so is the alternative-inducing cost .
- Guess a family of policies which yields the optimum for all possible values of the parameters, and compute its expected net present value (EPV) in terms of the scale functions.
- 2.
- Identify the optimal arguments with respect to the parameters of our policies. This step forced us to restrict to the case of exponential jumps, where the independence of ruin and ruin overshoots leads to simplifying factorizations related to the memoryless property of this distribution.
- 3.
- In the final step, confirm the optimality of the selected candidate optimal policy via a verification theorem (sufficient condition for optimality). If the conjectured value function were sufficiently smooth (this means in our problem), this would require only verifying that it satisfies an associated HJB equation (system of variational inequalities).
2. Optimizing Dividends and Capital Injections with Proportional Costs
2.1. The Framework
- given a couple : describing dividends and capital injection, the modified surplus process is defined (under the ) by setting
- the cumulative dividend strategy L is adapted, non-decreasing and càdlàg (right-continuous, left-limits), ;
- the cumulative capital injection process I is adapted, non-decreasing and càdlàg, ;
- the triplet satisfying the previous conditions is referred to as (general) strategy and the family of all such strategies is denoted by .
- (prior to ruin) for every , the dividends should satisfy
- (after the ruin time ), we set (The reader is invited to note that in this case one of the jumping times for N such that remain adapted, non-decreasing and càdlàg).
- the satisfying the dividends restriction is called an admissible strategy and the family of all such strategies is denoted by .
- for an admissible strategy ,
- (a)
- we consider the ruin time (if a “very large” claim occurs, bankruptcy is declared; as we will see in the main result, it is never optimal to take and modify the equity I accordingly).
- (b)
- the associated cost is
- (c)
- Every strategy can be replaced with by modifying if such that and improving the associated cost . From now on, whenever a policy is considered, we identify it with and reason accordingly.
- (d)
- In connection to this type of policies and the related cost, we set
2.2. The Value Function
- First, we focus on the regularity properties of the value function (lower and upper-bound and Lipschitz-continuity) in Proposition 2.
- Second, we prove the connection between this value function and the associated partial-integral differential system (of HJB-type) in Proposition 3.
- Third, we show in Theorem 4 that the value function is the lowest absolutely-continuous super-solution of the associated equation and, as a verification result, give the optimality condition for candidate policies.
2.3. Some Elementary Properties of the Value Function
- For every , the set of admissible strategies is non-empty. If , then and, for every , .
- For every and every , one has .
- If , then, for every and every policy , one has .
- For every and every , .
2.4. The HJB System
2.5. The Value Function as Smallest Super-Solution
- The value function is non-negative, for and .
- Every non-negative -regular of growth viscosity super-solution of (8), where α is introduced in Proposition 15, is greater than or equal to on .
- If is a family of admissible strategies such that the associated costs is an super-solution for (8), then is optimal and .
3. The Value Function for the Cramér-Lundberg Model
3.1. The Guess Step: Severity-Constrained Double Barrier Policies, for the Cramér-Lundberg Process with Exponential Jumps
- the inverse Laplace transform of , where is the Laplace exponent, and
- .
- ;
- ;
- ;
- , for all ;
- The cost function satisfies, for ,In particular, if we setthen
- Moreover, the cost can be explicitly written asand
- The first step of the previous proposition applies also to the perturbed Cramér-Lundberg process obtained by adding a Brownian term to (3). The second and third however use specific features available only if (in particular that ).
- Note thatwhere the last identity as well as the notation appear in [15]. Our formula interpolates between the de Finetti and Shreve, Lehoczky and Gaver cases:(Again, for details, the reader is guided to [15] and references therein).
- The Equation (20) may also be expressed as
- a term depending on b, which is multiplied by the scale function , and
- a term independent of b, , which has been called sometimes “smooth Gerber-Shiu function”—see for example [35], and also “smooth harmonic extension of , see [2]. It is striking that the scale function and the Gerber-Shiu function are the same for these three problems. This begs for a formula for the Gerber-Shiu function which does not depend on the problem, and such a candidate is offered by the “LRZ harmonic extension” obtained in [36,37]. However, this has only been rigorously proved in particular examples – see for example [12].
4. The Optimal for the Cramér-Lundberg Model with Exponential Claims
4.1. Preliminary Remarks
- As a consequence, picking can never be optimal.
- For fixed and , there exists a unique critical point satisfying
4.2. Determining the Candidate Maximal Arguments
- either , in which case (27) implies that satisfies
- or satisfiesand
5. From Guess to Optimality
- A.
- If , then, we have the following dichotomy.
- A1.
- The “cheap” equity regime, with , holds for , where is the unique solution of
- A2.
- For “expensive” equity i.e., ,
- B.
- In the remaining case , independently of , and the value function is got as in A1:
- A careful look at the upper-bound in 2.2.2 (in the proof) shows that . As a consequence:
- the heuristic intuition in ([14], 5.2) no longer holds true for our case ( and fails to hold true);
- is only absolutely continuous (but not ) such that the comparison in Theorem 4 must be given among absolutely continuous super-solutions.
- The equality in (50) gives a way to characterize when is known.
- As , the claims approach ∞ average. In a highly impacted market, the notion of equity expensiveness vanishes and it is not optimal to wait for any amount of time (Figure 2a).
- On the other hand (Figure 3a), as the discount parameter , for all p that may give a dichotomy (cf. A), such that the Equation (31) leads to . In other words, absence of inflation makes every equity expensive. Conversely (Figure 3c), high inflation (large q) makes the notion of equity expensiveness vanish.
6. Proofs of the Results
6.1. Proofs for the Value Function: Estimates, Dynamic Programming, Solution
- The classical (no dividend, no injection) policy is an admissible strategy. The second result is a mere consequence of the definition of .
- For , we consider the strategy consisting in no capital injection (), then paying and (continuously) the premium and declare bankruptcy at the first (positive) claim. For this admissible strategy , the ruin time (the first jumping time except on the 0-probability event ). We have
- One merely notes that, for every ,
- 4.
- The first inequality follows by modifying an arbitrary strategy into on to get . For the last inequality, one modifies (on ) an admissible strategy into .
- on ;
- on ;
- (pointwise a.e. on ).
6.2. Proofs for the Guess Step
6.3. Proofs for the Optimal Couple
6.4. Proof of the Main Theorem
- if and only if ;
- , for all .
- The continuous function satisfies and . We let denote the first solution of .
- We consider . One notes that . Moreover, such that , for all . In particular, . Owing to the monotonicity of , one deduces that .
Author Contributions
Funding
Conflicts of Interest
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Avram, F.; Goreac, D.; Li, J.; Wu, X. Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model. Mathematics 2021, 9, 931. https://doi.org/10.3390/math9090931
Avram F, Goreac D, Li J, Wu X. Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model. Mathematics. 2021; 9(9):931. https://doi.org/10.3390/math9090931
Chicago/Turabian StyleAvram, Florin, Dan Goreac, Juan Li, and Xiaochi Wu. 2021. "Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model" Mathematics 9, no. 9: 931. https://doi.org/10.3390/math9090931