Stochastic Optimization in Insurance

Azcue, Pablo; Muler, Nora

doi:10.1007/978-1-4939-0995-7

Cited by 52 publications

(19 citation statements)

References 49 publications

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“…Proof. The proof of the second inequality is similar to the one of Proposition 1.2 of [9]: for any initial surplus levels (x, y) ∈ R 2 + , x + p 1 t and y + p 2 t are upper bounds for the surplus levels of company one and company two at time t, respectively (even after the ruin time τ of one of the companies), so the cumulative dividends paid by company one and company two up to time t are less than or equal to x + p 1 t and y + p 2 t, respectively. So, since e −δs is a positive and decreasing function,…”

Section: Lemma 21 the Optimal Value Function Is Well Defined And Samentioning

confidence: 83%

“…In order to obtain the Hamilton-Jacobi-Bellman (HJB) equation associated to the optimization problem (2.4), we need to state the so called dynamic programming principle (DPP). The proof that this holds is similar to the one given in Lemma 2.1 of [9] and uses the fact that V is increasing and continuous in R 2 + .…”

Section: Hamilton-jacobi-bellman Equationmentioning

confidence: 84%

“…Arguing by contradiction, we assume that V is not a subsolution of (3.1) at (x 0 , y 0 ) with x 0 > 0 and y 0 > 0. With a similar proof to Proposition 3.1 of [9], but extending the definitions to two variables, we first show that there exists ε > 0, h ∈ (0, min{x 0 /2, y 0 /2}), and a continuously differentiable function ψ : R 2 + → R such that ψ is a test function for the subsolution of (3.1) at (x 0 , y 0 ) and satisfies…”

Section: Proposition 32 It Holds That V Is a Viscosity Subsolution mentioning

confidence: 93%

“…Optimal dividend strategies for two collaborating insurance companies 527 Proof. The proof follows by standard convolution arguments and is the extension to two variables of Lemma 4.1 of [9]. Proposition 4.1.…”

Section: Smallest Viscosity Solutionmentioning

confidence: 98%

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Optimal dividend strategies for two collaborating insurance companies

2017

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We consider a two-dimensional optimal dividend problem in the context of two insurance companies with compound Poisson surplus processes, who collaborate by paying each other's deficit when possible. We solve the stochastic control problem of maximizing the weighted sum of expected discounted dividend payments (among all admissible dividend strategies) until ruin of both companies, by extending results of univariate optimal control theory. In the case that the dividends paid by the two companies are equally weighted, the value function of this problem compares favorably with the one of merging the two companies completely. We identify this optimal value function as the smallest viscosity supersolution of the respective Hamilton-Jacobi-Bellman equation and provide an iterative approach to approximate it numerically. Curve strategies are identified as the natural analogue of barrier strategies in this two-dimensional context. A numerical example is given for which such a curve strategy is indeed optimal among all admissible dividend strategies, and for which this collaboration mechanism also outperforms the suitably weighted optimal dividend strategies of the two stand-alone companies.

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Section: Lemma 21 the Optimal Value Function Is Well Defined And Samentioning

confidence: 83%

Section: Hamilton-jacobi-bellman Equationmentioning

confidence: 84%

Section: Proposition 32 It Holds That V Is a Viscosity Subsolution mentioning

confidence: 93%

Section: Smallest Viscosity Solutionmentioning

confidence: 98%

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Optimal dividend strategies for two collaborating insurance companies

2017

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“…Its treatment goes back to Gerber [6] and is analyzed in terms of optimal stochastic control by Azcue & Muler [2, 3] and Schmidli [11]. The opposite extremal case resembles the situation of a possible capital injection at any point in time, which is discussed in Sect.…”

Section: Introduction and Some First Considerationsmentioning

confidence: 99%

On a dividend problem with random funding

Strini

Thonhauser

2019

Eur. Actuar. J.

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We consider a modification of the dividend maximization problem from ruin theory. Based on a classical risk process we maximize the difference of expected cumulated discounted dividends and total expected discounted additional funding (subject to some proportional transaction costs). For modelling dividends we use the common approach whereas for the funding opportunity we use the jump times of another independent Poisson process at which we choose an appropriate funding height. In case of exponentially distributed claims we are able to determine an explicit solution to the problem and derive an optimal strategy whose nature heavily depends on the size of the transaction costs. Furthermore, the optimal strategy identifies unfavourable surplus positions prior to ruin at which refunding is highly recommended.

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2017

Reinsurance

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Stochastic Optimization in Insurance

Cited by 52 publications

References 49 publications

Optimal dividend strategies for two collaborating insurance companies

Optimal dividend strategies for two collaborating insurance companies

On a dividend problem with random funding

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