Stochastic Control in Continuous Time

Schmidli, Hanspeter

doi:10.1007/978-1-84800-003-2_2

Cited by 91 publications

(149 citation statements)

References 6 publications

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“…Note that V (x) = 0 for x < 0. The derivation of (1) follows standard arguments, see Fleming & Soner [11] or Schmidli [17]. If we define ∆ as the death state, the generator of the killed risk reserve (as a part of (1)) is given by…”

Section: Exponential Time Horizonmentioning

confidence: 99%

“…using V (x, L) instead of V (x, L)) leads to the original problem without killing, but with the modified discounting factor δ + γ instead of δ. So in this case the effect of a finite random time horizon is just a simple shift of the discounting parameter (the solution to this problem can correspondingly be found in Azcue & Muler [7] and Schmidli [17]).…”

Section: Exponential Time Horizonmentioning

confidence: 99%

“…Like in the proof of Theorem 2.5, this condition together with f ′′′ (b 1 ,b * that at b 1 the derivative attains a minimum with value 1. This relates to the construction procedure for optimal band strategies in the infinite horizon case as described in Schmidli [17] , where one searches for minima equal to 1 of derivatives of solutions to the IDE part of the associated HJB equation.…”

Section: State-dependent Barrier Strategiesmentioning

confidence: 99%

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On optimal dividend strategies in insurance with a random time horizon

Albrecher¹,

Thonhauser²

2012

Advances in Statistics, Probability and Actuarial Science

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For the classical compound Poisson surplus process of an insurance portfolio we investigate the problem of how to optimally pay out dividends to shareholders if the criterion is to maximize the expected discounted dividend payments until the time of ruin or a random time horizon, whichever is smaller. We explicitly solve this problem for an exponential time horizon and exponential claim sizes. Furthermore, we study the case of an Erlang(2) time horizon by introducing an external state process and derive the solution under the assumption that the external state process is observable. The results are illustrated by numerical examples.

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Section: Exponential Time Horizonmentioning

confidence: 99%

Section: Exponential Time Horizonmentioning

confidence: 99%

Section: State-dependent Barrier Strategiesmentioning

confidence: 99%

See 1 more Smart Citation

On optimal dividend strategies in insurance with a random time horizon

Albrecher¹,

Thonhauser²

2012

Advances in Statistics, Probability and Actuarial Science

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“…It then follows from [16] that there is an optimal stopping boundary a * > 0 that fully characterises the solution of (5.39) and the stopping set is [a * , ∞) (an expression for a * can be found in Schmidli [43,Theorem 2.53] with the notation m = μ 1 and δ = ρ). We now notice that using Girsanov's theorem and (4.30), we obtain from (4.19) that…”

Section: Proposition 513mentioning

confidence: 99%

Optimal dividends with partial information and stopping of a degenerate reflecting diffusion

Angelis

2019

Finance Stoch

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We study the optimal dividend problem for a firm's manager who has partial information on the profitability of the firm. The problem is formulated as one of singular stochastic control with partial information on the drift of the underlying process and with absorption. In the Markovian formulation, we have a two-dimensional degenerate diffusion whose first component is singularly controlled. Moreover, the process is absorbed when its first component hits zero. The free boundary problem (FBP) associated to the value function of the control problem is challenging from the analytical point of view due to the interplay of degeneracy and absorption. We find a probabilistic way to show that the value function of the dividend problem is a smooth solution of the FBP and to construct an optimal dividend strategy. Our approach establishes a new link between multidimensional singular stochastic control problems with absorption and problems of optimal stopping with 'creation'. One key feature of the stopping problem is that creation occurs at a state-dependent rate of the 'local time' of an auxiliary two-dimensional reflecting diffusion.

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“…The above is obtained by simplifying the double integrals in the last two terms by using integration by 217 parts again and switching the order of integration using Fubini's Theorem [29]. Recall that F(0) = 0 and 218 F(x − ) = F(x) for x ∈ R, F being absolutely continuous with respect to Lebesgue measure.…”

mentioning

confidence: 99%

Minimizing an Insurer's Ultimate Ruin Probability by Noncheap Proportional Reinsurance Arrangements and Investments

Kasumo¹

2019

Preprint

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In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard Black-Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate and a single risky asset whose price process is modelled by a geometric Brownian motion. Additionally, the company is allowed to purchase noncheap proportional reinsurance priced via the expected value principle. Using the Hamilton-Jacobi-Bellman approach, we derive a second-order Volterra integrodifferential equation which we transform into a linear Volterra integral equation of the second kind. We proceed to solve this integral equation numerically using the block-by-block method for the optimal reinsurance retention level that minimizes the ultimate ruin probability. The numerical results based on light- and heavy-tailed distributions show that proportional reinsurance and investments play a vital role in enhancing the survival of insurance companies. But the ruin probability exhibits sensitivity to the volatility of the stock price.

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Stochastic Control in Continuous Time

Cited by 91 publications

References 6 publications

On optimal dividend strategies in insurance with a random time horizon

On optimal dividend strategies in insurance with a random time horizon

Optimal dividends with partial information and stopping of a degenerate reflecting diffusion

Minimizing an Insurer's Ultimate Ruin Probability by Noncheap Proportional Reinsurance Arrangements and Investments

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