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Measuring the suboptimality of dividend controls in a Brownian risk model

Part of: Stochastic analysis Mathematical economics Stochastic systems and control

Published online by Cambridge University Press:  07 June 2023

Julia Eisenberg  and
Paul Krühner
Julia Eisenberg*
Affiliation:
Technische Universität Wien
Paul Krühner*
Affiliation:
Wirtschaftsuniversität Wien
*
*Postal address: Wiedner Hauptstraße 8, 1040 Wien, Austria. Email address: jeisenbe@fam.tuwien.ac.at
**Postal address: Welthandelsplatz 1, 1020 Wien, Austria. Email address: peisenbe@wu.ac.at
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Abstract

We consider an insurance company modelling its surplus process by a Brownian motion with drift. Our target is to maximise the expected exponential utility of discounted dividend payments, given that the dividend rates are bounded by some constant. The utility function destroys the linearity and the time-homogeneity of the problem considered. The value function depends not only on the surplus, but also on time. Numerical considerations suggest that the optimal strategy, if it exists, is of a barrier type with a nonlinear barrier. In the related article of Grandits et al. (Scand. Actuarial J. 2, 2007), it has been observed that standard numerical methods break down in certain parameter cases, and no closed-form solution has been found. For these reasons, we offer a new method allowing one to estimate the distance from an arbitrary smooth-enough function to the value function. Applying this method, we investigate the goodness of the most obvious suboptimal strategies—payout on the maximal rate, and constant barrier strategies—by measuring the distance from their performance functions to the value function.

Keywords

Suboptimal controlHamilton–Jacobi–Bellman equationdividend payoutsexponential utility function

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 91B30: Risk theory, insurance 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 4 , December 2023 , pp. 1442 - 1472
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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