The Annals of Applied Probability

Optimal dividend and investment problems under Sparre Andersen model

Lihua Bai, Jin Ma, and Xiaojing Xing

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Abstract

In this paper, we study a class of optimal dividend and investment problems assuming that the underlying reserve process follows the Sparre Andersen model, that is, the claim frequency is a “renewal” process, rather than a standard compound Poisson process. The main feature of such problems is that the underlying reserve dynamics, even in its simplest form, is no longer Markovian. By using the backward Markovization technique, we recast the problem in a Markovian framework with expanded dimension representing the time elapsed after the last claim, with which we investigate the regularity of the value function, and validate the dynamic programming principle. Furthermore, we show that the value function is the unique constrained viscosity solution to the associated HJB equation on a cylindrical domain on which the problem is well defined.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 6 (2017), 3588-3632.

Dates
Received: June 2016
Revised: January 2017
First available in Project Euclid: 15 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1513328709

Digital Object Identifier
doi:10.1214/17-AAP1288

Mathematical Reviews number (MathSciNet)
MR3737933

Zentralblatt MATH identifier
06848274

Subjects
Primary: 91B30: Risk theory, insurance 93E20: Optimal stochastic control 60K05: Renewal theory 35D40: Viscosity solutions

Keywords
Optimal dividend problem Sparre Andersen model backward Markovization dynamic programming Hamilton–Jacobi–Bellman equation constrained viscosity solution

Citation

Bai, Lihua; Ma, Jin; Xing, Xiaojing. Optimal dividend and investment problems under Sparre Andersen model. Ann. Appl. Probab. 27 (2017), no. 6, 3588--3632. doi:10.1214/17-AAP1288. https://projecteuclid.org/euclid.aoap/1513328709


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