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A numerical approach to optimal dividend policies with capital injections and transaction costs. (English) Zbl 1360.91153
Summary: This work focuses on numerical methods for finding optimal dividend payment and capital injection policies to maximize the present value of the difference between the cumulative dividend payment and the possible capital injections. Using dynamic programming principle, the value function obeys a quasi-variational inequality (QVI). The state constraint of the impulsive control gives rise to a capital injection region with free boundary. Since the closed-form solutions are virtually impossible to obtain, we use Markov chain approximation techniques to construct a discrete-time controlled Markov chain to approximate the value function and optimal controls. Convergence of the approximation algorithms is proved.

MSC:
91G60 Numerical methods (including Monte Carlo methods)
65C30 Numerical solutions to stochastic differential and integral equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65C05 Monte Carlo methods
91B30 Risk theory, insurance (MSC2010)
93E20 Optimal stochastic control
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