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In this paper, we consider a company that wishes to determine the optimal reinsurance

strategy minimising the total expected discounted amount of capital injections

needed to prevent the ruin. The company's surplus process is assumed to follow a

Brownian motion with drift, and the reinsurance price is modelled by a continuous-time

Markov chain with two states. The presence of regime-switching complicates

substantially the optimal reinsurance problem, as the surplus-independent strategies

turn out to be suboptimal. We develop a recursive approach that allows to

represent a solution to the corresponding Hamilton-Jacobi-Bellman equation and

the corresponding reinsurance strategy as the unique limits of the sequence of

solutions to ordinary differential equations and their first and second order derivatives.

Via Ito's formula we prove the constructed function to be the value function.

Two examples illustrate the recursive procedure along with a numerical approach

yielding the direct solution to the HJB equation.


Classification: 60K10; 91B30; 60J65; 49K15