A numerical method for the expected penalty–reward function in a Markov-modulated jump–diffusion process

Abstract

JEL classification: G22 Keywords:Expected penalty-reward function Markov-modulated process Jump-diffusion process Volterra integro-differential system of equations Subject Category and Insurance BranchCategory: IM11 IM13 a b s t r a c t A generalization of the Cramér-Lundberg risk model perturbed by a diffusion is proposed. Aggregate claims of an insurer follow a compound Poisson process and premiums are collected at a constant rate with additional random fluctuation. The insurer is allowed to invest the surplus into a risky asset with volatility dependent on the level of the investment, which permits the incorporation of rational investment strategies as proposed by Berk and Green (2004). The return on investment is modulated by a Markov process which generalizes previously studied settings for the evolution of the interest rate in time.The Gerber-Shiu expected penalty-reward function is studied in this context, including ruin probabilities (a first-passage problem) as a special case. The second order integro-differential system of equations that characterizes the function of interest is obtained. As a closed-form solution does not exist, a numerical procedure based on the Chebyshev polynomial approximation through a collocation method is proposed. Finally, some examples illustrating the procedure are presented.

The Gerber-Shiu discounted penalty function: A review from practical perspectives

“…(2.14) for a finite time period T > 0 [17,51], where w(•) models the penalty in the case of ruin whereas P(•) models the reward for survival. This is further generalized with two variables [94,106,107] as in…”

“…The constant force of interest r is replaced by a suitable stochastic rate in [17,51,185]. Tax payment at a fixed rate γ is incorporated [125] together with a constant debit interest, where the surplus process is modelled as…”

“…As discussed in Section 3.3, the Gerber-Shiu function can often be formulated in the form of integro-differential equation with a suitable boundary condition, which certainly offers a valid approach of numerically solving the equation for the Gerber-Shiu function. Along this line, the Chebyshev polynomial approximation has attracted a great deal of attention in dealing with Markov-modulated jump-diffusion processes [51], where the Laplace-Carson transform in time of the penalty-reward Gerber-Shiu function (2.14) is expanded and truncated as +∞…”

“…Nevertheless, numerical methods on the Gerber-Shiu function do not seem to have been adequately explored in the literature. To the best of our knowledge, there exist only a handful of developments in the context of the Gerber-Shiu function: the numerical inversion of the Laplace transform [60,87] (Section 4.1), finite truncation of the Fourier expansion [30,109,175,176] (Section 4.2), the numerical solution to the integrodifferential equation with suitable boundary conditions [51,52,71,203,205] (Section 4.3), and simulation via random number generation [43,57] (Section 4.4). As is always the case, the method needs to be carefully chosen for each problem setting, because each of those methods above has both advantages and disadvantages with different technical prerequisites for implementation and convergence (Section 4.5).…”

On a Discrete Markov-Modulated Risk Model with Random Premium Income and Delayed Claims

“…Yuen et al [27] investigated the expected penalty functions for a discrete semi-Markov risk model with randomized dividends. Other related works can be found in Lu and Li [28], Diko and Usábel [29], Chen et al [30], Huo et al [31], etc.…”