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This paper studies some first passage time problems in a refracted jump diffusion process with hyper-exponential jumps. Closed-form expressions for four functions associated with the first passage time are obtained by solving some ordinary integro-differential equations. In addition, the obtained results are used to value equity-linked death benefit products with state-dependent fees. Specifically, we obtain the closed-form Laplace transform of the fair value of barrier option, which is further recovered by the bilateral Abate-Whitt algorithm. Numerical results confirm that the proposed approach is efficient. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new generalized Gerber–Shiu discounted penalty function for a compound Poisson risk model, which can be used to study the moments of the ruin time. First, by taking derivatives with respect to the original Gerber–Shiu discounted penalty function, we construct a relation between the original Gerber–Shiu discounted penalty function and our new generalized Gerber–Shiu discounted penalty function. Next, we use Laplace transform to derive a defective renewal equation for the generalized Gerber–Shiu discounted penalty function, and give a recursive method for solving the equation. Finally, when the claim amounts obey the exponential distribution, we give some explicit expressions for the generalized Gerber–Shiu discounted penalty function. Numerical illustrations are also given to study the effect of the parameters on the generalized Gerber–Shiu discounted penalty function. [ABSTRACT FROM AUTHOR]