Inspired by the consideration of some inside and future market information in financial market, a class of anticipated backward doubly stochastic Volterra integral equations (ABDSVIEs) are introduced to induce dynamic risk measures for risk quantification. The theory, including the existence, uniqueness and a comparison theorem for ABDSVIEs, is provided. Finally, dynamic convex risk measures by ABDSVIEs are discussed.

In this paper, a risk model driven by spectrally negative Lévy processes is considered where dividends until a general draw-down time are paid under a constant barrier strategy. In this model, the moments of the sum of the discounted dividend payments until the general draw-down time are derived through the corresponding scale functions. The explicit expressions for the Laplace transform of the discounted dividends are also obtained. Moreover, numerical examples are given to illustrate the impacts of barrier levels and draw-down functions on the results.

As a crucial modeling tool for stochastic financial markets, the Lévy risk model effectively characterizes the evolution of risks during enterprise operations. Through dynamic evaluation and quantitative analysis of risk indicators under specific dividend- distribution strategies, this model can provide theoretical foundations for optimizing corporate capital allocation. Addressing the inadequate adaptability of traditional single-period threshold strategies in time-varying market environments, this paper proposes a dividend strategy based on multiperiod dynamic threshold adjustments. By implementing periodic modifications of threshold parameters, this strategy enhances the risk model’s dynamic responsiveness to market fluctuations and temporal variations. Within the framework of the spectrally negative Lévy risk model, this paper constructs a stochastic control model for multiperiod threshold dividend strategies. We derive the integro-differential equations for the expected present value of aggregate dividend payments before ruin and the Gerber–Shiu function, respectively. Combining the methodologies of the discounted increment density, the operator introduced by Dickson and Hipp, and the inverse Laplace transforms, we derive the explicit solutions to these integro-differential equations. Finally, numerical simulations of the related results are conducted using given examples, thereby demonstrating the feasibility of the analytical method proposed in this paper.

This paper focuses on the De Finettis dividend problem for the spectrally negative Lévy risk process, where the dividend is deducted from the surplus process according to the racheting dividend strategy which was firstly introduced in Albrecher et al. (2018). A major feature of the racheting strategy lies in which the dividend rate never decreases. Unlike the conventional studies, the closed form expression for the expected, accumulated, and discounted dividend payments until the draw-down time (rather than the ruin time) is obtained in terms of the scale functions corresponding to the underlying Lévy process. The optimal barrier for the ratcheting strategy is also studied, where the dividend rate can be increased. Finally, two special cases, where the scale functions are explicitly known, i.e., the Brownian motion with drift and the compound Poisson model, are considered to illustrate the main result.

In this paper, we study the dynamic risk measures for processes induced by backward stochastic differential equations driven by Teugel’s martingales associated with Lévy processes (BSDELs). The representation theorem for generators of BSDELs is provided. Furthermore, the time consistency of the coherent and convex dynamic risk measures for processes is characterized by means of the generators of BSDELs. Moreover, the coherency and convexity of dynamic risk measures for processes are characterized by the generators of BSDELs. Finally, we provide two numerical examples to illustrate the proposed dynamic risk measures.