We introduce two models of taxation, the latent and natural tax processes, which have both been used to represent loss-carry-forward taxation on the capital of an insurance company. In the natural tax process, the tax rate is a function of the current level of capital, whereas in the latent tax process, the tax rate is a function of the capital that would have resulted if no tax had been paid. Whereas up to now these two types of tax processes have been treated separately, we show that, in fact, they are essentially equivalent. This allows a unified treatment, translating results from one model to the other. Significantly, we solve the question of existence and uniqueness for the natural tax process, which is defined via an integral equation. Our results clarify the existing literature on processes with tax.

Introduction and main results

Risk processes are a model for the evolution in time of the (economic) capital or surplus of an insurance company. Suppose that we have some model X = (Xt)t≥۰ for the risk process, in which Xt represents the capital of the company at time t; for instance, a common choice is for X to be a Lévy process with negative jumps. Any such model can be modified in order to incorporate desirable features. For instance, reflecting the path at a given barrier models the situation where the insurance company pays out any capital in excess of the barrier as dividends to shareholders. Similarly, ‘refracting’ the path at a given level and with a given angle corresponds to the case where dividends are paid out at a certain fixed rate whenever the capital is above the level or, equivalently, corresponds to a two-step premium rate. These modifications are described in more detail in Chapter 10 of Kyprianou (2014), in the Lévy process case. Between the reflected and refracted processes are a class of processes where partial reflection occurs whenever the process reaches a new maximum. The motivation in risk theory for these processes is that the times of partial reflection can be understood to correspond to tax payments associated with a so-called losscarry-forward regime in which taxes are paid only when the insurance company is in a profitable situation. In this paper we study tax processes of this kind.