# Optimal control of taxation for spectrally negative Levy processes

• Dalal Al Ghanim

Student thesis: Phd

### Abstract

In the context of loss-carry-forward taxation on the capital of an insurance company, we introduce two tax processes, latent and natural tax processes and show they are equivalent. This equivalence relation enables us to deal straightforward with the existence and uniqueness of the natural tax process, which is defined via an integral equation, and allows us to translate results from one model to the other. We clarify by our results the existing literature on tax processes. Using our equivalence relation, we derive an explicit expression for the expected deficit at ruin and the maximum surplus prior to ruin for the natural tax process when ruin happens before it reaches some positive level. We explain the relation of this expression with the draw-down literature. We introduce and solve two optimal control tax problems for a spectrally negative Levy risk process. The first one aims to find the maximum tax value function and the tax strategy that achieves this. We prove a value function is the optimal value by putting it through a verification lemma. We find that, when the Levy measure has a log-convex tail, the optimal tax strategy is a piecewise constant natural tax strategy. We show, on a special case, that our solution agrees with the solution of an optimal tax control problem considered in a previous literature. In the second optimal control tax problem, we add the bail-out concept to the model such that ruin is not allowed. An optimal strategy is defined as a tax and bail-out admissible strategy that maximises the net profit of taxation. In order to find the optimal tax value in this model, we introduce a new approach to find unknown fluctuation identities. Our work shows that the function representing the net present value of tax can be uniquely characterised by a PDE and a set of boundary conditions, and we use this to derive an explicit formula for this function. We verify that, on special cases, our results agree with existing results in the literature. We find, under no condition on the Levy measure, that the optimal strategy is a piecewise constant tax rate function and a bail-out process which allows the capital to be injected back to zero whenever it becomes strictly negative. We introduce a natural tax model with bail-outs when ruin is allowed if the deficit at ruin exceeds some pre-specified level. We derive a new fluctuation identities for the Levy process reflected at its infimum. We use these identities and our new approach, to find the net profit of taxation in this model. We do this under an assumption on the Levy process, that it has positive Gaussian coefficient in the unbounded variation case.
Date of Award 1 Aug 2021 English The University of Manchester Ronnie Loeffen (Supervisor) & Alexander R. Watson (Supervisor)

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