On an optimal consumption problem for p-integrable consumption plans

A generalization is presented of the existence results for an optimal consumption problem of Aumann and Perles [4] and Cox and Huang [10]. In addition, we present avery general optimality principle

Exit problems of Lévy processes with applications in finance

In this thesis we study the pricing of options of American type in a continuous time setting. We begin with a general introduction where we briefly sketch history and different aspects of the option pricing problem. In the first chapter we consider four perpetual options of American type driven by a geometric Brownian motion: the American put and call, the Russian option and the integral option. We derive their values exploiting properties of Brownian motion and Bessel processes. From a practical point of view perpetual options do not seem of much use, since in practice the time of expiration is always finite. However, following an appealing idea of Peter Carr, we build an approximating sequence of perpetual-type options and prove this converges pointwise to the value of the corresponding finite time American option. Next we compute for the mentioned options the first approximation. The second chapter proposes the class of ``phase type Lévy processes'' as a new model for the stock price. This is a class of jump-diffusions which is dense in all Lévy processes and whose positive and negative jumps form compound Poisson processes with jump distributions of phase type. We illustrate its analytical tractability by pricing the perpetual American put and Russian option under this model. In the third chapter we study the same problems but now for the class of Lévy processes without negative jumps. We restrict ourselves to this class, since it contains already a lot of the rich structure of Lévy processes while still being analytically tractable due to many available results exploiting the fact that the jumps of the Lévy process have one sign. A recent study of Carr and Wu offers empirical evidence supporting the case of a model where the risky asset is driven by a spectrally negative Lévy process. For this class of Lévy processes, we review theory on first exit times of finite and semi-infinite intervals. Subsequently, we determine the Laplace transform of the exit time and exit position from an interval containing the origin of the process reflected at its supremum. The proof relies on Itô -excursion theory. The fourth chapter complements the study of the previous chapter. We find the Laplace transform of the first exit time of a finite interval containing the origin of the process reflected at its infimum. Then we turn our attention to these reflected processes killed upon leaving a finite interval containing zero and determine their resolvent measures. Invoking the R-theory of irreducible Markov chains developed by Tuomen and Tweedie, we are able to give a relatively complete description of the ergodic behaviour of their transition probabilities. The obtained results on Lévy processes in chapters 3 and 4 also have applications in the context of the theories of queueing, dams and insurance risk. Finally, the fifth chapter considers the utility-optimisation problem of an agent that operates in a general semimartingale market and seeks to trade so as to maximise his utility from inter-temporal consumption and final wealth. In this setting existence is established following a direct variational approach. Also a characterisation for the optimal consumption and final wealth plan is given

Exit problems for spectrally negative Lévy processes and applications to Russian, American and Canadized options

We consider spectrally negative Lévy process and determine the joint Laplace trans- form of the exit time and exit position from an interval containing the origin of the process reflected in its supremum. In the literature of fluid models, this stopping time can be identified as the time to buffer-overflow. The Laplace transform is determined in terms of the scale functions that appear in the two sided exit problem of the given Lévy process. The obtained results together with existing results on two sided exit problems are applied to solving optimal stopping problems associated with the pricing of American and Russian options and their Canadized versions

Queues with Lévy input and hysteretic control

We consider a (doubly) reflected Lévy process where the Lévy exponent is controlled by a hysteretic policy consisting of two stages. In each stage there is typically a different service speed, drift parameter, or arrival rate. We determine the steady-state performance, both for systems with finite and infinite capacity. Thereby, we unify and extend many existing results in the literature, focusing on the special cases of M/G/1 queues and Brownian motion. © The Author(s) 2009

Smoothness of scale functions for spectrally negative Lévy processes

Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often require excursion theory rather than Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic

Smoothness of scale functions for spectrally negative Lévy processes

Abstract The purpose of this review article is to give an up to date account of the theory and applications of scale functions for spectrally negative Lévy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Lévy processes, in particular a reasonable understanding of the Lévy–Khintchine formula and its relationship to the Lévy–Itô decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Lév