Paavo Salminen

Optimal stopping of Hunt and Lévy processes

Infinite horizon (perpetual) optimal stopping problems for Hunt processes on R are studied via the representation theory of excessive functions. In particular, we focus on problems with one-sided structure, that is, there exists a point x* such that the stopping region is of the form . The main result states that if it is possible to find a Radon measure such that the excessive function induced by this measure via the spectral representation has some very intuitive properties then the constructed excessive function coincides with the value function of the problem. Corresponding results for two-sided problems are also indicated. Specializing to Lévy processes, we obtain, by applying the Wiener–Hopf factorization, a general representation of the value function in terms of the maximum of the Lévy process. To illustrate the results, an explicit expression for the Green kernel of Brownian motion with exponential jumps is computed and some optimal stopping problems for Poisson process with positive exponential jumps and negative drift are solved.

Optimal stopping, Appell polynomials, and Wiener–Hopf factorization

In this paper, we study the Novikov–Shiryaev optimal stopping problem for Lévy processes. In particular, we are interested in finding the representing measure of the value function. It is seen that this can be expressed in terms of the Appell polynomials. For spectrally one-sided Lévy processes, the results are appealing and explicit. An important tool in our approach and computations is the Wiener–Hopf factorization.

On Busy Periods of the Unbounded Brownian Storage

A stationary storage process with Brownian input and constant service rate is studied. Explicit formulae for quantities related to busy periods (excursions) are derived. In particular, we compute the distributions of the occupation times the process spends above and below, respectively, the present level during the on-going busy period, and make the surprising observation that these occupation times are identically distributed.

Brownian Excursions Revisited

There are two classical approaches to the theory of Brownian excursions. The first one goes back to Levy. His ideas were worked out in greater detail and extended by Ito and McKean (see [4], [5], and [9]). Also Chung’s and Knight’s contributions are of great importance (see [1], [7], and [8]). In this approach the lengths of the excursions are the basic objects. In the second approach, due to Williams (see [12], [14], and [15]), one works with excursions having a given maximum. In both approaches Ito’s theory of excursions (see [3]) plays an active part (see [5], and [12]).

On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes

Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary t ↦ a + bt, a ≥ 0, b ∈ ℝ, by a reflecting Brownian motion. The main tool hereby is Doob’s formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three-dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter δ > 0 and show that the probability that such a Bessel bridge crosses an affine boundary is equal to the probability that this Bessel bridge stays below some fixed value.

ANALYSIS OF STOCHASTIC FLUID QUEUES DRIVEN BY LOCAL-TIME PROCESSES

We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically (but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator), hence making the theory of Lévy processes applicable. Another important ingredient in our approach is the use of Palm calculus for stationary random point processes and measures.

On Dufresne's Perpetuity, Translated and Reflected

AbstractLet B(µ) denote a Brownian motion with drift µ. In this paper we study two perpetual integral functionals of B(µ). The first one, introduced and investigated by Dufresne in [5], is

IMPULSE CONTROL AND EXPECTED SUPREMA

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