This paper introduces a benchmark approach for the modelling of continuous, complete financial markets, when an equivalent risk-neutral measure does not exist. This approach is based on the unique characterization of a benchmark portfolio, the growth optimal portfolio, which is obtained via a generalization of the mutual fund theorem. The discounted growth optimal portfolio with minimum variance drift is shown to follow a Bessel process of dimension four. Some form of arbitrage can be explicitly modelled by arbitrage amounts. Fair contingent claim prices are derived as conditional expectations under the real world probability measure. The Heath-Jarrow-Morton forward rate equation remains valid despite the absence of an equivalent risk neutral measure.

Recently Bassan and Spizzichino (1999) have given some new concepts of multivariate ageing for exchangeable random variables, such as a special type of bivariate IFR, by comparing distributions of residual lifetimes of dependent components of different ages. In the same vein, we further study some properties of these concepts of IFR in the bivariate case. Then we introduce certain concepts of bivariate DMRL ageing and we develop a treatment that parallels those developed for bivariate IFR. For both the IFR and DMRL concepts, we analyse a weak and a strong version, and discuss some of the differences between them.

In this paper, we consider a Hilbert-space-valued autoregressive stochastic sequence (Xn) with several regimes. We suppose that the underlying process (In) which drives the evolution of (Xn) is stationary. Under some dependence assumptions on (In), we prove the existence of a unique stationary solution, and with a symmetric compact autocorrelation operator, we can state a law of large numbers with rates and the consistency of the covariance estimator. An overall hypothesis states that the regimes where the autocorrelation operator's norm is greater than 1 should be rarely visited.

A Markov chain model for a battle between two opposing forces is formulated, which is a stochastic version of one studied by F. W. Lanchester. Solutions of the backward equations for the final state yield martingales and stopping identities, but a more powerful technique is a time-reversal analogue of a known method for studying urn models. A general version of a remarkable result of Williams and McIlroy (1998) is proved.

We consider a variant of the house-selling problem in which the seller has n objects for sale and the buyers come sequentially one by one. The objective of this paper is to find the seller's optimal strategy for choosing his selling price so as to maximize the average reward per selling attempt until the nth house is sold.

In this paper we find conditions under which the epoch times of two nonhomogeneous Poisson processes are ordered in the multivariate dispersive order. Some consequences and examples of this result are given. These results extend a recent result of Brown and Shanthikumar (1998).

There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating- and fixed-strike Asian options. The proof involves a change of numéraire and time reversal of Brownian motion. Symmetries are very useful in option valuation, and in this case the result allows the use of more established fixed-strike pricing methods to price floating-strike Asian options.

The usual direct method of simulation for cluster processes requires the generation of the parent point process over a region larger than the actual observation window, since we have to allow for all possible parents giving rise to observed daughter points, and some of these parents may fall outwith the observation window. When there is no a priori bound on the distance between parent and child then we have to take care to control approximations arising from edge effects. In this paper, we present a simulation method which requires simulation only of those parent points actually giving rise to observed daughter points, thus avoiding edge effect approximation. The idea is to replace the cluster distribution by one which is conditioned to plant at least one daughter point in the observation window, and to modify the parent process to have an inhomogeneous intensity exactly balancing the effect of the conditioning. We furthermore show how the method extends to cases involving infinitely many potential parents, for example gamma-Poisson processes and shot-noise G-Cox processes, allowing us to avoid approximation due to truncation of the parent process.

Let (Sk)k≥0 be a Markov chain with state space E and (ξx)x∊E be a family of ℝp-valued random vectors assumed independent of the Markov chain. The ξx could be assumed independent and identically distributed or could be Gaussian with reasonable correlations. We study the large deviations of the sum

We study the contribution made by three or four points to certain areas associated with a typical polygon in a Voronoi tessellation of a planar Poisson process. We obtain some new results about moments and distributions and give simple proofs of some known results. We also use Robbins' formula to obtain the first three moments of the area of a typical polygon and hence the variance of the area of the polygon covering the origin.

We consider a network of dams to which the external input is a multivariate Markov additive process. For each state of the Markov chain modulating the Markov additive process, the release rates are linear (constant multiple of the content level). Each unit of material processed by a given station is then divided into fixed proportions each of which is routed to another station or leaves the system. For each state of the modulating process, this routeing is determined by some substochastic matrix. We identify simple conditions for stability and show how to compute transient and stationary characteristics of such networks.

The following problem in risk theory is considered. An insurance company, endowed with an initial capital a ≥ 0, receives premiums and pays out claims that occur according to a renewal process {N(t), t ≥ 0}. The times between consecutive claims are i.i.d. The sequence of successive claims is a sequence of i.i.d. random variables. The capital of the company is invested at interest rate α ∊ [0,1], claims increase at rate β ∊ [0,1]. The aim is to find the stopping time that maximizes the capital of the company. A dynamic programming method is used to find the optimal stopping time and to specify the expected capital at that time.

This paper is concerned with the correlation structure of a stationary discrete time-series with long memory or long-range dependence. Given a sequence of bounded variation, we obtain necessary and sufficient conditions for a function generated from the sequence to be a proper correlation function. These conditions are applied to derive various slowly decaying correlation models. To obtain correlation models with short-range dependence from an absolutely summable sequence, a simple method is introduced.

Asymptotic behaviour of the distribution of the absolute maximum of a process with independent increments is studied depending on the properties of the Lévy measure of the process. Some applications to the risk process are also considered.

In a sequence of Markov-dependent trials, the optimal strategy which maximizes the probability of stopping on the last success is considered. Both homogeneous Markov chains and nonhomogeneous Markov chains are studied. For the homogeneous case, the analysis is divided into two parts and both parts are realized completely. For the nonhomogeneous case, we prove a result which contains the result of Bruss (2000) under an independence structure.

For a real-valued ergodic process X with strictly stationary increments satisfying some measurability and continuity assumptions it is proved that the long-run ‘average behaviour’ of all its increments over finite intervals replicates the distribution of the corresponding increments of X in a strong sense. Moreover, every Lévy process has a version that possesses this ergodic path property.

Empirical point processes of exceedances play an important role in extreme value theory, and their limiting behaviour has been extensively studied. Here, we provide explicit bounds on the accuracy of approximating an exceedance process by a compound Poisson or Poisson cluster process, in terms of a Wasserstein metric that is generally more suitable for the purpose than the total variation metric. The bounds only involve properties of the finite, empirical sequence that is under consideration, and not of any limiting process. The argument uses Bernstein blocks and Lindeberg's method of compositions.

We consider a class of optimal stopping problems where the ability to stop depends on an exogenous Poisson signal process - we can only stop at the Poisson jump times. Even though the time variable in these problems has a discrete aspect, a variational inequality can be obtained by considering an underlying continuous-time structure. Depending on whether stopping is allowed at t = 0, the value function exhibits different properties across the optimal exercise boundary. Indeed, the value function is only 𝒞0 across the optimal boundary when stopping is allowed at t = 0 and 𝒞2 otherwise, both contradicting the usual 𝒞1 smoothness that is necessary and sufficient for the application of the principle of smooth fit. Also discussed is an equivalent stochastic control formulation for these stopping problems. Finally, we derive the asymptotic behaviour of the value functions and optimal exercise boundaries as the intensity of the Poisson process goes to infinity or, roughly speaking, as the problems converge to the classical continuous-time optimal stopping problems.

The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝd and a gauge body B ⊂ ℝd, such a generalized contact distribution is the conditional distribution of the random vector (dB(L,Z),uB(L,Z),pB(L,Z),lB(L,Z)) given that Z∩L = ∅, where Z is a Boolean model, dB(L,Z) is the distance of L from Z with respect to B, pB(L,Z) is the boundary point in L realizing this distance (if it exists uniquely), uB(L,Z) is the corresponding boundary point of B (if it exists uniquely) and lB(L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.

In this paper we obtain the rate of convergence to equilibrium of a class of interacting marked point processes, introduced by Kerstan, in two different situations. Indeed, we prove the exponential and subexponential ergodicity of such a class of stochastic processes. Our results are an extension of the corresponding results of Brémaud, Nappo and Torrisi. The generality of the dynamics which we take into account allows the application to the so-called loss networks, and multivariate birth and death processes.