In this paper we develop a constructive structure theory for a class of exponential functionals of Brownian motion which includes Asian option values. This is done in two stages of differing natures. As a first step, the functionals are represented as Laguerre reduction series obtained from main results of Schröder (2006), this paper's companion paper. These reduction series are new and given in terms of the negative moments of the integral of geometric Brownian motion, whose structure theory is developed in a second step. Providing a new angle on these processes, this is done by establishing connections with theta functions. Integral representations and computable formulae for the negative moments are thus derived and then shown to furnish highly efficient ways for computing the negative moments. Application of this paper's Laguerre reduction series in numerical examples suggests that one of the most efficient methods for the explicit valuation of Asian options is obtained. The paper also provides mathematical background results referred to in Schröder (2005c).

Let (Xt)t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E→ℝ\{0}, and let (φt)t≥0 be an additive functional defined by φt=∫0tv(Xs)d s. We consider the case in which the process (φt)t≥0 is oscillating and that in which (φt)t≥0 has a negative drift. In each of these cases, we condition the process (Xt,φt)t≥0 on the event that (φt)t≥0 is nonnegative until time T and prove weak convergence of the conditioned process as T→∞.

A forwards induction policy is a type of greedy algorithm for Markov decision processes. It is straightforward to implement and is optimal for a large class of models, especially in stochastic resource allocation. In this paper we consider a model for the optimal allocation of resources in pre-clinical pharmaceutical research. We show that although they are not always strictly optimal, forwards induction policies perform well.

Expectiles have received increasing attention as a risk measure in risk management because of their coherency and elicitability at the level $\alpha\geq1/2$. With a view to practical risk assessments, this paper delves into the worst-case expectile, where only partial information on the underlying distribution is available and there is no closed-form representation. We explore the asymptotic behavior of the worst-case expectile on two specified ambiguity sets: one is through the Wasserstein distance from a reference distribution and transforms this problem into a convex optimization problem via the well-known Kusuoka representation, and the other is induced by higher moment constraints. We obtain precise results in some special cases; nevertheless, there are no unified closed-form solutions. We aim to fully characterize the extreme behaviors; that is, we pursue an approximate solution as the level $\alpha $ tends to 1, which is aesthetically pleasing. As an application of our technique, we investigate the ambiguity set induced by higher moment conditions. Finally, we compare our worst-case expectile approach with a more conservative method based on stochastic order, which is referred to as ‘model aggregation’.

Two classical stochastic processes are considered, the Ehrenfest process, introduced in 1907 in the kinetic theory of gases to describe the heat exchange between two bodies, and the Engset process, one of the early (1918) stochastic models of communication networks. In this paper we investigate the asymptotic behavior of the distributions of hitting times of these two processes when the number of particles/sources goes to infinity. Results concerning the hitting times of boundaries in particular are obtained. We rely on martingale methods; a key ingredient is an important family of simple nonnegative martingales, an analogue, for the Ehrenfest process, of the exponential martingales used in the study of random walks or of Brownian motion.

We study the endemic behaviour of a homogeneously mixing SIS epidemic in a population of size N with a general infectious period, Q, by introducing a novel subcritical branching process with immigration approximation. This provides a simple but useful approximation of the quasistationary distribution of the SIS epidemic for finite N and the asymptotic Gaussian limit for the endemic equilibrium as N → ∞. A surprising observation is that the quasistationary distribution of the SIS epidemic model depends on Q only through

We study the asymptotic behavior of the tail probability of integrated stable processes exceeding power barriers. In the first part of the paper the limiting behavior of the integrals of stable processes generated by ergodic dissipative flows is established. In the second part an example with the integral of a stable process generated by a conservative flow is analyzed. Finally, the difference in the order of magnitude of the exceedance probability in the two cases is related to the dependence structure of the underlying stable process.

We present a stochastic algorithm which generates optimal probabilities for the chaos game to decompress an image represented by the fixed point of an IFS operator. The algorithm can be seen as a sort of time-inhomogeneous regenerative process. We prove that optimal probabilities exist and, by martingale methods, that the algorithm converges almost surely. The method holds for IFS operators associated with any arbitrary number of possibly overlapping affine contraction maps on the pixels space.

The annual premium for a life assurance contract is obtained by equating the expected discounted value of the sum payable on death to the expected discounted value of the series of premiums, and loading the resultant net premium for expenses and contingencies. The approach is essentially deterministic. An adequate stochastic model of mortality is readily available, but it is of limited practical value. A life office writes a large number of policies on independent lives and, apart from the effects of a few very large policies, the overall mortality behaviour of its portfolio is effectively deterministic.

We consider the N-oscillator system of the van der Pol type, which contains a small positive parameter ε multiplying the non-linear damping and the random disturbance. For a formulation of the output we take the solution X(t) = (Xi(t))i=1···,N of the system of 2N-dimensional stochastic differential equations. Rotating each component Xi(t) about the origin of the plane by an angle t, we find that on time scales of order 1/ε together with sufficiently large N each Xi(t) behaves as the equi-ultimately bounded solution of an equation of the McKean type admitting a stationary probability distribution.

With the general convergence theory for branching processes as basis a special problem is studied. An extra point process of events during life is assigned to each realised individual, and the behaviour of the superposition of such point processes in action is studied as the population grows. With the proper scaling and under some regularity conditions the superposition is shown to converge in distribution to a Poisson process. Another scaling gives rise to a mixed Poisson process as limit.

Established weak convergence techniques for point processes are applied, together with some recent strong convergence results for branching processes.

Motivated by applications in digital image processing, we discuss information-theoretic bounds to the amount of detail that can be recovered from a defocused, noisy signal. Mathematical models are constructed for test-pattern, defocusing and noise. Using these models, upper bounds are derived for the amount of detail that can be recovered from the degraded signal, using any method of image restoration. The bounds are used to assess the performance of the class of linear restorative procedures. Certain members of the class are shown to be optimal, in the sense that they attain the bounds, while others are shown to be sub-optimal. The effect of smoothness of point-spread function on the amount of resolvable detail is discussed concisely.

The theory of epidemics has been formulated as a deterministic theory and as a stochastic theory. There has been little work, except for a recent paper by Bailey [1], on the transition between the two formulations. In this paper we present an alternate formulation to that of Bailey that leads to a perturbation method more easily extendible to higher orders.

A general study of the asymptotic law of a skew-product of two real diffusions is proposed; a complete and elementary form of the Rosenkrantz theorem is given; the results are applied to three examples: the Brownian motions on SO3, on S3, and on SH3.

Correction terms are obtained for the Brownian motion approximation to one- and two-barrier first-passage probabilities. These approximations are given in terms of their Laplace transforms, which are formally (and non-rigorously) inverted. Applications to the one-sample Kolmogorov-Smirnov statistic are discussed.