We provide a set of verifiable sufficient conditions for proving in a number of practical examples the equivalence of the martingale and the PDE approaches to the valuation of derivatives. The key idea is to use a combination of analytic and probabilistic assumptions that covers typical models in finance falling outside the range of standard results from the literature. Applications include Heston's stochastic volatility model and the Black-Karasinski term structure model.

This short note shows that the Lundberg exponential upper bound in the ruin problem of non-life insurance with compound Poisson claims is also valid for the Poisson shot noise delayed-claims model, and that the optimal exponent depends only on the distribution of the total claim per accident, not on the time it takes to honour the claim. This result holds under Cramer's condition.

In various stochastic models the random equation of implicit renewal theory appears where the real random variable S and the stochastic process Ψ with index space and state space R are independent. By use of stochastic approximation the distribution function of S is recursively estimated on the basis of independent or ergodic copies of Ψ. Under integrability assumptions almost sure L1-convergence is proved. The choice of gains in the recursion is discussed. Applications are given to insurance mathematics (perpetuities) and queueing theory (stationary waiting and queueing times).

The bifurcating autoregressive model has been used previously to model cell lineage data. A feature of this model is that each line of descendants from an initial cell follows an AR(1) model, and that the environmental effects on sisters are correlated. However, this model concentrates on modelling the correlations between mother and daughter cells and between sister cells, and does not explain the large correlations between more distant relatives observed by some authors. Here the model is extended, firstly by allowing lines of descent to follow an ARMA(p,q) model rather than an AR(1) model, and secondly by allowing correlations between the environmental effects of relatives more distant than sisters. The models are applied to several data sets consisting of independent cell lineage trees.

An open hierarchical (manpower) system divided into a totally ordered set of k grades is discussed. The transitions occur only from one grade to the next or to an additional (k+1)th grade representing the external environment of the system. The model used to describe the dynamics of the system is a continuous-time homogeneous Markov chain with k+1 states and infinitesimal generator R = (rij) satisfying rij = 0 if i > j or i + 1 < j ≤ k (i, j = 1,…,k+1), the transition matrix P between times 0 and 1 being P = expR. In this paper, two-wave panel data about the hierarchical system are considered and the resulting fact that, in general, the maximum-likelihood estimated transition matrix cannot be written as an exponential of an infinitesimal generator R having the form described above. The purpose of this paper is to investigate when this can be ascribed to the effect of sampling variability.

We analyse a class of diffusion models that (i) allows an explicit expression for the likelihood function of discrete time observation, (ii) allows the possibility of heavy-tailed observations, and (iii) allows an analysis of the tails of the increments. The class simply consists of transformed Ornstein–Uhlenbeck processes and is of relevance for heavy-tailed time series. We also treat the question of the existence of an equivalent martingale measure for the class of models considered.

We consider a sequence matching problem involving the optimal alignment score for contiguous sequences; rewarding matches and penalizing for deletions and mismatches. Arratia and Waterman conjectured in [1] that the score constant a(μ, δ) is a strictly monotone function (i) in δ for all positive δ and (ii) in μ if 0 ≤ μ ≤ 2δ. Here we prove that (i) is true for all δ and (ii) is true for some μ.

We study the present value Z∞ = ∫0∞ e-Xt-dYt where (X,Y) is an integrable Lévy process. This random variable appears in various applications, and several examples are known where the distribution of Z∞ is calculated explicitly. Here sufficient conditions for Z∞ to exist are given, and the possibility of finding the distribution of Z∞ by Markov chain Monte Carlo simulation is investigated in detail. Then the same ideas are applied to the present value Z-∞ = ∫0∞ exp{-∫0tRsds}dYt where Y is an integrable Lévy process and R is an ergodic strong Markov process. Numerical examples are given in both cases to show the efficiency of the Monte Carlo methods.

In applied probability, the distribution of a sum of n independent Bernoulli random variables with success probabilities p1,p2,…, pn is often approximated by a Poisson distribution with parameter λ = p1 + p2 + pn. Popular bounds for the approximation error are excellent for small values, but less efficient for moderate values of p1,p2,…,pn.

Upper bounds for the total variation distance are established, improving conventional estimates if the success probabilities are of medium size. The results may be applied directly, e.g. to approximation problems in risk theory.

This paper considers a branching process generated by an offspring distribution F with mean m < ∞ and variance σ2 < ∞ and such that, at each generation n, there is an observed δ-migration, according to a binomial law Bpvn*Nnbef which depends on the total population size Nnbef. The δ-migration is defined as an emigration, an immigration or a null migration, depending on the value of δ, which is assumed constant throughout the different generations. The process with δ-migration is a generation-dependent Galton-Watson process, whereas the observed process is not in general a martingale. Under the assumption that the process with δ-migration is supercritical, we generalize for the observed migrating process the results relative to the Galton-Watson supercritical case that concern the asymptotic behaviour of the process and the estimation of m and σ2, as n → ∞. Moreover, an asymptotic confidence interval of the initial population size is given.

We formulate stochastic indicator parameters that characterize pollution levels in geographical regions with heterogeneous contaminant distributions. The indicator parameters are expressed in terms of the random fields representing the contaminant distributions and the critical threshold level specified by health and environmental standards. Certain theoretical results are proven regarding univariate and bivariate indicator parameters. The analytical expressions obtained are general and can be used in practice for various types of contaminant distributions. A test of ergodicity-breaking is suggested for scientific and engineering applications in terms of the indicator parameters. Fractal characteristics of the indicator parameters are discussed. The effects of modelling and observation scale on exceedance contamination analysis are examined. Indicator random field parameters are studied on both continuum and lattice domains using analytical means and numerical simulations.

This paper discusses the distribution of tumor size at detection derived within the framework of a new stochastic model of carcinogenesis. This distribution assumes a simple limiting form, with age at detection tending to infinity which is found to be a generalization of the distribution that arises in the length-biased sampling. Two versions of the model are considered with reference to spontaneous and induced carcinogenesis; both of them show similar asymptotic behavior. When the limiting distribution is applied to real data analysis its adequacy can be tested through testing the conditional independence of the size, V, and the age, A, at detection given A > t*, where the value of t* is to be estimated from the given sample. This is illustrated with an application to data on premenopausal breast cancer. The proposed distribution offers the prospect of the estimation of some biologically meaningful parameters descriptive of the temporal organization of tumor latency. An estimate of the model stability to the prior distribution of tumor size and some other stability results for the Bayes formula are given.

We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.

The theory of majorization is applied to investigate the effects of heterogeneity in a class of epidemics. In particular, the heterogeneity of the components of the contact vector of a given susceptible is studied. It is shown that if the heterogeneity can be partially ordered through majorization, then the probability functions of escaping infection can be similarly ordered. The results can be applied to AIDS research on the spread of the HIV virus.

Let ?(u) be the probability of eventual ruin in the classical Sparre Andersen model of risk theory if the initial risk reserve is u. For a large class of such models ?(u) behaves asymptotically like a multiple of exp (–Ru) where R is the adjustment coefficient; R depends on the premium income rate, the claim size distribution and the distribution of the time between claim arrivals. Estimation of R has been considered by many authors. In the present paper we deal with confidence bounds for R. A variety of methods is used, including jackknife estimation of asymptotic variances and the bootstrap. We show that, under certain assumptions, these procedures result in interval estimates that have asymptotically the correct coverage probabilities. We also give the results of a simulation study that compares the different techniques in some particular cases.

The multihit–one target model induces a stochastic ordering of cell survival with respect to the cell sensitivity characteristics. This property can be used for a description of cell killing effects in heterogeneous populations of cells on the basis of randomized versions of the model. In such versions, either the critical number of lesions or the mean number of hits per unit dose (sensitivity), or both, are assumed to be random. We give some new results specifying conditions under which the randomized multihit models are identifiable, with a focus on the following cases: (1) the critical number of radiation-induced lesions, m, is random; (2) the sensitivity parameter, x, is random given m is known or otherwise; (3) x and m form a pair of independent random variables.

We consider a dam in which the release rate depends both on the state and some modulating process. Conditions for the existence of a limiting distribution are established in terms of an associated risk process. The case where the release rate is a product of the state and the modulating process is given special attention, and in particular explicit formulas are obtained for a finite state space Markov modulation.

The corpuscle problem of Wicksell is discussed. We give a numerical quadrature of Gauss–Chebyshev type for Wicksell's integral equation which combines a size distribution of discs on a sectional plane with that of spheres. We also give an estimation procedure of three-dimensional size distributions based on this quadrature and examine its theoretical properties. In practice, we need a smoothing technique for empirical distribution functions before applying this estimator. Simulation results are given. Our idea also is applied to the thick section case and an analysis of microscopic data is given.

The theory of piecewise-deterministic Markov processes is used in order to investigate insurance risk models where borrowing, investment and inflation are present.

Obtaining good estimates for the distribution function of random variables like (‘perpetuity’) and (‘aggregate claim amount’), where the (Yi), (Zi) are independent i.i.d. sequences and (N(t)) is a general point process, is a key question in insurance mathematics. In this paper, we show how suitably chosen metrics provide a theoretical justification for bootstrap estimation in these cases. In the perpetuity case, we also give a detailed discussion of how the method works in practice.