In this article we study stochastic perturbations of partial differential equations describing forced-damped vibrations of a string. Two models of such stochastic disturbances are considered; one is triggered by an initial white noise, and the other is in the form of non-Gaussian random forcing. Let uε (t, x) be the displacement at time t of a point x on a string, where the time variable t ≧ 0, and the space variable . The small parameter ε controls the intensity of the random fluctuations. The random fields uε (t, x) are shown to satisfy a large deviations principle, and the random deviations of the unperturbed displacement function are analyzed as the noise parameter ε tends to zero.

We study a classical stochastic control problem arising in financial economics: to maximize expected logarithmic utility from terminal wealth and/or consumption. The novel feature of our work is that the portfolio is allowed to anticipate the future, i.e. the terminal values of the prices, or of the driving Brownian motion, are known to the investor, either exactly or with some uncertainty. Results on the finiteness of the value of the control problem are obtained in various setups, using techniques from the so-called enlargement of filtrations. When the value of the problem is finite, we compute it explicitly and exhibit an optimal portfolio in closed form.

The paper introduces an approach focused towards the modelling of dynamics of financial markets. It is based on the three principles of market clearing, exclusion of instantaneous arbitrage and minimization of increase of arbitrage information. The last principle is equivalent to the minimization of the difference between the risk neutral and the real world probability measures. The application of these principles allows us to identify various market parameters, e.g. the risk-free rate of return. The approach is demonstrated on a simple financial market model, for which the dynamics of a virtual risk-free rate of return can be explicitly computed.

We investigate the behaviour of P(R ≧ r) and P(R ≦ −r) as r → ∞for the random variable where is an independent, identically distributed sequence with P(− 1 ≦ M ≦ 1) = 1. Random variables of this type appear in insurance mathematics, as solutions of stochastic difference equations, in the analysis of probabilistic algorithms and elsewhere. Exponential and Poissonian tail behaviour can arise.

We study the dispersion of a collection of particles carried by an isotropic Brownian flow in Of particular interest are the center of mass and the centered spatial second moments. Their asymptotic behavior depends strongly on the spatial dimension and the largest Lyapunov exponent of the flow. We use estimates for the pair separation process to give a fairly complete picture of this behavior as t → ∞. In particular, for incompressible flows in two dimensions, we show that the variance of the center of mass grows sublinearly, while dispersion relative to the center of mass grows linearly.

A Cox risk process with a piecewise constant intensity is considered where the sequence (Li) of successive levels of the intensity forms a Markov chain. The duration σi of the level Li is assumed to be only dependent via Li. In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of Li contains a discrete component, a Cramér–Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).

We study the limiting behaviour of large systems of two types of Brownian particles undergoing bisexual branching. Particles of each type generate individuals of both types, and the respective branching law is asymptotically critical for the two-dimensional system, while being subcritical for each individual population.

The main result of the paper is that the limiting behaviour of suitably scaled sums and differences of the two populations is given by a pair of measure and distribution valued processes which, together, determine the limit behaviours of the individual populations.

Our proofs are based on the martingale problem approach to general state space processes. The fact that our limit involves both measure and distribution valued processes requires the development of some new methodologies of independent interest.

We give a unified presentation of stability results for stochastic vector difference equations based on various choices of binary operations and , assuming that are stationary and ergodic. In the scalar case, under standard addition and multiplication, the key condition for stability is E[log |A0|] < 0. In the generalizations, the condition takes the form γ< 0, where γis the limit of a subadditive process associated with . Under this and mild additional conditions, the process has a unique finite stationary distribution to which it converges from all initial conditions.

The variants of standard matrix algebra we consider replace the operations + and × with (max, +), (max,×), (min, +), or (min,×). In each case, the appropriate stability condition parallels that for the standard recursions, involving certain subadditive limits. Since these limits are difficult to evaluate, we provide bounds, thus giving alternative, computable conditions for stability.

The partially observed control problem is considered for stochastic processes with control entering into the diffusion and the observation. The maximum principle is proved for the partially observable optimal control. A pure probabilistic approach is used, and the adjoint processes are characterized as solutions of related backward stochastic differential equations in finite-dimensional spaces. Most of the derivation is identified with that of the completely observable case.

It is shown that the critical two-level (2, d, 1, 1)-superprocess is persistent in dimensions d greater than 4. This complements the extinction result of Wu (1994) and implies that the critical dimension is 4.

Let uε(t, x) be the position at time t of a point x on a string, where the time variable t varies in an interval I: = [0, T], T is a fixed positive time, and the space variable x varies in an interval J. The string is performing forced vibrations and also under the influence of small stochastic perturbations of intensity ε. We consider two kinds of random perturbations, one in the form of initial white noise, and the other is a nonlinear random forcing which involves the formal derivative of a Brownian sheet. When J has finite endpoints, a Dirichlet boundary condition is imposed for the solutions of the resulting non-linear wave equation. Assuming that the initial conditions are of sufficient regularity, we analyze the deviations uε(t, x) from u0(t, x), the unperturbed position function, as the intensity of perturbation ε ↓ 0 in the uniform topology. We also discuss some continuity properties of the realization of the solutions uε(t, x).

The paper proposes a general model for pricing of derivative securities. The underlying dynamics follows stochastic equations involving anticipative stochastic integrals. These equations are solved explicitly and structural properties of solutions are studied.

Continuous-time threshold autoregressive (CTAR) processes have been developed in the past few years for modelling non-linear time series observed at irregular intervals. Several approximating processes are given here which are useful for simulation and inference. Each of the approximating processes implicitly defines conditions on the thresholds, thus providing greater understanding of the way in which boundary conditions arise.

This paper examines the availability of a maintained system where the rate of deterioration is governed by an exogenous random environment. We provide a qualitative result that exposes the relationship between remaining lifetime, environment, and repairs. This result leads to simple bounds that can be used to choose inspection rates that guarantee a specified level of availability. The principal result requires no specific distributional assumptions, is intuitively appealing and can be directly applied by practitioners. Our development employs techniques from stochastic calculus.

In this paper, we consider several stochastic models arising from environmental problems. First, we study pollution in a domain where undesired chemicals are deposited at random times and locations according to Poisson streams. The chemical concentration can be modeled by a linear stochastic partial differential equation (SPDE) which is solved by applying a general result. Various properties, especially the limit behavior of the pollution process, are discussed. Secondly, we consider the pollution problem when a tolerance level is imposed. The chemical concentration can still be modeled by a SPDE which is no longer linear. Its properties are investigated in this paper. When the leakage rate is positive, it is shown that the pollution process has an equilibrium state given by the deterministic model treated in [2]. Finally, the linear filtering problem is considered based on the data of several observation stations.

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].

We study the scaling limit of random fields which are solutions of a non-linear partial differential equation, known as the Burgers equation, under stochastic initial conditions. These are assumed to be of a non-local shot noise type and driven by a Cox process. Previous work by Bulinskii and Molchanov (1991), Surgailis and Woyczynski (1993a), and Funaki et al. (1994) concentrated on the case of local shot noise data which permitted use of techniques from the theory of random fields with finite range dependence. Those are not available for the non-local case being considered in this paper.

Burgers' equation is known to describe various physical phenomena such as non-linear and shock waves, distribution of self-gravitating matter in the universe, and other flow satisfying conservation laws (see e.g. Woyczynski (1993)).

A measure-valued diffusion approximation to a two-level branching structure was introduced in Dawson and Hochberg (1991) where it was shown that conditioned on non-extinction at time t, and appropriately rescaled, the process converges as t → ∞to a non-trivial limiting distribution. Here we discuss a different approach to conditioning on non-extinction (popular in one-level branching) and relate the two limiting distributions.

Non-linear stochastic systems driven by white noise are analysed from the viewpoint of non-linear oscillation theory. Under various familiar hypotheses concerning dissipative and restorative dynamical forces, the existence and uniqueness, asymptotic growth, and oscillatory behavior of the solutions are demonstrated.

It is known (Weizsäcker and Winkler (1990)) that for bounded predictable functions H and a Poisson process with jump times exists almost surely, and that in this case both limits are equal. Here we relax the boundedness condition on H. Our tool is a law of large numbers for local L2-martingales. We show by examples that our condition is close to optimal. Furthermore we indicate a generalization to point processes on more general spaces. The above property is called PASTA (‘Poisson arrivals see time averages') and is heavily used in queueing theory.