Several random particle systems approaches were recently suggested to solve nonlinear filtering problems numerically. The present analysis is concerned with genetic-type interacting particle systems. Our aim is to study the fluctuations on path space of such particle-approximating models.

Given an (ordinary) super-Brownian motion (SBM) $\varrho$ on $\mathbf{R}^d$ of dimension $d = 2, 3$, we consider a (catalytic) SBM $X^{\varrho}$ on $\mathbf{R}^d$ with "local branching rates" $\varrho_s(dx)$. We show that $X_t^{\varrho}$ is absolutely continuous with a density function $\xi_t^{\varrho}$, say. Moreover, there exists a version of the map $(t, z) \mapsto \xi_t^{\varrho}(z)$ which is $\mathscr{C}^{\infty}$ and solves the heat equation off the catalyst $\varrho$; more precisely, off the (zero set of) closed support of the time-space measure $ds\varrho_s(dx)$. Using self-similarity, we apply this result to give the following answer to an open problem on the long-term behavior of $X^{\varrho}$ in dimension $d = 2$: If $\varrho$ and $X^{\varrho}$ start with a Lebesgue measure, then does $X_T^{\varrho}$ converge (persistently) as $T \to \infty$ toward a random multiple of Lebesgue measure?

We consider a gene with two alleles. Allele A is normal, allele S is abnormal. Individuals with genotype SS have a severe disease called sickle-cell disease. Individuals with genotype AS are not sick, and it is thought that they are more resistant to malarial infection than individuals with genotype AA. This could explain why the allele S has persisted in regions where malaria is endemic. We use a stochastic spatial process to test this hypothesis. For our model, we show that if the genotype AS has an advantage over the genotype AA, then the allele S will persist in the population even if the genotype SS is not viable.

We introduce a multitype interacting particle system, which is a natural generalization of the contact process. Here, individuals in the population have two life stages, young and adult. Only adults can give birth and each new offspring is young. Transition from young to adult occurs at constant rate, and individuals die at rates that depend on their life stage. Important to the analysis of this process is the construction of a multitype dual process.

We consider a fluid queue with sessions arriving according to a Poisson process. A long-tailed distribution of session lengths induces long range dependence in the system and causes its performance to deteriorate. The deterioration is due to occurrence of load regimes far from average ones. Nonetheless, the extent of this performance deterioration is shown to depend crucially on the average values of the system parameters.

Motivated by the problem of estimating the fractal dimension of a strange attractor, we prove weak consistency of U-statistics for stationary ergodic and mixing sequences when the kernel function is unbounded, extending by this earlier results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss. We apply the obtained results to show consistency of the Takens estimator for the correlation dimension.

In modeling particle transport through a medium, the path of a particle behaves as a transient Markov chain. We are interested in characteristics of the particle's movement conditional on its starting state, which take the form of a "score" accumulated with each transition. Importance sampling is an essential variance reduction technique in this setting, and we provide an adaptive (iteratively updated) importance sampling algorithm that converges exponentially to the solution. Examples illustrating this phenomenon are provided.

Given two independent realizations of the stationary processes $\mathbf{X} = {X_n;n \geq 1}$ and $\mathbf{Y} = {Y_n;n \geq 1}$, our main quantity of interest is the waiting time $W_n(D)$ until a D-close version of the initial string $(X_1, X_2,\dots, X_n)$ first appears as a contiguous substring in $(Y_1, Y_2, Y_3,\dots)$, where closeness is measured with respect to some "average distortion" criterion.

We study the asymptotics of $W_n(D)$ for large n under various mixing conditions on X and Y. We first prove a strong approximation theorem between $\logW_n(D)$ and the logarithm of the probability of a D-ball around $(X_1, X_2,\dots, X_n)$. Using large deviations techniques, we show that this probability can, in turn, be strongly approximated by an associated random walk, and we conclude that: (i) $n^{-1} \log W_n(D)$ converges almost surely to a constant R determined byan explicit variational problem; (ii) $[\log W_n(D) - R]$, properly normalized, satisfies a central limit theorem, a law of the iterated logarithm and, more generally, an almost sure invariance principle.

Consider a finite list of items $n = 1, 2,\dots, N$, that are requested according to an i.i.d. process. Each time an item is requested it is moved to the front of the list. The associated search cost $C^N$ for accessing an item is equal to its position before being moved. If the request distribution converges to a proper distribution as $N \to \infty$, then the stationary search cost $C^N$ converges in distribution to a limiting search cost C.

We show that, when the (limiting) request distribution has a heavy tail (e.g., generalized Zipf's law), $\mathbb{P}[R = n] \sim c/n^{\alpha}$ as $n \to \infty, \alpha > 1$, then the limiting stationary search cost distribution $\mathbb{P}[C > n]$, or, equivalently, the least-recently used (LRU) caching fault probability, satisfies $$\lim_{n \to \infty} \frac{\mathbb{P}[C > n]}{\mathbb{P}[R > n]}= (1 - \frac{1}{\alpha}) [\Gamma(1 - \frac{1}{\alpha})]^{\alpha} \nearrow e^{\gamma}$ \text{as}$\alpha \to \infty,$$ where $\Gamma$ is the Gamma function and $\gamma(=0.5772\dots)$ is Euler's constant.

When the request distribution has a light tail $\mathbb{P}[R = n] \sim c \exp(-\lambdan_{\beta})$ as $n \to \infty (c, \lambda, \beta > 0), then $$\lim_{n \to \infty} \frac{\mathbb{P}[C_f > n]}{\mathbb{P}[R > n]} = e^{\gamma},$$ independently of $c, \lambda, \beta$, where $C_f$ is a fluid approximation of C.

We experimentally demonstrate that the derived asymptotic formulas yield accurate results for lists of finite sizes. This should be contrasted with the exponential computational complexity of Burville and Kingman's exact expression for finite lists. The results also imply that the fault probability of LRU caching is asymptotically at most a factor $e^{\gamma} (\approx 1.78)$ greater than for the optimal static arrangement.

Let n particles be independently and uniformly distributed in a rectangle $\mathbf{A} \subset \mathbb{R}^2$. Each subset consisting of $k \leq n$ particles may possibly aggregate in such a way that it is covered by some translate of a given convex set $C \subset \mathbf{A}$. The number of k-subsets which actually are covered by translates of C is denoted by W. The positions of such subsets constitute a point process on A. Each point of this process can be marked with the smallest necessary "size" of a set, of the same shape and orientation as C, which covers the particles determining the point. This results in a marked point process.

The purpose of this paper is to consider Poisson process approximations of W and of the above point processes, by means of Stein's method. To this end, the exact probability for k specific particles to be covered by some translate of C is given.

The use of saddlepoint approximations in statistics is a well-established technique for computing the distribution of a random variable whose moment generating function is known. In this paper, we apply the methodology to computing the prices of various European-style options, whose returns processes are not the Brownian motion with drift assumed in the Black-Scholes paradigm. Through a number of examples, we show that the methodology is generally accurate and fast.

We consider the problem of pricing contingent claims on a stock whose price process is modelled by a geometric Lévy process, in exact analogy with the ubiquitous geometric Brownian motion model. Because the noise process has jumps of random sizes, such a market is incomplete and there is not a unique equivalent martingale measure. We study several approaches to pricing options which all make use of an equivalent martingale measure that is in different respects "closest" to the underlying canonical measure, the main ones being the Föllmer-Schweizer minimal measure and the martingale measure which has minimum relative entropy with respect to the canonical measure. It is shown that the minimum relative entropy measure is that constructed via the Esscher transform, while the Föllmer-Schweizer measure corresponds to another natural analogue of the classical Black-Scholes measure.

We propose a notion of market completeness which is invariant under change to an equivalent probability measure. Completeness means that an operator T acting on stopping time simple trading strategies has dense range in the weak* topology on bounded random variables. In our setup, the claims which can be approximated by attainable ones has codimension equal to the dimension of the kernel of the adjoint operator $T*$ acting on signed measures, which in most cases is equal to the "dimension of the space of martingale measures." From this viewpoint the example of Artzner and Heath is no longer paradoxical since all the dimensions are 1. We also illustrate how one can check for injectivity of $T*$ and hence for completeness in the case of price processes on a Brownian filtration (e.g., Black-Scholes, Heath-Jarrow-Morton) and price processes driven by a multivariate point process.