Let $I_1,I_2,\ldots,I_n$ be independent Bernoulli random variables with $\mathbb{P}(I_i=1)=1-\mathbb{P}(I_i=0) =p_i$, $1\le i\le n$, and $W=\sum_{i=1}^nI_i$, $\lambda=\mathbb{E}W=\sum_{i=1}^np_i$. It~is well known that if~$p_i$'s are the same, then~$W$ follows a~binomial distribution and if~$p_i$'s are small, then the distribution of~$W$, denoted by~$\mathcal{L} W$, can be well approximated by the $\mathop{\mathrm{Poisson}}(\lambda)$. Define $r=\lfloor\lambda\rfloor$, the greatest integer~$\le\lambda$, and set $\delta=\lambda-\lfloor \lambda \rfloor$, and~$\kappa$ be the least integer more than or equal to $\max\{\lambda^2/(r-1-(1+\delta)^2),n\}$. In this paper, we prove that, if $r>1+(1+\delta)^2$, then \[ d_\kappa<d_{\kappa+1}<d_{\kappa+2}<\cdots<d_{\mathit{TV}} (\mathcal{L} W,\mbox{Poisson}(\lambda)), \] where $d_{\mathit{TV}}$ denotes the total variation metric and $d_m=d_{\mathit{TV}}(\mathcal{L} W,\break\Bi(m,\lambda/m))$, $m\ge\kappa$. Hence, in modelling the distribution of the sum of Bernoulli trials, Binomial approximation is generally better than Poisson approximation.

We consider weak and strong Gaussian approximations for a two-color generalized Friedman's urn model with homogeneous and nonhomogeneous generating matrices. In particular, the functional central limit theorems and the laws of iterated logarithm are obtained. As an application, we obtain the asymptotic properties for the randomized-play-the-winner rule. Based on the Gaussian approximations, we also get some variance estimators for the urn model.

The purpose of this paper is to show that many integral stochastic orders have a generator consisting of infinitely differentiable functions. This especially holds for all stochastic orders with characterizations via difference operators. The usefulness of this result is demonstrated in two applications relating to stochastic ordering of multivariate normal distributions and ordering of random vectors in a random environment.

We address the problem of existence of the uniform value in recursive games. We give two existence results: (i) the uniform value is shown to exist if the state space is countable, the action sets are finite and if, for some $a>0$, there are finitely many states in which the limsup value is less than $a$; (ii) for games with nonnegative payoff function, it is sufficient that the action set of player 2 is finite. The finiteness assumption can be further weakened.

How should we invest capital into a sequence of investment opportunities, if, for reasons of external competition, our interest focuses on trying to invest in the very best opportunity? We introduce new models to answer such questions. Our objective is to formulate them in a way that makes results high-risk specific in order to present true alternatives to other models. At the same time we try to keep them applicable in quite some generality, also for different utility functions. Viewing high-risk situations we assume that an investment on the very best opportunity yields a lucrative, possibly time-dependent, rate of return, that uninvested capital keeps its risk-free value, whereas "wrong" investments lose their value. Several models are presented, mainly for the so-called rank-based case. Optimal strategies and values are found, also for different utility functions, and several examples are explicitly solved. We also include results for the so-called full-information case, where, in addition, the quality distribution of investment opportunities is supposed to be known. In addition we present tractable models for an unknown number of opportunities in terms of Pascal arrival processes. Effort is made throughout the article to justify assumptions in the view of applicability.

We consider a general class of optimization problems in financial markets with incomplete information and transaction costs. Under a no-arbitrage condition strictly weaker than the existence of a martingale measure, and when asset prices are quasi-left-continuous processes, we show the existence of optimal strategies. Applications include maximization of expected utility, minimization of coherent risk measures and hedging of contingent claims.

We consider a classical risk process compounded by another independent process. Both of these component processes are assumed to be Lévy processes. We show asymptotically that as initial capital $y$ increases the ruin probability will essentially behave as $y^{-\kappa}$, where $\kappa$ depends on one of the component processes.

We derive explicit formulas for barrier options of European type and touch-and-out options assuming that under a chosen equivalent martingale measure the stock returns follow a Lévy process from a wide class, which contains Brownian motions (BM), normal inverse Gaussian processes (NIG), hyperbolic processes (HP), normal tilted stable Lévy processes (NTS Lévy), processes of the KoBoL family and any finite mixture of independent BM, NIG, HP, NTS Lévy and KoBoL processes. In contrast to the Gaussian case, for a barrier option, a rebate must be specified not only at the barrier but for all values of the stock on the other side of the barrier. We consider options with a constant or exponentially decaying rebate and options which pay a fixed rebate when the first barrier has been crossed but the second one has not. We obtain pricing formulas by solving boundary problems for the generalized Black--Scholes equation. We use the representation of the $q$-order harmonic measure of a set relative to a point in terms of the $q$-potential measure, the Wiener--Hopf factorization method and elements of the theory of pseudodifferential operators.

Kauffman and Levin introduced a class of models for the evolution of hereditary systems which they called $NK$ fitness landscapes. Inspired by spinglasses, these models have the attractive feature of being tunable, with regard to both overall size (through the parameter $N$) and connectivity (through $K$). There are $N$ genes, each of which exists in two possible alleles [leading to a system indexed by $\{0,1\}^{N}$]; the fitness score of an allele at a given site is determined by the alleles of $K$ neighboring sites. Otherwise the fitnesses are as simple as possible, namely i.i.d., and the fitnesses of different sites are simply averaged.

Much attention has been focused on these fitness landscapes as paradigms for investigating the interaction between size and complexity in making evolution possible. In particular, the effect of the interaction parameter $K$ on the height of the global maximum and the heights of local maxima has attracted considerable interest, as well as the behavior of a "hill-climbing" walk from a random starting point. Nearly all of this work has relied on simulations, not on rigorous mathematics.

In this paper, some asymptotic features of $NK$ fitness landscapes are reduced to questions about eigenvalues and Lyapunov exponents. When $K$ is fixed, the expected number of local maxima grows exponentially with $N$ at a rate depending on the top eigenvalue of a kernel derived from the distribution of the fitnesses, and the average height of a local maximum converges to a value determined by the corresponding eigenfunction.

The global maximum converges in probability as $N \to \infty$ to a constant given by the top Lyapunov exponent for a system of i.i.d. max-plus random matrices, and this constant is nondecreasing with $K$. Various such quantities are computed for certain special cases when $K$ is small, and these calculations can, in principle, be extended to larger $K$.

Kesten has observed that the known reconstruction methods of random sceneries seem to strongly depend on the one-dimensional setting of the problem and asked whether a construction still is possible in two dimensions. In this paper we answer this question in the affirmative under the condition that the number of colors in the scenery is large enough.

We study the stepping stone model on the two-dimensional torus. We prove several new hitting time results for random walks from which we derive some simple approximation formulas for the homozygosity in the stepping stone model as a function of the separation of the colonies and for Wright's genetic distance $F_{\mathit{ST}}$. These results confirm a result of Crow and Aoki (1984) found by simulation: in the usual biological range of parameters $F_{\mathit{ST}}$ grows like the $\log$ of the number of colonies. In the other direction, our formulas show that there is significant spatial structure in parts of parameter space where Maruyama and Nei (1971) and Slatkin and Barton (1989) have called the stepping model "effectively panmictic."

We consider a model for the control of a satellite--fuel is expended in a linear fashion to move a satellite following a diffusion--the aim being to keep the satellite above a critical level. Under suitable assumptions on the drift and diffusion coefficients, it is shown that the probability of the satellite falling below the critical level is minimized by a policy that moves the satellite a certain distance above the critical level and then imposes a reflecting boundary at this higher level until the fuel is exhausted.

In this paper we investigate a class of backward stochastic differential equations (BSDE) whose terminal values are allowed to depend on the history of a forward diffusion. We first establish a probabilistic representation for the spatial gradient of the viscosity solution to a quasilinear parabolic PDE in the spirit of the Feynman--Kac formula, without using the derivatives of the coefficients of the corresponding BSDE. Such a representation then leads to a closed-form representation of the martingale integrand of a BSDE, under only a standard Lipschitz condition on the coefficients. As a direct consequence we prove that the paths of the martingale integrand of such BSDEs are at least c\`{a}dl\`{a}g, which not only extends the existing path regularity results for solutions to BSDEs, but contains the cases where existing methods are not applicable. The BSDEs in this paper can be considered as the nonlinear wealth processes appearing in finance models; our results could lead to efficient Monte Carlo methods for computing both price and optimal hedging strategy for options with nonsmooth, path-dependent payoffs in the situation where the wealth is possibly nonlinear.

Additive white noise may significantly increase the response of bistable systems to a periodic driving signal. We consider the overdamped motion of a Brownian particle in two classes of double-well potentials, symmetric and asymmetric ones. These potentials are modulated periodically in time with period $1/\eps$, where $\eps$ is a moderately (not exponentially) small parameter. We show that the response of the system changes drastically when the noise intensity $\sigma$ crosses a threshold value. Below the threshold, paths are concentrated in one potential well, and have an exponentially small probability to jump to the other well. Above the threshold, transitions between the wells occur with probability exponentially close to $1/2$ in the symmetric case, and exponentially close to $1$ in the asymmetric case. The transition zones are localized in time near the instants of minimal barrier height. We give a mathematically rigorous description of the behavior of individual paths, which allows us, in particular, to determine the power-law dependence of the critical noise intensity on $\eps$ and on the minimal barrier height, as well as the asymptotics of the transition and nontransition probabilities.

We prove a simple asymptotic formula for partial autocorrelation functions of fractional ARIMA processes.

In 1999 Catoni determined the critical rate $H_3$ for the relaxation time of generalized Metropolis algorithms, models for which the speed of convergence to equilibrium can be strongly influenced by the effects of a possible almost periodicity. We recover this result with the help of Dobrushin's coefficient and give characterizations of $H_3$ in terms of other ergodic constants. In particular, we prove that it also governs the large deviation behavior of the singular gap for a sufficiently large but finite number of iterations of the underlying kernel at low temperature.