We consider a dynamic control problem associated with a generalized Brownian network, the objective being to minimize expected discounted cost over an infinite planning horizon. In this Brownian control problem (BCP), both the system manager’s control and the associated cumulative cost process may be locally of unbounded variation. Due to this aspect of the cost process, both the precise statement of the problem and its analysis involve delicate technical issues. We show that the BCP is equivalent, in a certain sense, to a reduced Brownian control problem (RBCP) of lower dimension. The RBCP is a singular stochastic control problem, in which both the controls and the cumulative cost process are locally of bounded variation.

Atlas-type models are constant-parameter models of uncorrelated stocks for equity markets with a stable capital distribution, in which the growth rates and variances depend on rank. The simplest such model assigns the same, constant variance to all stocks; zero rate of growth to all stocks but the smallest; and positive growth rate to the smallest, the Atlas stock. In this paper we study the basic properties of this class of models, as well as the behavior of various portfolios in their midst. Of particular interest are portfolios that do not contain the Atlas stock.

We show how Rio’s method [Probab. Theory Related Fields 104 (1996) 255–282] can be adapted to establish a rate of convergence in ${\frac{1}{\sqrt{n}}}$ in the multidimensional central limit theorem for some stationary processes in the sense of the Kantorovich metric. We give two applications of this general result: in the case of the Knudsen gas and in the case of the Sinai billiard.

We consider a general discrete-time financial market with proportional transaction costs as in [Kabanov, Stricker and Rásonyi Finance and Stochastics 7 (2003) 403–411] and [Schachermayer Math. Finance 14 (2004) 19–48]. In addition to the usual investment in financial assets, we assume that the agents can invest part of their wealth in industrial projects that yield a nonlinear random return. We study the problem of maximizing the utility of consumption on a finite time period. The main difficulty comes from the nonlinearity of the nonfinancial assets’ return. Our main result is to show that existence holds in the utility maximization problem. As an intermediary step, we prove the closedness of the set A_T of attainable claims under a robust no-arbitrage property similar to the one introduced in [Schachermayer Math. Finance 14 (2004) 19–48] and further discussed in [Kabanov, Stricker and Rásonyi Finance and Stochastics 7 (2003) 403–411]. This allows us to provide a dual formulation for A_T.

We describe a new, surprisingly simple algorithm, that simulates exact sample paths of a class of stochastic differential equations. It involves rejection sampling and, when applicable, returns the location of the path at a random collection of time instances. The path can then be completed without further reference to the dynamics of the target process.

In this paper a general class of tree algorithms is analyzed. It is shown that, by using an appropriate probabilistic representation of the quantities of interest, the asymptotic behavior of these algorithms can be obtained quite easily without resorting to the usual complex analysis techniques. This approach gives a unified probabilistic treatment of these questions. It simplifies and extends some of the results known in this domain.

We study the small time path behavior of double stochastic integrals of the form ∫₀^t(∫₀^rb(u) dW(u))^T dW(r), where W is a d-dimensional Brownian motion and b is an integrable progressively measurable stochastic process taking values in the set of d×d-matrices. We prove a law of the iterated logarithm that holds for all bounded progressively measurable b and give additional results under continuity assumptions on b. As an application, we discuss a stochastic control problem that arises in the study of the super-replication of a contingent claim under gamma constraints.

In this paper an original interacting particle system approach is developed for studying Markov chains in rare event regimes. The proposed particle system is theoretically studied through a genealogical tree interpretation of Feynman–Kac path measures. The algorithmic implementation of the particle system is presented. An estimator for the probability of occurrence of a rare event is proposed and its variance is computed, which allows to compare and to optimize different versions of the algorithm. Applications and numerical implementations are discussed. First, we apply the particle system technique to a toy model (a Gaussian random walk), which permits to illustrate the theoretical predictions. Second, we address a physically relevant problem consisting in the estimation of the outage probability due to polarization-mode dispersion in optical fibers.

Random geometric graphs result from taking n uniformly distributed points in the unit cube, [0,1]^d, and connecting two points if their Euclidean distance is at most r, for some prescribed r. We show that monotone properties for this class of graphs have sharp thresholds by reducing the problem to bounding the bottleneck matching on two sets of n points distributed uniformly in [0,1]^d. We present upper bounds on the threshold width, and show that our bound is sharp for d=1 and at most a sublogarithmic factor away for d≥2. Interestingly, the threshold width is much sharper for random geometric graphs than for Bernoulli random graphs. Further, a random geometric graph is shown to be a subgraph, with high probability, of another independently drawn random geometric graph with a slightly larger radius; this property is shown to have no analogue for Bernoulli random graphs.

We study some stochastic models of physical mapping of genomic sequences. Our starting point is a global construction of the process of the clones and of the process of the anchors which are used to map the sequence. This yields explicit formulas for the moments of the proportion occupied by the anchored clones, even in inhomogeneous models. This also allows to compare, in this respect, inhomogeneous models to homogeneous ones. Finally, for homogeneous models, we provide nonasymptotic bounds of the variance and we prove functional invariance results.

We study a maturity randomization technique for approximating optimal control problems. The algorithm is based on a sequence of control problems with random terminal horizon which converges to the original one. This is a generalization of the so-called Canadization procedure suggested by Carr [Review of Financial Studies II (1998) 597–626] for the fast computation of American put option prices. In addition to the original application of this technique to optimal stopping problems, we provide an application to another problem in finance, namely the super-replication problem under stochastic volatility, and we show that the approximating value functions can be computed explicitly.

A multiclass queueing system is considered, with heterogeneous service stations, each consisting of many servers with identical capabilities. An optimal control problem is formulated, where the control corresponds to scheduling and routing, and the cost is a cumulative discounted functional of the system’s state. We examine two versions of the problem: “nonpreemptive,” where service is uninterruptible, and “preemptive,” where service to a customer can be interrupted and then resumed, possibly at a different station. We study the problem in the asymptotic heavy traffic regime proposed by Halfin and Whitt, in which the arrival rates and the number of servers at each station grow without bound. The two versions of the problem are not, in general, asymptotically equivalent in this regime, with the preemptive version showing an asymptotic behavior that is, in a sense, much simpler. Under appropriate assumptions on the structure of the system we show: (i) The value function for the preemptive problem converges to V, the value of a related diffusion control problem. (ii) The two versions of the problem are asymptotically equivalent, and in particular nonpreemptive policies can be constructed that asymptotically achieve the value V. The construction of these policies is based on a Hamilton–Jacobi–Bellman equation associated with V.

We extend classical results by A. V. Nagaev [Izv. Akad. Nauk UzSSR Ser. Fiz.–Mat. Nauk 6 (1969) 17–22, Theory Probab. Appl. 14 (1969) 51–64, 193–208] on large deviations for sums of i.i.d. regularly varying random variables to partial sum processes of i.i.d. regularly varying vectors. The results are stated in terms of a heavy-tailed large deviation principle on the space of càdlàg functions. We illustrate how these results can be applied to functionals of the partial sum process, including ruin probabilities for multivariate random walks and long strange segments. These results make precise the idea of heavy-tailed large deviation heuristics: in an asymptotic sense, only the largest step contributes to the extremal behavior of a multivariate random walk.

We study the approximation of $\mathbb{E}f(X_{T})$ by a Monte Carlo algorithm, where X is the solution of a stochastic differential equation and f is a given function. We introduce a new variance reduction method, which can be viewed as a statistical analogue of Romberg extrapolation method. Namely, we use two Euler schemes with steps δ and δ^β,0<β<1. This leads to an algorithm which, for a given level of the statistical error, has a complexity significantly lower than the complexity of the standard Monte Carlo method. We analyze the asymptotic error of this algorithm in the context of general (possibly degenerate) diffusions. In order to find the optimal β (which turns out to be β=1/2), we establish a central limit type theorem, based on a result of Jacod and Protter for the asymptotic distribution of the error in the Euler scheme. We test our method on various examples. In particular, we adapt it to Asian options. In this setting, we have a CLT and, as a by-product, an explicit expansion of the discretization error.

A Milstein-type scheme was proposed to improve the rate of convergence of its approximation of the solution to a stochastic differential equation driven by a vector of continuous semimartingales. A necessary and sufficient condition was provided for this rate to be 1/n when the SDE is driven by a vector of continuous local martingales, or continuous semimartingales under an additional assumption on their finite variation part. The asymptotic behavior (weak convergence) of the normalized error processes was also studied.

Herein, we analyze an efficient branching particle method for asymptotic solutions to a class of continuous-discrete filtering problems. Suppose that t→X_t is a Markov process and we wish to calculate the measure-valued process t→μ_t(⋅)≐P{X_t∈⋅|σ{Y_tk, t_k≤t}}, where t_k=kɛ and Y_tk is a distorted, corrupted, partial observation of X_tk. Then, one constructs a particle system with observation-dependent branching and n initial particles whose empirical measure at time t, μ_tⁿ, closely approximates μ_t. Each particle evolves independently of the other particles according to the law of the signal between observation times t_k, and branches with small probability at an observation time. For filtering problems where ɛ is very small, using the algorithm considered in this paper requires far fewer computations than other algorithms that branch or interact all particles regardless of the value of ɛ. We analyze the algorithm on Lévy-stable signals and give rates of convergence for E^1/2{‖μⁿ_t−μ_t‖_γ²}, where ‖⋅‖_γ is a Sobolev norm, as well as related convergence results.

We introduce a theory of stochastic integration with respect to a family of semimartingales depending on a continuous parameter, as a mathematical background to the theory of bond markets. We apply our results to the problem of super-replication and utility maximization from terminal wealth in a bond market. Finally, we compare our approach to those already existing in literature.