We consider a ballistic random walk in an i.i.d. random environment that does not allow retreating in a certain fixed direction. We prove an invariance principle (functional central limit theorem) under almost every fixed environment. The assumptions are nonnestling, at least two spatial dimensions, and a 2+ɛ moment for the step of the walk uniformly in the environment. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively.

We study Brownian motion in a drifted Brownian potential. Kawazu and Tanaka [J. Math. Soc. Japan 49 (1997) 189–211] exhibited two speed regimes for this process, depending on the drift. They supplemented these laws of large numbers by central limit theorems, which were recently completed by Hu, Shi and Yor [Trans. Amer. Math. Soc. 351 (1999) 3915–3934] using stochastic calculus. We studied large deviations [Ann. Probab. 29 (2001) 1173–1204], showing among other results that the rate function in the annealed setting, that is, after averaging over the potential, has a flat piece in the ballistic regime. In this paper we focus on this subexponential regime, proving that the probability of deviating below the almost sure speed has a polynomial rate of decay, and computing the exponent in this power law. This provides the continuous-time analogue of what Dembo, Peres and Zeitouni proved for the transient random walk in random environment [Comm. Math. Phys. 181 (1996) 667–683]. Our method takes a completely different route, making use of Lamperti’s representation together with an iteration scheme.

We study branching random walks in random i.i.d. environment in ℤ^d, d≥1. For this model, the population size cannot decrease, and a natural definition of recurrence is introduced. We prove a dichotomy for recurrence/transience, depending only on the support of the environmental law. We give sufficient conditions for recurrence and for transience. In the recurrent case, we study the asymptotics of the tail of the distribution of the hitting times and prove a shape theorem for the set of lattice sites which are visited up to a large time.

In this article, we study linearly edge-reinforced random walk on general multi-level ladders for large initial edge weights. For infinite ladders, we show that the process can be represented as a random walk in a random environment, given by random weights on the edges. The edge weights decay exponentially in space. The process converges to a stationary process. We provide asymptotic bounds for the range of the random walker up to a given time, showing that it localizes much more than an ordinary random walker. The random environment is described in terms of an infinite-volume Gibbs measure.

On étudie la covariance (pour la mesure de Gibbs) des spins en deux sites dans le cas d’un modèle de Sherrington–Kirkpatrick avec champ externe; lorsque le nombre de sites du modèle tend vers l’infini, une évaluation asymptotique des moments d’ordre p de cette covariance permet d’obtenir un théorème limite faible avec une loi limite en général non gaussienne.

We study the covariance (for Gibbs measure) of spins at two sites in the case of a Sherrington–Kirkpatrick model with an external field. When the number of sites of the model grows to infinity, an asymptotic evaluation of the p moments of that covariance allows us to obtain a weak limit theorem, with a generally non-Gaussian limit law.

We consider stochastic differential equations driven by Wiener processes. The vector fields are supposed to satisfy only local Lipschitz conditions. The Lipschitz constants of the drift vector field, valid on balls of radius R, are supposed to grow not faster than log R, while those of the diffusion vector fields are supposed to grow not faster than $\sqrt{\log R}$. We regularize the stochastic differential equations by associating with them approximating ordinary differential equations obtained by discretization of the increments of the Wiener process on small intervals. By showing that the flow associated with a regularized equation converges uniformly to the solution of the stochastic differential equation, we simultaneously establish the existence of a global flow for the stochastic equation under local Lipschitz conditions.

Time-inhomogeneous controlled diffusion processes in both cylindrical and noncylindrical domains are considered. Bellman’s principle and its applications to proving the continuity of value functions are investigated.

In this paper, we derive comparison results for terminal values of d-dimensional special semimartingales and also for finite-dimensional distributions of multivariate Lévy processes. The comparison is with respect to nondecreasing, (increasing) convex, (increasing) directionally convex and (increasing) supermodular functions. We use three different approaches. In the first approach, we give sufficient conditions on the local predictable characteristics that imply ordering of terminal values of semimartingales. This generalizes some recent convex comparison results of exponential models in [Math. Finance 8 (1998) 93–126, Finance Stoch. 4 (2000) 209–222, Proc. Steklov Inst. Math. 237 (2002) 73–113, Finance Stoch. 10 (2006) 222–249]. In the second part, we give comparison results for finite-dimensional distributions of Lévy processes with infinite Lévy measure. In the first step, we derive a comparison result for Markov processes based on a monotone separating transition kernel. By a coupling argument, we get an application to the comparison of compound Poisson processes. These comparisons are then extended by an approximation argument to the ordering of Lévy processes with infinite Lévy measure. The third approach is based on mixing representations which are known for several relevant distribution classes. We discuss this approach in detail for the comparison of generalized hyperbolic distributions and for normal inverse Gaussian processes.

In this paper we discuss existence and uniqueness for a one-dimensional time inhomogeneous stochastic differential equation directed by an $\mathbb{F}$-semimartingale M and a finite cubic variation process ξ which has the structure Q+R, where Q is a finite quadratic variation process and R is strongly predictable in some technical sense: that condition implies, in particular, that R is weak Dirichlet, and it is fulfilled, for instance, when R is independent of M. The method is based on a transformation which reduces the diffusion coefficient multiplying ξ to 1. We use generalized Itô and Itô–Wentzell type formulae. A similar method allows us to discuss existence and uniqueness theorem when ξ is a Hölder continuous process and σ is only Hölder in space. Using an Itô formula for reversible semimartingales, we also show existence of a solution when ξ is a Brownian motion and σ is only continuous.

We study the extremal behavior of a stochastic integral driven by a multivariate Lévy process that is regularly varying with index α>0. For predictable integrands with a finite (α+δ)-moment, for some δ>0, we show that the extremal behavior of the stochastic integral is due to one big jump of the driving Lévy process and we determine its limit measure associated with regular variation on the space of càdlàg functions.

Given a standard Brownian motion B^μ=(B_t^μ)_0≤t≤T with drift μ∈ℝ and letting S_t^μ=max_0≤s≤t B_s^μ for 0≤t≤T, we consider the optimal prediction problem: $$V=\inf_{0\le \tau \le T}\mathsf{E}(B_{\tau}^{\mu}-S_{T}^{\mu})^{2}$$ where the infimum is taken over all stopping times τ of B^μ. Reducing the optimal prediction problem to a parabolic free-boundary problem we show that the following stopping time is optimal: $$τ_*=\inf \{t_*≤t≤T|b_1(t)≤S_t^μ−B_t-^μ≤b_2(t)\}$$ where t_*∈[0, T) and the functions t↦b₁(t) and t↦b₂(t) are continuous on [t_*, T] with b₁(T)=0 and b₂(T)=1/2μ. If μ>0, then b₁ is decreasing and b₂ is increasing on [t_*, T] with b₁(t_*)=b₂(t_*) when t_*≠0. Using local time-space calculus we derive a coupled system of nonlinear Volterra integral equations of the second kind and show that the pair of optimal boundaries b₁ and b₂ can be characterized as the unique solution to this system. This also leads to an explicit formula for V in terms of b₁ and b₂. If μ≤0, then t_*=0 and b₂≡+∞ so that τ_* is expressed in terms of b₁ only. In this case b₁ is decreasing on [z_*, T] and increasing on [0, z_*) for some z_*∈[0, T) with z_*=0 if μ=0, and the system of two Volterra equations reduces to one Volterra equation. If μ=0, then there is a closed form expression for b₁. This problem was solved in [Theory Probab. Appl. 45 (2001) 125–136] using the method of time change (i.e., change of variables). The method of time change cannot be extended to the case when μ≠0 and the present paper settles the remaining cases using a different approach.

Suppose $F$ is a distribution on the half-line $[0,∞)$. We study the limits of the ratios of tails $\overline{F*F}(x)/\overline F(x)$ as $x→∞$. We also discuss the classes of distributions ${\mathscr{S}}$, ${\mathscr{S}}(\gamma)$ and ${\mathscr{S}}^{*}$.

We prove a multivariable approximate Carleman theorem on the determination of complex measures on ℝⁿ and ℝⁿ₊ by their moments. This is achieved by means of a multivariable Denjoy–Carleman maximum principle for quasi-analytic functions of several variables. As an application, we obtain a discrete Phragmén–Lindelöf-type theorem for analytic functions on ℂ₊ⁿ.