We study the Euler scheme for a stochastic differential equation driven by a Lévy process Y. More precisely, we look at the asymptotic behavior of the normalized error process u_n(Xⁿ−X), where X is the true solution and Xⁿ is its Euler approximation with stepsize 1/n, and u_n is an appropriate rate going to infinity: if the normalized error processes converge, or are at least tight, we say that the sequence (u_n) is a rate, which, in addition, is sharp when the limiting process (or processes) is not trivial.

We suppose that Y has no Gaussian part (otherwise a rate is known to be $u_{n}=\sqrt {n}$ ). Then rates are given in terms of the concentration of the Lévy measure of Y around 0 and, further, we prove the convergence of the sequence u_n(Xⁿ−X) to a nontrivial limit under some further assumptions, which cover all stable processes and a lot of other Lévy processes whose Lévy measure behave like a stable Lévy measure near the origin. For example, when Y is a symmetric stable process with index α∈(0,2), a sharp rate is u_n=(n/logn)^1/α; when Y is stable but not symmetric, the rate is again u_n=(n/logn)^1/α when α>1, but it becomes u_n=n/(logn)² if α=1 and u_n=n if α<1.

We refine the conditions for the lower bound in an abstract large deviation result with nonconvex rate function we had previously introduced. We apply the results to certain stochastic recursive schemes.

Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several exponential and moment inequalities, particularly those related to laws of the iterated logarithm, for self-normalized random variables including martingales. Tail probability bounds are also derived. For random variables B_t>0 and A_t, let Y_t(λ)=exp{λA_t−λ²B_t²/2}. We develop inequalities for the moments of A_t/B_t or sup_t≥0A_t/{B_t(log logB_t)^1/2} and variants thereof, when EY_t(λ)≤1 or when Y_t(λ) is a supermartingale, for all λ belonging to some interval. Our results are valid for a wide class of random processes including continuous martingales with A_t=M_t and $B_{t}=\sqrt {\langle M\rangle _{t}}$ , and sums of conditionally symmetric variables d_i with A_t=∑_i=1^td_i and $B_{t}=\sqrt{\sum_{i=1}^{t}d_{i}^{2}}$. A sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in R^m, m≥1, is also established. Another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self-normalized by the square root of the sum of squares. The key ingredient in this development is a new exponential supermartingale involving ∑_i=1^td_i and ∑_i=1^td_i². A compact law of the iterated logarithm for self-normalized martingales is also derived in this connection.

Let M be a noncompact metric space in which every closed ball is compact, and let G be a semigroup of Lipschitz mappings of M. Denote by (Y_n)_n≥1 a sequence of independent G-valued, identically distributed random variables (r.v.’s), and by Z an M-valued r.v. which is independent of the r.v. Y_n, n≥1. We consider the Markov chain (Z_n)_n≥0 with state space M which is defined recursively by Z₀=Z and Z_n+1=Y_n+1Z_n for n≥0. Let ξ be a real-valued function on G×M. The aim of this paper is to prove central limit theorems for the sequence of r.v.’s (ξ(Y_n,Z_n−1))_n≥1. The main hypothesis is a condition of contraction in the mean for the action on M of the mappings Y_n; we use a spectral method based on a quasi-compactness property of the transition probability of the chain mentioned above, and on a special perturbation theorem.

We establish both uniform and nonuniform error bounds of the Berry–Esseen type in normal approximation under local dependence. These results are of an order close to the best possible if not best possible. They are more general or sharper than many existing ones in the literature. The proofs couple Stein’s method with the concentration inequality approach.

A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (X_n)_n≥1 is said to be conditionally identically distributed (c.i.d.), with respect to a filtration $(\mathcal{G}_{n})_{n\geq 0}$ , if it is adapted to $(\mathcal{G}_{n})_{n\geq 0}$ and, for each n≥0, (X_k)_k>n is identically distributed given the past $\mathcal{G}_{n}$ . In case $\mathcal{G}_{0}=\{\varnothing,\Omega\}$ and $\mathcal{G}_{n}=\sigma(X_{1},\ldots,X_{n})$ , a result of Kallenberg implies that (X_n)_n≥1 is exchangeable if and only if it is stationary and c.i.d. After giving some natural examples of nonexchangeable c.i.d. sequences, it is shown that (X_n)_n≥1 is exchangeable if and only if (X_τ(n))_n≥1 is c.i.d. for any finite permutation τ of {1,2,…}, and that the distribution of a c.i.d. sequence agrees with an exchangeable law on a certain sub-σ-field. Moreover, (1/n)∑_k=1ⁿX_k converges a.s. and in L¹ whenever (X_n)_n≥1 is (real-valued) c.i.d. and E[|X₁|]<∞. As to the CLT, three types of random centering are considered. One such centering, significant in Bayesian prediction and discrete time filtering, is $E[X_{n+1}\vert \mathcal{G}_{n}]$. For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.

Let ξ_t, t∈[0,T], be a strong Markov process with values in a complete separable metric space (X,ρ) and with transition probability function P_s,t(x,dy), 0≤s≤t≤T, x∈X. For any h∈[0,T] and a>0, consider the function $$α(h,a)=\sup\{P_{s,t}(x,\{y:ρ(x,y)≥a\}):x∈X,0≤s≤t≤(s+h)∧T\}.$$ It is shown that a certain growth condition on α(h,a), as a↓0 and h stays fixed, implies the almost sure boundedness of the p-variation of ξ_t, where p depends on the rate of growth.

We study Girsanov’s theorem in the context of symmetric Markov processes, extending earlier work of Fukushima–Takeda and Fitzsimmons on Girsanov transformations of “gradient type.” We investigate the most general Girsanov transformation leading to another symmetric Markov process. This investigation requires an extension of the forward–backward martingale method of Lyons–Zheng, to cover the case of processes with jumps.

We give general sufficient conditions which imply upper and lower bounds for the probability that a multiparameter process hits a given set E in terms of a capacity of E related to the process. This extends a result of Khoshnevisan and Shi [Ann. Probab. 27 (1999) 1135–1159], where estimates for the hitting probabilities of the (N,d) Brownian sheet in terms of the (d−2N) Newtonian capacity are obtained, and readily applies to a wide class of Gaussian processes. Using Malliavin calculus and, in particular, a result of Kohatsu-Higa [Probab. Theory Related Fields 126 (2003) 421–457], we apply these general results to the solution of a system of d nonlinear hyperbolic stochastic partial differential equations with two variables. We show that under standard hypotheses on the coefficients, the hitting probabilities of this solution are bounded above and below by constants times the (d−4) Newtonian capacity. As a consequence, we characterize polar sets for this process and prove that the Hausdorff dimension of its range is min (d,4) a.s.

The purpose of this note is to provide proofs for some facts about the NK model of evolution proposed by Kauffman and Levin. In the case of normally distributed fitness summands, some of these facts have been previously conjectured and heuristics given. In particular, we provide rigorous asymptotic estimates for the number of local fitness maxima in the case when K is unbounded. We also examine the role of the individual fitness distribution and find the model to be quite robust with respect to this.

In a supercritical branching particle system, the trimmed tree consists of those particles which have descendants at all times. We develop this concept in the superprocess setting. For a class of continuous superprocesses with Feller underlying motion on compact spaces, we identify the trimmed tree, which turns out to be a binary splitting particle system with a new underlying motion that is a compensated h-transform of the old one. We show how trimmed trees may be estimated from above by embedded binary branching particle systems.

We study stationary stable processes related to periodic and cyclic flows in the sense of Rosiński [Ann. Probab. 23 (1995) 1163–1187]. These processes are not ergodic. We provide their canonical representations, consider examples and show how to identify them among general stationary stable processes. We conclude with the unique decomposition in distribution of stationary stable processes into the sum of four major independent components: 1. A mixed moving average component. 2. A harmonizable (or “trivial”) component. 3. A cyclic component 4. A component which is different from these.

Expected utility maximization problems in mathematical finance lead to a generalization of the classical definition of entropy. It is demonstrated that a necessary and sufficient condition for the second law of thermodynamics to operate is that any one of the generalized entropies should tend to its minimum value of zero.

We extend to Riemannian manifolds the theory of conditioned stochastic differential equations. We also provide some enlargement formulas for the Brownian filtration in this nonflat setting.