Oscillation of harmonic functions for subordinate Brownian motion and its applications

In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X$X$ in a bounded κ$\kappa$-fat open set; if u$u$ is a positive harmonic function with respect to X$X$ in a bounded κ$\kappa$-fat open set D$D$ and h$h$ is a positive harmonic function in D$D$ vanishing on D^c$D^c$, then the non-tangential limit of u/h$u/h$ exists almost everywhere with respect to the Martin-representing measure of h$h$.     

Mean-square convergence of stochastic multi-step methods with variable step-size

We study mean-square consistency, stability in the mean-square sense and mean-square convergence of drift-implicit linear multi-step methods with variable step-size for the approximation of the solution of Itô stochastic differential equations. We obtain conditions that depend on the step-size ratios and that ensure mean-square convergence for the special case of adaptive two-step-Maruyama schemes. Further, in the case of small noise we develop a local error analysis with respect to the h$h$–ε$\epsilon$ approach and we construct some stochastic linear multi-step methods with variable step-size that have order 2 behaviour if the noise is small enough.     

Power variation for Gaussian processes with stationary increments

We develop the asymptotic theory for the realised power variation of the processes X=?•G$X=?•G$, where G$G$ is a Gaussian process with stationary increments. More specifically, under some mild assumptions on the variance function of the increments of G$G$ and certain regularity conditions on the path of the process ?$?$ we prove the convergence in probability for the properly normalised realised power variation. Moreover, under a further assumption on the Hölder index of the path of ?$?$, we show an associated stable central limit theorem. The main tool is a general central limit theorem, due essentially to Hu and Nualart [Y. Hu, D. Nualart, Renormalized self-intersection local time for fractional Brownian motion, Ann. Probab. (33) (2005) 948–983], Nualart and Peccati [D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. (33) (2005) 177–193] and Peccati and Tudor [G. Peccati, C.A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals, in: M. Emery, M. Ledoux, M. Yor (Eds.), Seminaire de Probabilites XXXVIII, in: Lecture Notes in Math, vol. 1857, Springer-Verlag, Berlin, 2005, pp. 247–262], for sequences of random variables which admit a chaos representation.     

A converse comparison theorem for anticipated BSDEs and related non-linear expectations

The converse comparison theorem has received much attention in the theory of backward stochastic differential equations (BSDEs). However, no such theorem has been proved for anticipated BSDEs. In this paper, we derive a converse comparison theorem by first giving an existence and uniqueness theorem for adapted solutions of anticipated BSDEs with a stopping time and then related to (f,δ)$(f,\delta )$-expectations induced by anticipated BSDEs.     

On free stochastic processes and their derivatives

We study a family of free stochastic processes whose covariance kernels K$K$ may be derived as a transform of a tempered measure σ$\sigma$. These processes arise, for example, in consideration of non-commutative analysis involving free probability. Hence our use of semi-circle distributions, as opposed to Gaussians. In this setting we find an orthonormal basis in the corresponding non-commutative L²$L^2$ of sample-space. We define a stochastic integral for our family of free processes.     

Strong convergence theorems for finitely many nonexpansive mappings

Under the assumption that E$E$ is a reflexive Banach space whose norm is uniformly Gêteaux differentiable and which has a weakly continuous duality mapping _Jφ$J_{\phi }$ with gauge function φ$\phi$, Ceng–Cubiotti–Yao [Strong convergence theorems for finitely many nonexpansive mappings and applications, Nonlinear Analysis 67 (2007) 1464–1473] introduced a new iterative scheme for a finite commuting family of nonexpansive mappings, and proved strong convergence theorems about this iteration. In this paper, only under the hypothesis that E$E$ is a reflexive Banach space which has a weakly continuous duality mapping _Jφ$J_{\phi }$ with gauge function φ$\phi$, and several control conditions about the iterative coefficient are removed, we present a short and simple proof of the above theorem.     

Quadratic reflected BSDEs with unbounded obstacles

In this paper, we analyze a real-valued reflected backward stochastic differential equation (RBSDE) with an unbounded obstacle and an unbounded terminal condition when its generator f$f$ has quadratic growth in the z$z$-variable. In particular, we obtain existence, uniqueness, and stability results, and consider the optimal stopping for quadratic g$g$-evaluations. As an application of our results we analyze the obstacle problem for semi-linear parabolic PDEs in which the non-linearity appears as the square of the gradient. Finally, we prove a comparison theorem for these obstacle problems when the generator is concave in the z$z$-variable.     

Some results on general quadratic reflected BSDEs driven by a continuous martingale

We study the well-posedness of general reflected BSDEs driven by a continuous martingale, when the coefficient f$f$ of the driver has at most quadratic growth in the control variable Z$Z$, with a bounded terminal condition and a lower obstacle which is bounded above. We obtain the basic results in this setting: comparison and uniqueness, existence, stability. For the comparison theorem and the special comparison theorem for reflected BSDEs (which allows one to compare the increasing processes of two solutions), we give intrinsic proofs which do not rely on the comparison theorem for standard BSDEs. This allows to obtain the special comparison theorem under minimal assumptions. We obtain existence by using the fixed point theorem and then a series of perturbations, first in the case where f$f$ is Lipschitz in the primary variable Y$Y$, and then in the case where f$f$ can have slightly-superlinear growth and the case where f$f$ is monotonous in Y$Y$ with arbitrary growth. We also obtain a local Lipschitz estimate in BMO$BMO$ for the martingale part of the solution.     

Subgeometric rates of convergence of f-ergodic strong Markov processes

We provide a condition in terms of a supermartingale property for a functional of the Markov process, which implies (a) f$f$-ergodicity of strong Markov processes at a subgeometric rate, and (b) a moderate deviation principle for an integral (bounded) functional. An equivalent condition in terms of a drift inequality on the extended generator is also given. Results related to (f,r)$(f,r)$-regularity of the process, of some skeleton chains and of the resolvent chain are also derived. Applications to specific processes are considered, including elliptic stochastic differential equations, Langevin diffusions, hypoelliptic stochastic damping Hamiltonian systems and storage models.     

A finite dimensional approximation of the effective diffusivity for a symmetric random walk in a random environment

We consider a nearest neighbor, symmetric random walk on a homogeneous, ergodic random lattice Z^d${\mathbb{Z}}^d$. The jump rates of the walk are independent, identically Bernoulli distributed random variables indexed by the bonds of the lattice. A standard result from the homogenization theory, see [A. De Masi, P.A. Ferrari, S. Goldstein, W.D. Wick, An invariance principle for reversible Markov processes, Applications to random walks in random environments, J. Statist. Phys. 55(3/4) (1989) 787–855], asserts that the scaled trajectory of the particle satisfies the functional central limit theorem. The covariance matrix of the limiting normal distribution is called the effective diffusivity of the walk. We use the duality structure corresponding to the product Bernoulli measure to construct a numerical scheme that approximates this parameter when d?3$d?3$. The estimates of the convergence rates are also provided.     

Some limit theorems for Hawkes processes and application to financial statistics

In the context of statistics for random processes, we prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [0,T]$[0,T]$ when T→∞$T\to \infty$. We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with mesh Δ$\Delta$ over [0,T]$[0,T]$ up to some further time shift τ$\tau$. The behaviour of this functional depends on the relative size of Δ$\Delta$ and τ$\tau$ with respect to T$T$ and enables to give a full account of the second-order structure. As an application, we develop our results in the context of financial statistics. We introduced in Bacry et al. (2013) [7] a microscopic stochastic model for the variations of a multivariate financial asset, based on Hawkes processes and that is confined to live on a tick grid. We derive and characterise the exact macroscopic diffusion limit of this model and show in particular its ability to reproduce the important empirical stylised fact such as the Epps effect and the lead–lag effect. Moreover, our approach enables to track these effects across scales in rigorous mathematical terms.     

Multivariate generalized Ornstein–Uhlenbeck processes

De Haan and Karandikar (1989) [7] introduced generalized Ornstein–Uhlenbeck processes as one-dimensional processes _(Vt)t≥0${\left(V_t\right)}_{t\ge 0}$ which are basically characterized by the fact that for each h>0$h>0$ the equidistantly sampled process _(Vnh)n∈N0${\left(V_{nh}\right)}_{n\in {\mathbb{N}}_0}$ satisfies the random recurrence equation _Vnh=_{A(n−1)h,nhV(n−1)h}+_B(n−1)h,nh$V_{nh}=A_{(n-1)h,nh}{V}_{(n-1)h}+B_{(n-1)h,nh}$, n∈N$n\in \mathbb{N}$, where _{(A(n−1)h,nh,B(n−1)h,nh)n∈N}${(A_{(n-1)h,nh},B_{(n-1)h,nh})}_{n\in \mathbb{N}}$ is an i.i.d. sequence with positive _A0,h$A_{0,h}$ for each h>0$h>0$. We generalize this concept to a multivariate setting and use it to define multivariate generalized Ornstein–Uhlenbeck (MGOU) processes which occur to be characterized by a starting random variable and some Lévy process (X,Y)$(X,Y)$ in R^m×m×R^m${\mathbb{R}}^{m\times m}\times {\mathbb{R}}^m$. The stochastic differential equation an MGOU process satisfies is also derived. We further study invariant subspaces and irreducibility of the models generated by MGOU processes and use this to give necessary and sufficient conditions for the existence of strictly stationary MGOU processes under some extra conditions.     

A comparison of a posteriori error estimates for biharmonic problems solved by the FEM

The classical a posteriori error estimates are mostly oriented to the use in the finite element h$h$-methods while the contemporary higher-order hp$hp$-methods usually require new approaches in a posteriori error estimation. These methods hold a very important position among adaptive numerical procedures for solving ordinary as well as partial differential equations arising from various technical applications.     

Abelian theorems for stochastic volatility models with application to the estimation of jump activity

In this paper, we prove a kind of Abelian theorem for a class of stochastic volatility models (X,V)$(X,V)$ where both the state process X$X$ and the volatility process V$V$ may have jumps. Our results relate the asymptotic behavior of the characteristic function of _XΔ$X_{\Delta }$ for some Δ>0$\Delta >0$ in a stationary regime to the Blumenthal–Getoor indexes of the Lévy processes driving the jumps in X$X$ and V$V$. The results obtained are used to construct consistent estimators for the above Blumenthal–Getoor indexes based on low-frequency observations of the state process X$X$. We derive convergence rates for the corresponding estimator and show that these rates cannot be improved in general.     

Ergodicity for time-changed symmetric stable processes

In this paper we study ergodicity and related semigroup property for a class of symmetric Markov jump processes associated with time-changed symmetric α$\alpha$-stable processes. For this purpose, explicit and sharp criteria for Poincaré type inequalities (including Poincaré, super Poincaré and weak Poincaré inequalities) of the corresponding non-local Dirichlet forms are derived. Moreover, our main results, when applied to a class of one-dimensional stochastic differential equations driven by symmetric α$\alpha$-stable processes, yield sharp criteria for their various ergodic properties and corresponding functional inequalities.     

The Schauder fixed point theorem in geodesic spaces

We give a direct proof of Schauder's fixed point theorem in the setting of geodesic metric spaces, generalizing the classical Schauder's theorem and improving a recent version of this theorem in CAT(κ)$\mathrm{CAT}(\kappa )$ spaces. As an application we prove an existence result for a variational inequality in the setting of CAT(κ)$\mathrm{CAT}(\kappa )$ spaces.