Strong approximations of three-dimensional Wiener sausages

We prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall [10]șs fine L ²-norm estimates between the Wiener sausage and the Brownian intersection local times. Research supported by the Hungarian National Foundation for Scientific Research, Grants T 037886, T 043037 and K 61052.     

SOME LIMIT PROPERTIES OF LOCAL TIME FOR RANDOM WALK

Let X,X1,X2 be i. i. d. random variables with EX^2＋δ〈∞ （for some δ〉0）. Consider a one dimensional random walk S={Sn}n≥0, starting from S0 =0. Let ζ＊ （n）=supx∈zζ（x,n）,ζ（x,n） =#{0≤k≤n：[Sk]=x}. A strong approximation of ζ（n） by the local time for Wiener process is presented and the limsup type and liminf-type laws of iterated logarithm of the maximum local time ζ＊（n） are obtained. Furthermore,the precise asymptoties in the law of iterated logarithm of ζ＊（n） is proved.     

Laws of the iterated logarithm for high-dimensional wiener sausage

Let {β(s), s ≥ 0} be the standard Brownian motion in ℝ ^d with d ≥ 4 and let |W _r (t)| be the volume of the Wiener sausage associated with {β(s), s ≥ 0} observed until time t. From the central limit theorem of Wiener sausage, we know that when d ≥ 4 the limit distribution is normal. In this paper, we study the laws of the iterated logarithm for | W_r (t) | - \mathbbE| W_r (t) |\left| {W_r (t)} \right| - \mathbb{E}\left| {W_r (t)} \right| in this case.     

Stochastic anticipative calculus for functionals of a gaussian generalized random field

We give a construction of Skorohod integrals with respect to a Gaussian D'-valued random field W The method is based on the multiple Wiener integral expansion for L ₂-functionals of W We also give a representation of the Malliavin derivative operator of L ²-functionals of W     

The occupation time of Brownian motion in a ball

LetW be a Wiener process of dimensiond3, starting from 0, and letX(t) be the total time spent byW in the ball centered at 0 with radiust. We give an affirmative answer to a conjecture of Taylor and Tricot⁽¹⁶⁾ on the tail distribution ofX(t). Lévy's lower functions ofX(t) are characterized by an integral test.     

On a problem posed by Orey and Pruitt related to the range of the N-parameter wiener process in R d

Let W ^(N,d) be the N-parameter Wiener process with values in R ^d .It is shown that the range of W^(N,d) is all of R^d when d<2N, answering a question raised earlier by Orey and Pruitt.     

Invertibility of the Gabor frame operator on the Wiener amalgam space

We use a generalization of Wiener's 1/f theorem to prove that for a Gabor frame with the generator in the Wiener amalgam space W(L^∞,?¹)(R^d), the corresponding frame operator is invertible on this space. Therefore, for such a Gabor frame, the canonical dual belongs also to W(L^∞,?¹)(R^d).     

Rooted Tree Analysis for Order Conditions of Stochastic Runge-Kutta Methods for the Weak Approximation of Stochastic Differential Equations

Abstract

A general class of stochastic Runge-Kutta methods for the weak approximation of Itô and Stratonovich stochastic differential equations with a multi-dimensional Wiener process is introduced. Colored rooted trees are used to derive an expansion of the solution process and of the approximation process calculated with the stochastic Runge-Kutta method. A theorem on general order conditions for the coefficients and the random variables of the stochastic Runge-Kutta method is proved by rooted tree analysis. This theorem can be applied for the derivation of stochastic Runge-Kutta methods converging with an arbitrarily high order.     

Towards an example of a nonconvex monotone follower control problem

A one-dimensional monotone follower control problem with a nonconvex Lagrangian is considered. The control problem consists in tracking a standard Wiener process by an adapted nondecreasing process starting at 0. The verification theorem for the problem is presented. The optimal control and the value function are explicitly defined. For some values of parameters of the problem, it is shown that the value function belongs to C ². An interesting feature of the optimally controlled state process is that for some initial states it has jumps at times other than the inital time. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

Information Complexity of Multivariate Fredholm Integral Equations in Sobolev Classes

In this paper, the complexity of full solution of Fredholm integral equations of the second kind with data from the Sobolev classW^r₂is studied. The exact order of information complexity is derived. The lower bound is proved using a Gelfand number technique. The upper bound is shown by providing a concrete algorithm of optimal order, based on a specific hyperbolic cross approximation of the kernel function. Numerical experiments are included, comparing the optimal algorithm with the standard Galerkin method.     

Quadratic variation of functionals of two-parameter Wiener process

The quadratic variation of functionals of the two-parameter Wiener process of the form f(W(s, t)) is investigated, where W(s, t) is the standard two-parameter Wiener process and f is a function on the reals. The existence of the quadratic variation is obtained under the condition that f′ is locally absolutely continuous and f^N is locally square integrable.     

Generalized stochastic integrals and the malliavin calculus

Summary The paper first reviews the Skorohod generalized stochastic integral with respect to the Wiener process over some general parameter space T and it's relation to the Malliavin calculus as the adjoint of the Malliavin derivative. Some new results are derived and it is shown that every sufficiently smooth process {u_t, tT} can be decomposed into the sum of a Malliavin derivative of a Wiener functional, and a process whose generalized integral over T vanishes. Using the results on the generalized integral, the Bismut approach to the Malliavin calculus is generalized by allowing non adapted variations of the Wiener process yielding sufficient conditions for the existence of a density which is considerably weaker than the previously known conditions.Let e _i be a non-random complete orthonormal system on T, the Ogawa integral u W is defined as _i (e _i u) e _i dW where the integrals are Wiener integrals. Conditions are given for the existence of an intrinsic Ogawa integral i.e. independent of the choice of the orthonormal system and results on it's relation to the Skorohod integral are derived.The transformation of measures induced by (W + u d u non adapted is discussed and a Girsanov-type theorem under certain regularity conditions is derived.The work of M.Z. was supported by the Fund for Promotion of Research at the Technion     

Nonanticipative transformations of the two-parameter Wiener process and a Girsanov theorem

Nonanticipative linear transformations of the two-parameter Wiener process W are studied. It is shown that they induce measures equivalent to two-parameter Wiener measure and the corresponding Radon-Nikodym derivatives are calculated. A two-parameter extension of Girsanov's theorem is established for a class of nonanticipative, possibly nonlinear transformations of W.     

Wiener Integrals with Respect to the Hermite Process and a Non-Central Limit Theorem

Abstract

We introduce Wiener integrals with respect to the Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an example, we study a generalization of the fractional Ornstein–Uhlenbeck process.     

Nonlinear Filtering for Diffusions in Random Environments

Suppose that the signal X to be estimated is a diffusion process in a random medium W and the signal is correlated with the observation noise. We study the historical filtering problem concerned with estimating the signal path up until the current time based upon the back observations. Using Dirichlet form theory, we introduce a filtering model for general rough signal X ^W and establish a multiple Wiener integrals representation for the unnormalized pathspace filtering process. Then, we construct a precise nonlinear filtering model for the process X itself and give the corresponding Wiener chaos decomposition.     

Average error bounds of trigonometric approximation on periodic Wiener spaces

In this paper, we study the approximation of identity operator and the convolution integral operator Bm by Fourier partial sum operators, Fejr operators, Valle-Poussin operators, Cesárooperators and Abel mean operators, respectively, on the periodic Wiener space (C1(R),Wo) and obtainthe average error estimations.     

The local time of iterated Brownian motion

We define and study the local time process {L ^*(x,t);x¹,t0} of the iterated Brownian motion (IBM) {H(t):=W ₁(|W ₂ (t)|); t0}, whereW ₁(·) andW ₂(·) are independent Wiener processes.Research supported by Hungarian National Foundation for Scientific Research, Grant No. T 016384.Research supported by an NSERC Canada Grant at Carleton University, Ottawa.Research supported by a PSC CUNY Grant, No. 6-66364.     

Differentiation formulas for stochastic integrals in the plane

For a one-parameter process of the form X_t=X₀+∫^t₀φ_sdW_s+∫^t₀ψ_sds, where W is a Wiener process and ∫φdW is a stochastic integral, a twice continuously differentiable function f(X_t) is again expressible as the sum of a stochastic integral and an ordinary integral via the Ito differentiation formula. In this paper we present a generalization for the stochastic integrals associated with a two-parameter Wiener process.Let {W₂, z∈R²₊} be a Wiener process with a two-dimensional parameter. Ertwhile, we have defined stochastic integrals ∫ φdWand ∫ψdWdW, as well as mixed integrals ∫h dz dW and ∫gdW dz. Now let X_z be a two-parameter process defined by the sum of these four integrals and an ordinary Lebesgue integral. The objective of this paper is to represent a suitably differentiable function f(X_z) as such a sum once again. In the process we will also derive the (basically one-dimensional) differentiation formulas of f(X_z) on increasing paths in $\text{R}{}^2{}_{\mathrm{+}}$.