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     

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     

Statistically Efficient Thinning of a Markov Chain Sampler

It is common to subsample Markov chain output to reduce the storage burden. Geyer shows that discarding k ? 1 out of every k observations will not improve statistical efficiency, as quantified through variance in a given computational budget. That observation is often taken to mean that thinning Markov chain Monte Carlo (MCMC) output cannot improve statistical efficiency. Here, we suppose that it costs one unit of time to advance a Markov chain and then θ > 0 units of time to compute a sampled quantity of interest. For a thinned process, that cost θ is incurred less often, so it can be advanced through more stages. Here, we provide examples to show that thinning will improve statistical efficiency if θ is large and the sample autocorrelations decay slowly enough. If the lag ? ? 1 autocorrelations of a scalar measurement satisfy ρ_? > ρ_{? + 1} > 0, then there is always a θ < ∞ at which thinning becomes more efficient for averages of that scalar. Many sample autocorrelation functions resemble first order AR(1) processes with ρ_? = ρ^|?| for some ? 1 < ρ < 1. For an AR(1) process, it is possible to compute the most efficient subsampling frequency k. The optimal k grows rapidly as ρ increases toward 1. The resulting efficiency gain depends primarily on θ, not ρ. Taking k = 1 (no thinning) is optimal when ρ ? 0. For ρ > 0, it is optimal if and only if θ ? (1 ? ρ)²/(2ρ). This efficiency gain never exceeds 1 + θ. This article also gives efficiency bounds for autocorrelations bounded between those of two AR(1) processes. Supplementary materials for this article are available online.     

Multiparameter acoustic imaging in the born approximation

Samples of biological tissue are modelled as inhomogeneous fluids with density ?(X) and sound speed c(x) at point x. The samples are contained in the sphere |x| ? δ and it is assumed that ?(x) ? ?₀ = 1 and c(x) ? c₀ = 1 for |x| ? δ, and |γ_n(x)| ? 1 and |?γ?(x)| ? 1 where γ?(x) = ?(x) ? 1 and γ_n(x) = c^?2(x) ? 1. The samples are insonified by plane pulses s(x · θ₀ – t) where x = |θ₀| = 1 and the scattered pulse is shown to have the form |x|^?1 e_s(|x| – t, θ, θ₀) in the far field, where x = |x| θ. The response e_s(τ, θ, θ₀) is measurable. The goal of the work is to construct the sample parameters γn and γ? from e_s(τ, θ, θ₀) for suitable choiches of s, θ and θ₀. In the limiting case of constant density: γ?(x)_? 0 it is shown that Where δ represents the Dirac δ and S² is the unit sphere |θ| = 1. Analogous formulas, based on two sets of measurements, are derived for the case of variable c(x) and ?(x).     

Bayesian estimation of the product of two proportions

Suppose that for i = 1,2, a Bernoulli random variable with success probability θ_i is observable from population i. The problem is to estimate θ = θ₁θ₂ using a Bayesian approach with squared error estimation loss in θ. For estimating θ, the best nonrandom sampling scheme, the two-stage sampling scheme, and the optimal sampling scheme are discussed. It is shown that the two-stage sampling scheme is typically asymptotically optimal, and can improve the Bayes risk (over the best nonrandom allocation) up to fifty percent     

Asymptotic normality of H-L estimators based on dependent data

LetX ₁,X ₂, ... be a strictly stationary φ-mixing sequence of r.v.'s with a common continuous cdfF. Let θ be a location parameter ofF. We prove the asymptotic normality of a class of Hodges-Lehmann estimators of θ under various regularity conditions on the mixing number φ and the underlyingF. We also establish the asymptotic linearity of signed rank statistics in the parameter θ. Our results also enable us to study the effect of φ-dependence on the asymptotic power of signed rank tests for testingH ₀: θ=0 againstH _n :θ=θ ₀ n ^?1/2,θ ₀≠0. Finally these results are shown to remain valid for strongly mixing processes {X _i } also.     

Admissibility of an estimate obtained by the method of moments in the presence of a nuisance shift parameter

Let x₁,..., x_n be a repeated sample from a one-dimensional population with distribution function (d.f.) F(x?η, θ), depending on a structure parameter θ∈^Θ?R ¹ and a nuisance shift parameter η^∈ R¹. The estimator which eliminates ν In a natural manner, has the form \(\sum\limits_1^n {\psi (x_i - \overline x ,\theta ) = 0,\overline x = (x_1 + ... + x_n )/n}\) and the simplest among them, corresponding to a functionψ (u, θ), quadratic in u, leads to the estimate θ (m₂), where \(m_2 = \sum\limits_1^n {(x_i - \overline x )^2 /n}\) which has to be considered as an estimate of θ by the method of moments with the elimination of the nuisance parameter n. If for some integer k ≥ 1, 1°) the d.f. F(x, θ) has a finite moment of order 2k, 2°) its central moments μ₂(θ), ..., μk(θ) are three times and μk+1(9).... μ₂k(θ) are twice continuously differentiable in the domain Θ and μ₂′(θ) ≠ 0, 3° as n → ∞, the limit covariance matrix of the centralized and normalized vector √n ∥ m₂-μ₂(θ) ...,m_R-μ_R(θ)∥ of the central sample moments m_j is nonsingular, θ∈^Θ, then the estimate θ(m₂) is asymptotically admissible (and optimal) in the class of estimates defined by the estimators λ_o(θ) + λ₂(θ)m₂ + ... + λ_k(θ)m_k=0 if and only if the moments μ₅(θ),..., μk+2 (θ) are determined in terms of μ₂(θ), μ₃(θ), μ₄(θ) in the following recurrent manner; $$\begin{array}{*{20}c} {\mu _{j + 2} (\theta ) = \mu _2 (\theta )\mu _j (\theta ) + j\mu _3 (\theta )\mu _{j - 1} (\theta ) + [\mu _4 (\theta ) - \mu _2 (\theta )^2 ]\mu _j ^\prime (\theta )/\mu _2 ^\prime (\theta ),} \\ {j \leqslant k,\theta ^\Theta .} \\ \end{array}$$ The asymptotic admissibility is understood in the same generally accepted sense as in [1], where a similar result has been obtained for families of d.f. containing only a structure parameter.     

Commuting Traces of Biadditive Maps on Invertible Elements

Let R be a simple unital ring. Under a mild technical restriction on R, we will characterize biadditive mappings G: R² → R satisfying G(u, u)u = uG(u, u), and G(1, r) = G(r, 1) = r for all unit u ∈ R and r ∈ R, respectively. As an application, we describe bijective linear maps θ: R → R satisfying θ(xyx^?1y^?1) = θ(x)θ(y)θ(x)^?1θ(y)^?1 for all invertible x, y ∈ R. This solves an open problem of Herstein on multiplicative commutators. More precisely, we will show that θ is an isomorphism. Furthermore, we shall see the existence of a unital simple ring R′ without nontrivial idempotents, that admits a bijective linear map f: R′ → R′, preserving multiplicative commutators, that is not an isomorphism.     

A study of random Weyl trees

We study binary search trees constructed from Weyl sequences {nθ}, n≥1, where θ is an irrational and {·} denotes “mod 1.” We explore various properties of the structure of these trees, and relate them to the continued fraction expansion of θ. If H_n is the height of the tree with n nodes when θ is chosen at random and uniformly on [0, 1], then we show that in probability, H_n∼(12/π²)log n log log n. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 271–295, 1998     

On a forward-backward parabolic equation

Boundary value problems for the equation $$\operatorname{sgn} (x) \cdot u_y - u_{xx} + ku = f(x,y)$$ (where k is a positive constant and ? is a given function) are investigated. The domain of the solutions will be the whole upper half-plane y>0, or the half-plane y>0 cut along the positive y-axis. We are interested in square integrable solutions u, with square integrable generalized derivatives u_y and u_xx. Existence theorems are proved, with an integral equations technique. Thus a theory is developed of Wiener-Hopf integral equations of the first kind with solutions belonging to Sobolev spaces.     

Prime links in some skew-polynomial and skew-laurent rings

Let R be a Noetherian commutative ring and a α₁,…,α_n commuting automorphisms of R. Define T = R[θ₁,…,θ_n;α₁,…,α_n] to be the skew-polynomial ring with θ_ir = α_i(r)θ_i and θ_iθ_j= θ_jθ_i, for all i,j ? (1,…,n) and r ? R, and let S = Rθ₁,θ₁:^-1,…,θ_n:,θ_n;^-1;α₁:,…,α_n] be the corresponding skew-Laurent ring. In this paper we show that S and T satisfy the strong second layer condition and characterize the links between prime ideals in these rings.     

Anticipative Stochastic Differential Equations with Non-smooth Diffusion Coefficient

In this paper we prove the existence and uniqueness of the solutions to the one-dimensional linear stochastic differential equation with Skorohod integral Xt（ω）=η（w）＋∫^t 0 asXs（ω）dWs＋∫^t 0 bsXs（ω）ds, t∈[0,1] where （Ws） is the canonical Wiener process defined on the standard Wiener space （W,H,u）, a is non-smooth and adapted, but η and b may be anticipating to the filtration generated by （Ws）. The intention of the paper is to eliminate the regularity of the diffusion coefficient a in the Malliavin sense, in the existing literature. The idea is to approach the non-smooth diffusion coefficient a by smooth ones.     

Occupation densities for stochastic integral processes in the second Wiener chaos

Summary The two-point distributions of Skorohod integral processes in the second Wiener chaos are mainly described by a Hilbert-Schmidt operatorT giving the mutual interaction of infinitely many Gaussian components and by simple multiplication operators. So are the Fourier transforms of their occupation measures. This enables us to use the well known Fourier analytic criterion discovered and elaborated by Berman to derive integral conditions for the existence of their occupation densities in terms of associated Hilbert-Schmidt operators. IfT is a trace class operator, we get a necessary and sufficient criterion, if it is not, still a sufficient one. In a case in which the interaction is particularly simple, we verify the appropriate integral condition and show that the results are essentially beyond the reach of enlargement of filtrations techniques of semimartingale theory.     

Weak Solutions to Stochastic Wave Equations with Values in Riemannian Manifolds

Let M be a compact Riemannian manifold. We prove existence of a global weak solution of the stochastic wave equation D _t u _t  = D _x u _x  + (X _u  + λ₀(u)u _t  + λ₁(u)u _x )[Wdot] where X is a continuous vector field on M, λ₀ and λ₁ are continuous vector bundles homomorphisms from TM to TM, and W is a spatially homogeneous Wiener process on ? with finite spectral measure. We use recently introduced general method of constructing weak solutions of SPDEs that does not rely on any martingale representation theorem.     

On the Elementary Retarded,Ultrahyperbolic Solution of the Klein-Gordon Operator,Iterated k Times

Let t = (t₁,…,t_n) be a point of ?ⁿ. We shall write . We put, by the definition, W_α(u, m) = (m^?2u)^{(α ? n)/4}[π^{(n ? 2)/2}2^{(α + n ? 2)/2}Г(α/2)]J_{(α ? n)/2}(m²u)^1/2; here α is a complex parameter, m a real nonnegative number, and n the dimension of the space. W_α(u, m), which is an ordinary function if Re α ≥ n, is an entire distributional function of α. First we evaluate {□ + m²}W_{α + 2}(u, m) = W_α(u, m), where {□ + m²} is the ultrahyperbolic operator. Then we express W_α(u, m) as a linear combination of R_α(u) of differntial orders; R_α(u) is Marcel Riesz's ultrahyperbolic kernel. We also obtain the following results: W_?2k(u, m) = {□ + m²}^kδ, k = 0, 1,…; W₀(u, m) = δ; and {□ + m²}^kW_2k(u, m) = δ. Finally we prove that W_α(u, m = 0) = R_α(u). Several of these results, in the particular case µ = 1, were proved earlier by a completely different method.     

Cantor spectrum for the almost Mathieu equation

For a dense G_δ of pairs (λ, α) in R², we prove that the operator (Hu)(n) = u(n + 1) + u(n ?1) + λ cos(2παn + θ) u(n) has a nowhere dense spectrum. Along the way we prove several interesting results about the case $\text{α =}\text{p}\text{q}$ of which we mention: (a) If qθ is not an integral multiple of π, then all gaps are open, and (b) If q is even and θ is chosen suitably, then the middle gap is closed for all λ.     

Non‐well‐founded extensions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {V}$\end{document}

This paper describes a new and user‐friendly method for constructing models of non‐well‐founded set theory. Given a sufficiently well‐behaved system θ of non‐well‐founded set‐theoretic equations, we describe how to construct a model M_θ for $\mathsf {ZFC}^-$ in which θ has a non‐degenerate solution. We shall prove that this M_θ is the smallest model for $\mathsf {ZFC}^-$ which contains $\mathbf {V}$ and has a non‐degenerate solution of θ.     

On the time continuity of entropy solutions

We show that any entropy solution u of a convection diffusion equation ?_t u + div F(u)-Df(u) = b{\partial_t u + {\rm div} F(u)-\Delta\phi(u) =b} in Ω × (0, T) belongs to C([0,T),L¹_loc(W)){C([0,T),L^1_{\rm loc}({\Omega}))} . The proof does not use the uniqueness of the solution.     

Regularity of Skorohod integral processes based on integrands in a finite Wiener chaos

Summary Letf be a square integrable kernel on them-dimensional unit cube,U the Skorohod integral process in them ^th Wiener chaos associated with it. Isoperimetric inequalities for functions on Wiener space yield the exponential integrability of the increments ofU. To this result we apply the majorizing measure technique to show thatU possesses a continuous version and give an upper bound of its modulus of continuity.