Picard dimension of signed radial Kato measures

The Picard dimension dimμ of a signed local Kato measure μ on the punctured unit ball in R^d, d ≥ 2, is the cardinal number of the set of extremal rays of the convex cone of all continuous solutions u ≥ 0 of the time-independent SchrSdinger equation Δu -- uμ = 0 on the punctured ball 0 〈 ｜｜x｜｜ 〈 1, with vanishing boundary values on the sphere ｜｜x｜｜ = 1. Using potential theory associated with the Schrodinger operator we prove, in this paper, that the dimμ for a signed radial Kato measure is 0, 1 or ＋∞. In particular, we obtain the Picard dimension of locally Holder continuous functions P proved by Nakai and Tada by other methods.     

<Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript> weights and <Emphasis Type="Italic">d</Emphasis>-homogeneous measures on ahlfors <Emphasis Type="Italic">d</Emphasis>-regular space

Let X be an Ahlfors d-regular space and mad-regular measure on X . We prove that a measure μ on X is d-homogeneous if and only if μ is mutually absolutely continuous with respect to m and the derivative Dmμ(x) is an A1 weight. Also, we show by an example that every Ahlfors d-regular space carries a measure which is d-homogeneous but not d-regular.     

Differentiability of dividends function on jump-diffusion risk process with a barrier dividend strategy

We consider a dividends model with a stochastic jump perturbed by diffusion. First, we prove that the expected discounted dividends function is twice continuously differentiable under the condition that the claim distribution function has continuous density. Then we show that the expected discounted dividends function under a barrier strategy satisfies some integro-differential equation of defective renewal type, and the solution of which can be explicitly expressed as a convolution formula. Finally, we study the Laplace transform of ruin time on the modified surplus process.     

Exponential change of measure for general piecewise deterministic Markov processes

We consider a general piecewise deterministic Markov process(PDMP) X = {X_t}_(t≥0) with a measure-valued generator A, for which the conditional distribution function of the inter-occurrence time is not necessarily absolutely continuous. A general form of the exponential martingales that are associated with X is given by■By considering this exponential martingale to be a likelihood-ratio process, we define a new probability measure and show that the process X is still a general PDMP under the new probability measure. We additionally find the new measure-valued generator and its domain. To illustrate our results, we investigate the continuous-time compound binomial model.     

ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES

In this paper, we focus on anticipated backward stochastic Volterra integral equations(ABSVIEs) with jumps. We solve the problem of the well-posedness of so-called M-solutions to this class of equation, and analytically derive a comparison theorem for them and for the continuous equilibrium consumption process. These continuous equilibrium consumption processes can be described by the solutions to this class of ABSVIE with jumps.Motivated by this, a class of dynamic risk measures induced by ABSV...     

Application of Moore-Penrose inverse in deciding the minimal martingale measure

The Moore-Penrose inverse is an important tool in algebra.This paper shows that the MoorePenrose inverse is also an effcient technique in determining the minimal martingale measure if a security price follows a semi-martingale which satisfies some structure condition.We extend a result of Dzhaparidze and Spreij concerning the Moore-Penrose inverse to the case that the Moore-Penrose inverse of any matrix-valued predictable process is still predictable.Furthermore,we obtain an explicit formula of the minimal martingale measure by employing the Moore-Penrose inverse.Specifically,the minimal martingale measure in a generalized Black-Scholes model is found.     

Dependence of solutions and eigenvalues of third order linear measure differential equations on measures

This paper deals with a complex third order linear measure differential equation id(y')~·+ 2 iq(x)y'dx + y(idq(x) + dp(x)) = λydx on a bounded interval with boundary conditions presenting a mixed aspect of the Dirichlet and the periodic problems. The dependence of eigenvalues on the coefficients p and q is investigated. We prove that the n-th eigenvalue is continuous in p and q when the norm topology of total variation and the weak*topology are considered. Moreover, the Fr′echet differentiability of the n-th eigenvalue in p and q with the norm topology of total variation is also considered. To deduce these conclusions, we investigate the dependence of solutions of the above equation on the coefficients p and q with different topologies and establish the counting lemma of eigenvalues according to the estimates of solutions.     

ON DONSKER AND VARADHAN''''S ASYMPTOTIC EVALUATION WITHOUT COMPACTNESS

Donsker and Varadhan have developed in a series of papers [1—3] a newtechnique to estimate the rate of exponential decay for certain probabilities associatedwith occupation times for Markov processes.We wish to investigate to what extentthe compactness condition is actually necessary.As a first step,we consider here thefollowing question.Let X be a metric space and let dx be a measure on X.Let C(X)be the space ofbounded continuous functions on X and p_t be a strongly continuous semigroup such     

The defocusing energy-supercritical Hartree equation

In this paper,we study the global well-posedness and scattering problem for the energysupercritical Hartree equation iut+Δu.(|x|.γ.|u|2)u=0 with γ4 in dimension d γ.We prove that if the solution u is apriorily bounded in the critical Sobolev space,that is,u ∈Lt∞(I;Hxsc(Rd)) with sc:= γ/2.11,then u is global and scatters.The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation(NLW) and nonlinear Schrdinger equation(NLS).We utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios:finite time blowup;soliton-like solution and low to high frequency cascade.Making use of the No-waste Duhamel formula,we deduce that the energy of the finite time blow-up solution is zero and so get a contradiction.Finally,we adopt the double Duhamel trick,the interaction Morawetz estimate and interpolation to kill the last two scenarios.     

FINITE DIFFERENCE APPROXIMATION FOR PRICING THE AMERICAN LOOKBACK OPTION

In this paper we are concerned with the pricing of lookback options with American type constrains. Based on the differential linear complementary formula associated with the pricing problem, an implicit difference scheme is constructed and analyzed. We show that there exists a unique difference solution which is unconditionally stable. Using the notion of viscosity solutions, we also prove that the finite difference solution converges uniformly to the viscosity solution of the continuous problem. Furthermore, by means of the variational inequality analysis method, the O（△t ＋ △x^2）-order error estimate is derived in the discrete L2-norm provided that the continuous problem is sufficiently regular. In addition, a numerical example is provided to illustrate the theoretical results.     

On the maximal operator associated with certain rotational invariant measures

The aim of this work is to investigate the integrability properties of the maximal operator Mu,associated with a non-doubling measure μ defined on Rn. We start by establishing for a wide class of radial and increasing measures μ that Mu is bounded on all the spaces Lu^p（R^n）,P〉1.Also,we show that there is a radial and increasing measure p for which Mμ does not map Lμ^p（R^n） into weak Lμ^p（R^n）,1≤p〈∞.     

AN ITERATIVE PROCESS FOR FINDING APPROXIMATE SOLUTIONS TO NONLINEAR EQUATIONS OF STRONGLY ACCRETIVE OPERATORS

In this paper, we investigate the Ishikawa iteration process in a p-uniformly smooth Banach space X. We prove that the Ishikawa iteration process converges strongly to the unique solution of the equation Tx=f when T is a Lipschitzian and strongly accretive operator frow X to X, or to the unique fixed point of T when T is a Lipschitzian and strictly pseudocontractive mapping from a nonempty closed convex subset K of X into itself. Our results are the extension and improvements of the earlier and recent results in this field.     

NATURAL BOUNDARY ELEMENT METHOD FOR THREE DIMENSIONAL EXTERIOR HARMONIC PROBLEM WITH AN INNER PROLATE SPHEROID BOUNDARY

In this paper, we study natural boundary reduction for Laplace equation with Dirichletor Neumann boundary condition in a three-dimensional unbounded domain, which is theoutside domain of a prolate spheroid. We express the Poisson integral formula and naturalintegral operator in a series form explicitly. Thus the original problem is reduced to aboundary integral equation on a prolate spheroid. The variational formula for the reducedproblem and its well-posedness are discussed. Boundary element approximation for thevariational problem and its error estimates, which have relation to the mesh size andthe terms after the series is truncated, are also presented. Two numerical examples arepresented to demonstrate the effectiveness and error estimates of this method.     

On initial data of the monopole equation

The space-time monopole equation is the reduction of anti-self-dual Yang-Mills equations in R2,2 to R2,1. This equation is a non-linear wave equation, and can be encoded in a Lax pair. An equivalent Lax pair is used by Dai and Terng to construct monopoles with continuous scattering data, and then the equation can be linearized by the scattering data, allowing one to use the inverse scattering method to solve the Cauchy problem with rapidly decaying small initial data. In this paper, we use the terminology of holomorphic bundle and transversality of certain maps, parametrized by initial data, to give more initial data, with which we can use scattering method to solve the Cauchy problem of the monopole equation up to gauge transformation.     

New Calderón reproducing formulae with exponential decay on spaces of homogeneous type

Assume that(X,d,μ) is a space of homogeneous type in the sense of Coifman and Weiss(1971,1977). In this article, motivated by the breakthrough work of Auscher and Hyt(o|¨)nen(2013) on orthonormal bases of regular wavelets on spaces of homogeneous type, we introduce a new kind of approximations of the identity with exponential decay(for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Han et al.(2018) to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, we establish the homogeneous continuous/discrete Calderón reproducing formulae on(X, d,μ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure μ a doubling measure,not necessary to satisfy the reverse doubling condition. It is well known that Calderón reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.     

Existence of Semiclassical States for a Quasilinear Schr?dinger Equation on R~N with Exponential Critical Growth

We study a quasilinear Schr?dinger equation{-ε~NΔNu+V(x)|u|~(N-2)u=Q(x)f(u) in R~N,0u∈W~(1,N)(R~N),u(x)|x|→∞→0,where V,Q are two continuous real functions on R~N and ε0 is a real parameter.Assume that the nonlinearity f is of exponential critical growth in the sense of Trudinger–Moser inequality,we are able to establish the existence and concentration of the semiclassical solutions by variational methods.     

Gradient estimates and coupling property for semilinear SDEs driven by jump processes

Let L be a L′evy process with characteristic measureν,which has an absolutely continuous lower bound w.r.t.the Lebesgue measure on Rn.By using Malliavin calculus for jump processes,we investigate Bismut formula,gradient estimates and coupling property for the semigroups associated to semilinear SDEs forced by L′evy process L.     

Fast-slow stochastic dynamical system with singular coefficients

This paper aims to study the asymptotic behavior of a fast-slow stochastic dynamical system with singular coefficients, where the fast motion is given by a continuous diffusion process while the slow component is driven by an α-stable noise with α ∈ [1, 2). Using Zvonkin’s transformation and the technique of the Poisson equation, we have that both the strong and weak convergences in the averaging principle are established, which can be viewed as a functional law of large numbers. Then we study t...     

Testing long memory based on a discretely observed process

In this paper we consider the problem of testing long memory for a continuous time process based on high frequency data. We provide two test statistics to distinguish between a semimartingale and a fractional integral process with jumps, where the integral is driven by a fractional Brownian motion with long memory. The small–sample performances of the statistics are evidenced by means of simulation studies. The real data analysis shows that the fractional integral process with jumps can capture the long memory of some financial data.     

Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations

In this paper, we establish a bang-bang principle of time optimal controls for a controlled parabolic equation of fractional order evolved in a bounded domain Ω of R^n, with a controller w to be any given nonempty open subset of Ω. The problem is reduced to a new controllability property for this equation, i.e. the null controllability of the system at any given time T 〉 0 when the control is restricted to be active in ω× E, where E is any given subset of [0, T] with positive （Legesgue） measure. The desired controllability result is established by means of a sharp observability estimate on the eigenfunctions of the Dirichlet Laplacian due to Lebeau and Robbiano, and a delicate result in the measure theory due to Lions.