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We consider a multidimensional diffusion X with drift coefficient b(α,Xt) and diffusion coefficient ?σ(β,Xt). The diffusion sample path is discretely observed at times tk=kΔ for k=1…n on a fixed interval [0,T]. We study minimum contrast estimators derived from the Gaussian process approximating X for small ?. We obtain consistent and asymptotically normal estimators of α for fixed Δ and ?→0 and of (α,β) for Δ→0 and ?→0 without any condition linking ? and Δ. We compare the estimators obtained with various methods and for various magnitudes of Δ and ? based on simulation studies. Finally, we investigate the interest of using such methods in an epidemiological framework. 相似文献
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Let K be a closed convex subset of a q-uniformly smooth separable Banach space, T:K→K a strictly pseudocontractive mapping, and f:K→K an L-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1), let xt be the unique fixed point of tf+(1-t)T. We prove that if T has a fixed point, then {xt} converges to a fixed point of T as t approaches to 0. 相似文献
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Paul-Emile Maing 《Nonlinear Analysis: Theory, Methods & Applications》2008,68(12):3913-3922
This paper is concerned with the Cauchy problem for the fast diffusion equation ut−Δum=αup1 in RN (N≥1), where m∈(0,1), p1>1 and α>0. The initial condition u0 is assumed to be continuous, nonnegative and bounded. Using a technique of subsolutions, we set up sufficient conditions on the initial value u0 so that u(t,x) blows up in finite time, and we show how to get estimates on the profile of u(t,x) for small enough values of t>0. 相似文献
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We consider the Mosco convergence of the sets of fixed points for one-parameter strongly continuous semigroups of nonexpansive mappings. One of our main results is the following: Let C be a closed convex subset of a Hilbert space E. Let {T(t):t≥0} be a strongly continuous semigroup of nonexpansive mappings on C. The set of all fixed points of T(t) is denoted by F(T(t)) for each t≥0. Let τ be a nonnegative real number and let {tn} be a sequence in R satisfying τ+tn≥0 and tn≠0 for n∈N, and limntn=0. Then {F(T(τ+tn))} converges to ?t≥0F(T(t)) in the sense of Mosco. 相似文献
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Let ηt be a Poisson point process of intensity t≥1 on some state space Y and let f be a non-negative symmetric function on Yk for some k≥1. Applying f to all k-tuples of distinct points of ηt generates a point process ξt on the positive real half-axis. The scaling limit of ξt as t tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m-th smallest point of ξt is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener–Itô chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen–Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k-flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry. 相似文献
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Given a point A in the real Grassmannian, it is well-known that one can construct a soliton solution uA(x,y,t) to the KP equation. The contour plot of such a solution provides a tropical approximation to the solution when the variables x, y, and t are considered on a large scale and the time t is fixed. In this paper we use several decompositions of the Grassmannian in order to gain an understanding of the contour plots of the corresponding soliton solutions. First we use the positroid stratification of the real Grassmannian in order to characterize the unbounded line-solitons in the contour plots at y?0 and y?0. Next we use the Deodhar decomposition of the Grassmannian–a refinement of the positroid stratification–to study contour plots at t?0. More specifically, we index the components of the Deodhar decomposition of the Grassmannian by certain tableaux which we call Go-diagrams , and then use these Go-diagrams to characterize the contour plots of solitons solutions when t?0. Finally we use these results to show that a soliton solution uA(x,y,t) is regular for all times t if and only if A comes from the totally non-negative part of the Grassmannian. 相似文献
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We consider a multidimensional diffusion X with drift coefficient b(Xt,α) and diffusion coefficient εa(Xt,β) where α and β are two unknown parameters, while ε is known. For a high frequency sample of observations of the diffusion at the time points k/n, k=1,…,n, we propose a class of contrast functions and thus obtain estimators of (α,β). The estimators are shown to be consistent and asymptotically normal when n→∞ and ε→0 in such a way that ε−1n−ρ remains bounded for some ρ>0. The main focus is on the construction of explicit contrast functions, but it is noted that the theory covers quadratic martingale estimating functions as a special case. In a simulation study we consider the finite sample behaviour and the applicability to a financial model of an estimator obtained from a simple explicit contrast function. 相似文献
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Let (Ut,Vt) be a bivariate Lévy process, where Vt is a subordinator and Ut is a Lévy process formed by randomly weighting each jump of Vt by an independent random variable Xt having cdf F. We investigate the asymptotic distribution of the self-normalized Lévy process Ut/Vt at 0 and at ∞. We show that all subsequential limits of this ratio at 0 (∞) are continuous for any nondegenerate F with finite expectation if and only if Vt belongs to the centered Feller class at 0 (∞). We also characterize when Ut/Vt has a non-degenerate limit distribution at 0 and ∞. 相似文献
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In this article, we consider a jump diffusion process (Xt)t≥0 observed at discrete times t=0,Δ,…,nΔ. The sampling interval Δ tends to 0 and nΔ tends to infinity. We assume that (Xt)t≥0 is ergodic, strictly stationary and exponentially β-mixing. We use a penalised least-square approach to compute two adaptive estimators of the drift function b. We provide bounds for the risks of the two estimators. 相似文献
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Consider a graph G with a minimal edge cut F and let G1, G2 be the two (augmented) components of G−F. A long-open question asks under which conditions the crossing number of G is (greater than or) equal to the sum of the crossing numbers of G1 and G2—which would allow us to consider those graphs separately. It is known that crossing number is additive for |F|∈{0,1,2} and that there exist graphs violating this property with |F|≥4. In this paper, we show that crossing number is additive for |F|=3, thus closing the final gap in the question. 相似文献
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This paper considers the short- and long-memory linear processes with GARCH (1,1) noises. The functional limit distributions of the partial sum and the sample autocovariances are derived when the tail index α is in (0,2), equal to 2, and in (2,∞), respectively. The partial sum weakly converges to a functional of α-stable process when α<2 and converges to a functional of Brownian motion when α≥2. When the process is of short-memory and α<4, the autocovariances converge to functionals of α/2-stable processes; and if α≥4, they converge to functionals of Brownian motions. In contrast, when the process is of long-memory, depending on α and β (the parameter that characterizes the long-memory), the autocovariances converge to either (i) functionals of α/2-stable processes; (ii) Rosenblatt processes (indexed by β, 1/2<β<3/4); or (iii) functionals of Brownian motions. The rates of convergence in these limits depend on both the tail index α and whether or not the linear process is short- or long-memory. Our weak convergence is established on the space of càdlàg functions on [0,1] with either (i) the J1 or the M1 topology (Skorokhod, 1956); or (ii) the weaker form S topology (Jakubowski, 1997). Some statistical applications are also discussed. 相似文献
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We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to d real numbers α1,…,αd. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with parameter t as t↓0. Suggestions in this direction have been made several times over in the literature, e.g. Chevallier (2013) [4] or Bosma and Smeets (2013) [2]. The new idea in this paper is that checking the LLL-conditions consists of solving linear equations in t. 相似文献
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We consider N independent stochastic processes (Xj(t),t∈[0,T]), j=1,…,N, defined by a one-dimensional stochastic differential equation with coefficients depending on a random variable ?j and study the nonparametric estimation of the density of the random effect ?j in two kinds of mixed models. A multiplicative random effect and an additive random effect are successively considered. In each case, we build kernel and deconvolution estimators and study their L2-risk. Asymptotic properties are evaluated as N tends to infinity for fixed T or for T=T(N) tending to infinity with N. For T(N)=N2, adaptive estimators are built. Estimators are implemented on simulated data for several examples. 相似文献
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Let M=(Mt)t≥0 be any continuous real-valued stochastic process. We prove that if there exists a sequence (an)n≥1 of real numbers which converges to 0 and such that M satisfies the reflection property at all levels an and 2an with n≥1, then M is an Ocone local martingale with respect to its natural filtration. We state the subsequent open question: is this result still true when the property only holds at levels an? We prove that this question is equivalent to the fact that for Brownian motion, the σ-field of the invariant events by all reflections at levels an, n≥1 is trivial. We establish similar results for skip free Z-valued processes and use them for the proof in continuous time, via a discretization in space. 相似文献
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We show that the equality m1(f(x))=m2(g(x)) for x in a neighborhood of a point a remains valid for all x provided that f and g are open holomorphic maps, f(a)=g(a)=0 and m1,m2 are Minkowski functionals of bounded balanced domains. Moreover, a polynomial relation between f and g is obtained. 相似文献