Nonparametric estimation for stochastic differential equations with random effects

We consider N$N$ independent stochastic processes (_Xj(t),t∈[0,T])$(X_j\left(t\right),t\in [0,T])$, j=1,…,N$j=1,\dots ,N$, defined by a one-dimensional stochastic differential equation with coefficients depending on a random variable _?j${?}_j$ and study the nonparametric estimation of the density of the random effect _?j${?}_j$ 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$L^2$-risk. Asymptotic properties are evaluated as N$N$ tends to infinity for fixed T$T$ or for T=T(N)$T=T\left(N\right)$ tending to infinity with N$N$. For T(N)=N²$T\left(N\right)=N^2$, adaptive estimators are built. Estimators are implemented on simulated data for several examples.     

Coalescence in the recent past in rapidly growing populations

In a rapidly growing population one expects that two individuals chosen at random from the n$n$th generation are unlikely to be closely related if n$n$ is large. In this paper it is shown that for a broad class of rapidly growing populations this is not the case. For a Galton–Watson branching process with an offspring distribution {_pj}$\left\{p_j\right\}$ such that _p0=0$p_0=0$ and ψ(x)=_{∑jpjI{j≥x}}$\psi \left(x\right)={\sum }_j{p}_j{I}_{\{j\ge x\}}$ is asymptotic to x^−αL(x)$x^{-\alpha }L\left(x\right)$ as x→∞$x\to \infty$ where L(⋅)$L(\cdot )$ is slowly varying at ∞$\infty$ and 0<α<1$0<\alpha <1$ (and hence the mean m=∑j_pj=∞$m=\sum j{p}_j=\infty$) it is shown that if _Xn$X_n$ is the generation number of the coalescence of the lines of descent backwards in time of two randomly chosen individuals from the n$n$th generation then n−_Xn$n-X_n$ converges in distribution to a proper distribution supported by N={1,2,3,…}$\mathbb{N}=\{1,2,3,\dots \}$. That is, in such a rapidly growing population coalescence occurs in the recent past rather than the remote past. We do show that if the offspring mean m$m$ satisfies 1<m≡∑j_pj<∞$1<m\equiv \sum j{p}_j<\infty$ and _p0=0$p_0=0$ then coalescence time _Xn$X_n$ does converge to a proper distribution as n→∞$n\to \infty$, i.e., coalescence does take place in the remote past.     

Parametric inference for discretely observed multidimensional diffusions with small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(α,_Xt)$b(\alpha ,X_t)$ and diffusion coefficient ?σ(β,_Xt)$?\sigma (\beta ,X_t)$. The diffusion sample path is discretely observed at times _tk=kΔ$t_k=k\Delta$ for k=1…n$k=1\dots n$ on a fixed interval [0,T]$[0,T]$. We study minimum contrast estimators derived from the Gaussian process approximating X$X$ for small ?$?$. We obtain consistent and asymptotically normal estimators of α$\alpha$ for fixed Δ$\Delta$ and ?→0$?\to 0$ and of (α,β)$(\alpha ,\beta )$ for Δ→0$\Delta \to 0$ and ?→0$?\to 0$ without any condition linking ?$?$ and Δ$\Delta$. We compare the estimators obtained with various methods and for various magnitudes of Δ$\Delta$ and ?$?$ based on simulation studies. Finally, we investigate the interest of using such methods in an epidemiological framework.     

Crossing number additivity over edge cuts

Consider a graph G$G$ with a minimal edge cut F$F$ and let _G1$G_1$, _G2$G_2$ be the two (augmented) components of G−F$G-F$. A long-open question asks under which conditions the crossing number of G$G$ is (greater than or) equal to the sum of the crossing numbers of _G1$G_1$ and _G2$G_2$—which would allow us to consider those graphs separately. It is known that crossing number is additive for |F|∈{0,1,2}$\left|F\right|\in \{0,1,2\}$ and that there exist graphs violating this property with |F|≥4$\left|F\right|\ge 4$. In this paper, we show that crossing number is additive for |F|=3$\left|F\right|=3$, thus closing the final gap in the question.     

Blow-up and propagation of disturbances for fast diffusion equations

This paper is concerned with the Cauchy problem for the fast diffusion equation _ut−Δu^m=αu^_p1$u_t-\Delta u^m=\alpha u^{p_1}$ in R^N${\mathbb{R}}^N$ (N≥1$N\ge 1$), where m∈(0,1)$m\in (0,1)$, _p1>1$p_1>1$ and α>0$\alpha >0$. The initial condition _u0$u_0$ is assumed to be continuous, nonnegative and bounded. Using a technique of subsolutions, we set up sufficient conditions on the initial value _u0$u_0$ so that u(t,x)$u(t,x)$ blows up in finite time, and we show how to get estimates on the profile of u(t,x)$u(t,x)$ for small enough values of t>0$t>0$.     

Minimum degree and disjoint cycles in generalized claw-free graphs

For s≥3$s\ge 3$ a graph is _K1,s$K_{1,s}$-free if it does not contain an induced subgraph isomorphic to _K1,s$K_{1,s}$. Cycles in _K1,3$K_{1,3}$-free graphs, called claw-free graphs, have been well studied. In this paper we extend results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions to _K1,s$K_{1,s}$-free graphs, normally called generalized claw-free graphs. In particular, we prove that if G$G$ is _K1,s$K_{1,s}$-free of sufficiently large order n=3k$n=3k$ with δ(G)≥n/2+c$\delta \left(G\right)\ge n/2+c$ for some constant c=c(s)$c=c\left(s\right)$, then G$G$ contains k$k$ disjoint triangles. Analogous results with the complete graph _K3$K_3$ replaced by a complete graph _Km$K_m$ for m≥3$m\ge 3$ will be proved. Also, the existence of 2$2$-factors for _K1,s$K_{1,s}$-free graphs with minimum degree conditions will be shown.     

On first-fit coloring of ladder-free posets

Bosek and Krawczyk exhibited an on-line algorithm for partitioning an on-line poset of width w$w$ into w^14lgw$w^{14lgw}$ chains. They also observed that the problem of on-line chain partitioning of general posets of width w$w$ could be reduced to First-Fit chain partitioning of 2w²+1$2w^2+1$-ladder-free posets of width w$w$, where an m$m$-ladder is the transitive closure of the union of two incomparable chains _x1≤?≤_xm$x_1\le ?\le x_m$, _y1≤?≤_ym$y_1\le ?\le y_m$ and the set of comparabilities {_x1≤_y1,…,_xm≤_ym}$\{x_1\le y_1,\dots ,x_m\le y_m\}$. Here, we provide a subexponential upper bound (in terms of w$w$ with m$m$ fixed) for the performance of First-Fit chain partitioning on m$m$-ladder-free posets, as well as an exact quadratic bound when m=2$m=2$, and an upper bound linear in m$m$ when w=2$w=2$. Using the Bosek–Krawczyk observation, this yields an on-line chain partitioning algorithm with a somewhat improved performance bound. More importantly, the algorithm and the proof of its performance bound are much simpler.     

An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.     

Holomorphic mappings preserving Minkowski functionals

We show that the equality _m1(f(x))=_m2(g(x))$m_1\left(f\left(x\right)\right)=m_2\left(g\left(x\right)\right)$ for x$x$ in a neighborhood of a point a$a$ remains valid for all x$x$ provided that f$f$ and g$g$ are open holomorphic maps, f(a)=g(a)=0$f\left(a\right)=g\left(a\right)=0$ and _m1,_m2$m_1,m_2$ are Minkowski functionals of bounded balanced domains. Moreover, a polynomial relation between f$f$ and g$g$ is obtained.     

The relaxed Newton method derivative: Its dynamics and non-linear properties

The dynamic behaviour of the one-dimensional family of maps f(x)=_c2[(a−1)x+_c1]^{−λ/(α−1)}$f\left(x\right)=c_2{[(a-1)x+c_1]}^{-\lambda /(\alpha -1)}$ is examined, for representative values of the control parameters a,_c1$a,c_1$, _c2$c_2$ and λ$\lambda$. The maps under consideration are of special interest, since they are solutions of the relaxed Newton method derivative being equal to a constant a$a$. The maps f(x)$f\left(x\right)$ are also proved to be solutions of a non-linear differential equation with outstanding applications in the field of power electronics. The recurrent form of these maps, after excessive iterations, shows, in an _xn$x_n$ versus λ$\lambda$ plot, an initial exponential decay followed by a bifurcation. The value of λ$\lambda$ at which this bifurcation takes place depends on the values of the parameters a,_c1$a,c_1$ and _c2$c_2$. This corresponds to a switch to an oscillatory behaviour with amplitudes of f(x)$f\left(x\right)$ undergoing a period doubling. For values of a$a$ higher than 1 and at higher values of λ$\lambda$ a reverse bifurcation occurs. The corresponding branches converge and a bleb is formed for values of the parameter _c1$c_1$ between 1 and 1.20. This behaviour is confirmed by calculating the corresponding Lyapunov exponents.     

Estimation for stochastic differential equations with a small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(_Xt,α)$b(X_t,\alpha )$ and diffusion coefficient εa(_Xt,β)$\epsilon a(X_t,\beta )$ where α$\alpha$ and β$\beta$ are two unknown parameters, while ε$\epsilon$ is known. For a high frequency sample of observations of the diffusion at the time points k/n$k/n$, k=1,…,n$k=1,\dots ,n$, we propose a class of contrast functions and thus obtain estimators of (α,β)$(\alpha ,\beta )$. The estimators are shown to be consistent and asymptotically normal when n→∞$n\to \infty$ and ε→0$\epsilon \to 0$ in such a way that ε⁻¹n^−ρ${\epsilon }^{-1}{n}^{-\rho }$ remains bounded for some ρ>0$\rho >0$. 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.     

A unified proof of Brooks’ theorem and Catlin’s theorem

Brooks’ theorem is a fundamental result in the theory of graph coloring. Catlin proved the following strengthening of Brooks’ theorem: Let d$d$ be an integer at least 3, and let G$G$ be a graph with maximum degree d$d$. If G$G$ does not contain _Kd+1$K_{d+1}$ as a subgraph, then G$G$ has a d$d$-coloring in which one color class has size α(G)$\alpha \left(G\right)$. Here α(G)$\alpha \left(G\right)$ denotes the independence number of G$G$. We give a unified proof of Brooks’ theorem and Catlin’s theorem.