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This IMA Volume in Mathematics and its Applications TOPICS IN STOCHASTIC ANALYSIS AND NONPARAMETRIC ESTIMATION contains papers that were presented at the IMA Participating Institution conference on "Asymptotic Analysis in Stochastic Processes, Nonparamet­ ric Estimation, and Related Problems" held on September 15-17, 2006 at Wayne State University. The conference, which was one of approximately ten selected each year for partial support by the IMA through its affiliates program, was dedicated to Professor Rafail Z. Khasminskii on the occasion th of his 75 birthday, in recognition of his profound contributions to the field of stochastic processes and nonparametric estimation theory. We are grateful to the participants and, especially, to the conference organizers, for making the event so successful. Pao-Liu Chow, Boris Mor­ dukhovich, and George Yin of the Department of Mathematics at Wayne State University did a superb job organizing this first-rate event and in editing these proceedings. We take this opportunity to thank the Nation al Science Foundation for its support of the IMA.

Inhaltsverzeichnis

Frontmatter

Asymptotic Analysis Involving Stochastic Differential Equations

Frontmatter

Some Recent Results on Averaging Principle

Averaging principle is one of the main methods in perturbation theory. It came into being more than two centuries ago in celestial mechanics, but even now there are many open questions having to do with applications of this principle. And these questions are not just about rigorous justification of procedures applied, but even it is unclear sometimes how to apply averaging procedures.
Mark Freidlin, Alexander Wentzell

Cramer’s Theorem for Nonnegative Multivariate Point Processes with Independent Increments

We consider a continuous time version of Cramer’s theorem with non-negative summands \( S_t = \tfrac{1} {t}\sum\nolimits_{i:\tau _i \leqslant t} {\xi _i ,} {\text{ }}t \to \infty \), where (τi, ξi)i≥1 is a sequence of random variables such that tS t is a random process with independent increments.
Fima Klebaner, Robert Liptser

On Bounded Solutions of the Balanced Generalized Pantograph Equation

The question about the existence and characterization of bounded solutions to linear functional-differential equations with both advanced and delayed arguments was posed in the early 1970s by T. Kato in connection with the analysis of the pantograph equation, y′(x)=ay(qx)+by(x). In the present paper, we answer this question for the balanced generalized pantograph equation of the form −a 2 y″(x + a 1 y′(x) + y(x) = ∫ 0 μ (dα), where a 1 ≥ 0, a 2 ≥ 0 a 1 2 + a 2 2 > 0, and μ is a probability measure. By setting K:=∫ 0 ln ± μ(dα), we prove that if K≦0 then the equation does not have nontrivial (i.e., nonconstant) bounded solutions, while if K > 0 then such a solution exists. The result in the critical case, K=0, settles a long-standing problem. The proof exploits the link with the theory of Markov processes, in that any solution of the balanced pantograph equation is an L-harmonic function relative to the generator L of a certain diffusion process with “multiplication” jumps. The paper also includes three “elementary” proofs for the simple prototype equation y′(x)+y(x)=1/2y(qx)+1/2y(x/q), based on perturbation, analytical, and probabilistic techniques, respectively, which may appear useful in other situations as efficient exploratory tools.
Leonid Bogachev, Gregory Derfel, Stanislav Molchanov, John Ochendon

Numerical Methods for Non-Zero-Sum Stochastic Differential Games: Convergence of the Markov Chain Approximation Method

The Markov chain approximation method is an efficient and popular collection of methods for the numerical solution of stochastic control problems in continuous time, for reflected-jump-diffusion-type models and the convergence proofs have been extended to zero-sum stochastic differential games. We apply it to a class of non-zero-sum stochastic differential games with a diffusion system model where the controls for the two players are separated, It is shown that equilibrium values for the approximating chain converge to equilibrium values for the original process and that any equilibrium value for the original process can be approximated by an ε-equilibrium for the chain for arbitrarily small ε > 0. The actual numerical algorithm is that for a stochastic game for a finite-state Markov chain.
Harold J. Kushner

Nonparametric Estimation

Frontmatter

On the Estimation of an Analytic Spectral Density Outside of the Observation Band

We consider a Gaussian stationary process X(t) with an integer analytic spectral density f(λ) and study a problem of its estimation. The process X(t) is non-observable. Instead of it we observe a linear transformation Y(t), 0 ≤ t ≤ T, of X(t) with a transfer function a(A), ∣a(λ)∣=1 if λ belongs to an interval I. We study how far from I the consistent estimation of f(λ) is possible, T → ∞.
Ildar A. Ibragimov

On Oracle Inequalities Related to High Dimensional Linear Models

We consider the problem of estimating an unknown vector θ from the noisy data Y=Aθ+ε, where A is a known m × n matrix and e is a white Gaussian noise. It is assumed that n is large and A is ill-posed. Therefore in order to estimate θ, a spectral regularization method is used and our goal is to choose a spectral regularization parameter with the help of the data Y. We study data-driven regularization methods based on the empirical risk minimization principle and provide some new oracle inequalities related to this approach.
Yuri Golubev

Hypothesis Testing under Composite Functions Alternative

In this paper, we consider the problem of the minimax hypothesis testing in the multivariate white gaussian noise model. We want to test the hypothesis about the absence of the signal against the alternative belonging to the set of smooth composite functions separated away from zero in sup-norm. We propose the test procedure and show that it is optimal in view of the minimax criterion if the smoothness parameters of the composition obey some special assumption. In this case we also present the explicit formula for minimax rate of testing. If this assumption does not hold, we give the explicit upper and lower bounds for minimax rate of testing which differ each other only by some logarithmic factor. In particular, it implies that the proposed test procedure is “almost” minimax. In both cases the minimax rate of testing as well as its upper and lower bounds are completely determined by the smoothness parameters of the composition.
Oleg V. Lepski, Christophe F. Pouet’t

Stochastic Partial Differential Equations

Frontmatter

On Parabolic Pdes and Spdes in Sobolev Spaces W P 2 without and with Weights

We present a “streamlined” theory of solvability of parabolic PDEs and SPDEs in half spaces in Sobolev spaces with weights. The approach is based on interior estimates for equations in the whole space and is easier than and quite different from the standard one.
Nicolai V. Krylov

Stochastic Parabolic Equations of Full Second Order

A procedure is described for defining a generalized solution for stochastic differential equations using the Cameron-Martin version of the Wiener Chaos expansion. Existence and uniqueness of this Wiener Chaos solution is established for parabolic stochastic PDEs such that both the drift and the diffusion operators are of the second order.
Sergey V. Lototsky, Boris L. Rozovskii
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