Optimal estimation for semimartingales

1986 ◽  
Vol 23 (02) ◽  
pp. 409-417 ◽  
Author(s):  
A. Thavaneswaran ◽  
M. E. Thompson
Keyword(s):  
Continuous Time ◽  
Asymptotic Properties ◽  
Optimal Estimation ◽  
Parametric Estimation ◽  
Local Martingale ◽  
Regularity Conditions ◽  
Probability Measures ◽  
Simple Process ◽  
Special Cases ◽  
Least Squares Estimates

This paper extends a result of Godambe's theory of parametric estimation for discrete-time stochastic processes to the continuous-time case. LetP={P} be a family of probability measures such that (Ω,F, P) is complete, (Ft, t≧0) is a standard filtration, andX = (XtFt, t ≧ 0)is a semimartingale for everyP ∈ P. For a parameterθ(Ρ), supposeXt=Vt,θ+Ht,θwhere theVθprocess is predictable and locally of bounded variation and theHθprocess is a local martingale. Consider estimating equations forθof the formprocess is predictable. Under regularity conditions, an optimal form forαθin the sense of Godambe (1960) is determined. WhenVt,θis linear inθthe optimal, corresponds to certain maximum likelihood or least squares estimates derived previously in special cases. Asymptotic properties of, are discussed.

Optimal estimation for semimartingales

1986 ◽  
Vol 23 (2) ◽  
pp. 409-417 ◽  
Author(s):  
A. Thavaneswaran ◽  
M. E. Thompson
Keyword(s):  
Continuous Time ◽  
Asymptotic Properties ◽  
Optimal Estimation ◽  
Parametric Estimation ◽  
Local Martingale ◽  
Regularity Conditions ◽  
Probability Measures ◽  
Optimal Form ◽  
Special Cases ◽  
Least Squares Estimates

This paper extends a result of Godambe's theory of parametric estimation for discrete-time stochastic processes to the continuous-time case. Let P ={P} be a family of probability measures such that (Ω, F, P) is complete, (Ft, t≧0) is a standard filtration, and X = (Xt Ft, t ≧ 0) is a semimartingale for every P ∈ P. For a parameter θ (Ρ), suppose Xt = Vt,θ + Ht,θ where the Vθ process is predictable and locally of bounded variation and the Hθ process is a local martingale. Consider estimating equations for θ of the form process is predictable. Under regularity conditions, an optimal form for α θ in the sense of Godambe (1960) is determined. When Vt,θ is linear in θ the optimal , corresponds to certain maximum likelihood or least squares estimates derived previously in special cases. Asymptotic properties of , are discussed.

A characterization of the set of local martingale measures

2018 ◽  
Vol 18 (05) ◽  
pp. 1850042 ◽  
Author(s):  
Abdelkarem Berkaoui
Keyword(s):  
Continuous Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Local Martingale ◽  
Probability Measures ◽  
Martingale Measures ◽  
Necessary And Sufficient ◽  
Adapted Process ◽  
Vector Valued

We generalize the results of [1] to continuous time case by stating necessary and sufficient conditions on a set of probability measures to be the set of local martingale measures for a vector valued, locally bounded and adapted process.

Optimal Consumption and Insurance: A Continuous-time Markov Chain Approach

2008 ◽  
Vol 38 (01) ◽  
pp. 231-257 ◽  
Author(s):  
Holger Kraft ◽  
Mogens Steffensen
Keyword(s):  
Continuous Time ◽  
Optimal Solution ◽  
Decision Problems ◽  
Optimal Consumption ◽  
Continuous Time Markov Chain ◽  
Financial Decision Making ◽  
Markov Chain Approach ◽  
Special Cases ◽  
Financial Decision ◽  
The Individual

Personal financial decision making plays an important role in modern finance. Decision problems about consumption and insurance are in this article modelled in a continuous-time multi-state Markovian framework. The optimal solution is derived and studied. The model, the problem, and its solution are exemplified by two special cases: In one model the individual takes optimal positions against the risk of dying; in another model the individual takes optimal positions against the risk of losing income as a consequence of disability or unemployment.

scholarly journals Antisymmetry of the Stochastical Order on all Ordered Topological Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 250-252 ◽  
Author(s):  
Tobias Fritz
Keyword(s):  
Topological Space ◽  
General Result ◽  
Elementary Proof ◽  
Short Note ◽  
Stochastic Order ◽  
Topological Spaces ◽  
Probability Measures ◽  
Ordered Topological Space ◽  
Special Cases

Abstract In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.

Asymptotic properties of stereological estimators of volume fraction for stationary random sets

1982 ◽  
Vol 19 (1) ◽  
pp. 111-126 ◽  
Author(s):  
Shigeru Mase
Keyword(s):  
Volume Fraction ◽  
Asymptotic Variance ◽  
Asymptotic Properties ◽  
Parametric Estimation ◽  
Random Sets ◽  
Stationary Time Series ◽  
Boolean Models ◽  
Window Functions ◽  
Spectral Density Functions ◽  
Point Estimators

We shall discuss asymptotic properties of stereological estimators of volume (area) fraction for stationary random sets (in the sense of Matheron) under natural and general assumptions. Results obtained are strong consistency, asymptotic normality, and asymptotic unbiasedness and consistency of asymptotic variance estimators. The method is analogous to the non-parametric estimation of spectral density functions of stationary time series using window functions. Proofs are given for areal estimators, but they are also valid for lineal and point estimators with slight modifications. Finally we show that stationary Boolean models satisfy the relevant assumptions reasonably well.

scholarly journals Optimal errors and phase transitions in high-dimensional generalized linear models

2019 ◽  
Vol 116 (12) ◽  
pp. 5451-5460 ◽  
Author(s):  
Jean Barbier ◽  
Florent Krzakala ◽  
Nicolas Macris ◽  
Léo Miolane ◽  
Lenka Zdeborová
Keyword(s):  
Phase Transitions ◽  
Generalized Linear Models ◽  
Message Passing ◽  
Statistical Physics ◽  
Linear Models ◽  
Optimal Estimation ◽  
Error Correcting Codes ◽  
Data Matrix ◽  
High Dimensional ◽  
Special Cases

Generalized linear models (GLMs) are used in high-dimensional machine learning, statistics, communications, and signal processing. In this paper we analyze GLMs when the data matrix is random, as relevant in problems such as compressed sensing, error-correcting codes, or benchmark models in neural networks. We evaluate the mutual information (or “free entropy”) from which we deduce the Bayes-optimal estimation and generalization errors. Our analysis applies to the high-dimensional limit where both the number of samples and the dimension are large and their ratio is fixed. Nonrigorous predictions for the optimal errors existed for special cases of GLMs, e.g., for the perceptron, in the field of statistical physics based on the so-called replica method. Our present paper rigorously establishes those decades-old conjectures and brings forward their algorithmic interpretation in terms of performance of the generalized approximate message-passing algorithm. Furthermore, we tightly characterize, for many learning problems, regions of parameters for which this algorithm achieves the optimal performance and locate the associated sharp phase transitions separating learnable and nonlearnable regions. We believe that this random version of GLMs can serve as a challenging benchmark for multipurpose algorithms.

scholarly journals Estimation for Nonnegative Lévy-Driven Ornstein-Uhlenbeck Processes

2007 ◽  
Vol 44 (04) ◽  
pp. 977-989 ◽  
Author(s):  
Peter J. Brockwell ◽  
Richard A. Davis ◽  
Yu Yang
Keyword(s):  
Continuous Time ◽  
Lévy Process ◽  
Moving Average ◽  
Asymptotic Properties ◽  
Estimation Procedure ◽  
Financial Econometrics ◽  
Levy Process ◽  
Efficient Estimation ◽  
Autoregressive Moving Average ◽  
Continuous Time Processes

Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a nondecreasing Lévy process constitute a very general class of stationary, nonnegative continuous-time processes. In financial econometrics a stationary Ornstein-Uhlenbeck (or CAR(1)) process, driven by a nondecreasing Lévy process, was introduced by Barndorff-Nielsen and Shephard (2001) as a model for stochastic volatility to allow for a wide variety of possible marginal distributions and the possibility of jumps. For such processes, we take advantage of the nonnegativity of the increments of the driving Lévy process to study the properties of a highly efficient estimation procedure for the parameters when observations are available of the CAR(1) process at uniformly spaced times 0,h,…,Nh. We also show how to reconstruct the background driving Lévy process from a continuously observed realization of the process and use this result to estimate the increments of the Lévy process itself when h is small. Asymptotic properties of the coefficient estimator are derived and the results illustrated using a simulated gamma-driven Ornstein-Uhlenbeck process.

On the GIX/G/∞ system

1990 ◽  
Vol 27 (3) ◽  
pp. 671-683 ◽  
Author(s):  
L. Liu ◽  
B. R. K. Kashyap ◽  
J. G. C. Templeton
Keyword(s):  
Steady State ◽  
Shot Noise ◽  
Continuous Time ◽  
Transient State ◽  
System Size ◽  
Noise Process ◽  
Shot Noise Process ◽  
Special Cases

By using a shot noise process, general results on system size in continuous time are given both in transient state and in steady state with discussion on some interesting results concerning special cases. System size before arrivals is also discussed.

scholarly journals Stackelberg equilibria in a continuous-time vertical contracting model with uncertain demand and delayed information

2014 ◽  
Vol 51 (A) ◽  
pp. 213-226 ◽  
Author(s):  
Bernt Øksendal ◽  
Leif Sandal ◽  
Jan Ubøe
Keyword(s):  
Continuous Time ◽  
Variable Coefficients ◽  
Uncertain Demand ◽  
Uhlenbeck Process ◽  
Delayed Information ◽  
Equilibrium Prices ◽  
Special Cases ◽  
Explicit Formulae ◽  
Contracting Model ◽  
Vertical Contracting

We consider explicit formulae for equilibrium prices in a continuous-time vertical contracting model. A manufacturer sells goods to a retailer, and the objective of both parties is to maximize expected profits. Demand is an Itô-Lévy process, and to increase realism, information is delayed. We provide complete existence and uniqueness proofs for a series of special cases, including geometric Brownian motion and the Ornstein-Uhlenbeck process, both with time-variable coefficients. Moreover, explicit solution formulae are given, so these results are operational. An interesting finding is that information that is more precise may be a considerable disadvantage for the retailer.

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