Convergence Rates for Diffusions on Continuous-Time Lattices

2007 ◽  
Author(s):  
Claudio Albanese ◽  
Aleksandar Mijatovic
Keyword(s):  
Continuous Time ◽  
Convergence Rates

scholarly journals CONVERGENCE RATES OF SUMS OF α-MIXING TRIANGULAR ARRAYS: WITH AN APPLICATION TO NONPARAMETRIC DRIFT FUNCTION ESTIMATION OF CONTINUOUS-TIME PROCESSES

2016 ◽  
Vol 33 (5) ◽  
pp. 1121-1153
Author(s):  
Shin Kanaya
Keyword(s):  
Continuous Time ◽  
Convergence Rates ◽  
Moving Average ◽  
Function Estimation ◽  
Triangular Arrays ◽  
Large Numbers ◽  
Strongly Mixing ◽  
Time Distance ◽  
Continuous Time Processes ◽  
Near Unit Root

The convergence rates of the sums of α-mixing (or strongly mixing) triangular arrays of heterogeneous random variables are derived. We pay particular attention to the case where central limit theorems may fail to hold, due to relatively strong time-series dependence and/or the nonexistence of higher-order moments. Several previous studies have presented various versions of laws of large numbers for sequences/triangular arrays, but their convergence rates were not fully investigated. This study is the first to investigate the convergence rates of the sums of α-mixing triangular arrays whose mixing coefficients are permitted to decay arbitrarily slowly. We consider two kinds of asymptotic assumptions: one is that the time distance between adjacent observations is fixed for any sample size n; and the other, called the infill assumption, is that it shrinks to zero as n tends to infinity. Our convergence theorems indicate that an explicit trade-off exists between the rate of convergence and the degree of dependence. While the results under the infill assumption can be seen as a direct extension of those under the fixed-distance assumption, they are new and particularly useful for deriving sharper convergence rates of discretization biases in estimating continuous-time processes from discretely sampled observations. We also discuss some examples to which our results and techniques are useful and applicable: a moving-average process with long lasting past shocks, a continuous-time diffusion process with weak mean reversion, and a near-unit-root process.

UNIFORM CONVERGENCE RATES OF KERNEL-BASED NONPARAMETRIC ESTIMATORS FOR CONTINUOUS TIME DIFFUSION PROCESSES: A DAMPING FUNCTION APPROACH

2016 ◽  
Vol 33 (4) ◽  
pp. 874-914 ◽  
Author(s):  
Shin Kanaya
Keyword(s):  
Uniform Convergence ◽  
Continuous Time ◽  
Convergence Rates ◽  
Diffusion Processes ◽  
Kernel Functions ◽  
Damping Function ◽  
Nonparametric Estimators ◽  
Empirical Process Theory ◽  
Convergence Results ◽  
The Moment

In this paper, we derive uniform convergence rates of nonparametric estimators for continuous time diffusion processes. In particular, we consider kernel-based estimators of the Nadaraya–Watson type, introducing a new technical device called adamping function. This device allows us to derive sharp uniform rates over an infinite interval with minimal requirements on the processes: The existence of the moment of any order is not required and the boundedness of relevant functions can be significantly relaxed. Restrictions on kernel functions are also minimal: We allow for kernels with discontinuity, unbounded support, and slowly decaying tails. Our proofs proceed by using the covering-number technique from empirical process theory and exploiting the mixing and martingale properties of the processes. We also present new results on the path-continuity property of Brownian motions and diffusion processes over an infinite time horizon. These path-continuity results, which should also be of some independent interest, are used to control discretization biases of the nonparametric estimators. The obtained convergence results are useful for non/semiparametric estimation and testing problems of diffusion processes.

scholarly journals Derivation of ensemble Kalman–Bucy filters with unbounded nonlinear coefficients

Nonlinearity ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 1061-1092
Author(s):  
Theresa Lange
Keyword(s):  
Kalman Filter ◽  
Discrete Time ◽  
Ensemble Kalman Filter ◽  
Continuous Time ◽  
Convergence Rates ◽  
Well Posedness ◽  
Square Root ◽  
Rigorous Derivation ◽  
Step Size ◽  
Ensemble Transform

Abstract We provide a rigorous derivation of the ensemble Kalman–Bucy filter as well as the ensemble transform Kalman–Bucy filter in case of nonlinear, unbounded model and observation operators. We identify them as the continuous time limit of the discrete-time ensemble Kalman filter and the ensemble square root filters, respectively, together with concrete convergence rates in terms of the discretisation step size. Simultaneously, we establish well-posedness as well as accuracy of both the continuous-time and the discrete-time filtering algorithms.

Empirical convergence rates for continuous-time Markov chains

2001 ◽  
Vol 38 (1) ◽  
pp. 262-269 ◽  
Author(s):  
Geoffrey Pritchard ◽  
David J. Scott
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Rate Of Convergence ◽  
Continuous Time ◽  
Transition Matrix ◽  
Convergence Rates ◽  
Parametric Model ◽  
Continuous Time Markov Chains ◽  
Finite State ◽  
Second Largest Eigenvalue

We consider the problem of estimating the rate of convergence to stationarity of a continuous-time, finite-state Markov chain. This is done via an estimator of the second-largest eigenvalue of the transition matrix, which in turn is based on conventional inference in a parametric model. We obtain a limiting distribution for the eigenvalue estimator. As an example we treat an M/M/c/c queue, and show that the method allows us to estimate the time to stationarity τ within a time comparable to τ.

Empirical convergence rates for continuous-time Markov chains

2001 ◽  
Vol 38 (01) ◽  
pp. 262-269 ◽  
Author(s):  
Geoffrey Pritchard ◽  
David J. Scott
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Rate Of Convergence ◽  
Continuous Time ◽  
Transition Matrix ◽  
Convergence Rates ◽  
Parametric Model ◽  
Continuous Time Markov Chains ◽  
Finite State ◽  
Second Largest Eigenvalue

We consider the problem of estimating the rate of convergence to stationarity of a continuous-time, finite-state Markov chain. This is done via an estimator of the second-largest eigenvalue of the transition matrix, which in turn is based on conventional inference in a parametric model. We obtain a limiting distribution for the eigenvalue estimator. As an example we treat an M/M/c/c queue, and show that the method allows us to estimate the time to stationarity τ within a time comparable to τ.

Discounted Continuous-Time Controlled Markov Chains: Convergence of Control Models

2012 ◽  
Vol 49 (04) ◽  
pp. 1072-1090
Author(s):  
Tomás Prieto-Rumeau ◽  
Onésimo Hernández-Lerma
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Convergence Rates ◽  
Control Model ◽  
Population System ◽  
Controlled Markov Chains ◽  
Control Models ◽  
Finite State ◽  
Definition Of ◽  
Optimal Reward

We are interested in continuous-time, denumerable state controlled Markov chains (CMCs), with compact Borel action sets, and possibly unbounded transition and reward rates, under the discounted reward optimality criterion. For such CMCs, we propose a definition of a sequence of control models {ℳ n } converging to a given control model ℳ, which ensures that the discount optimal reward and policies of ℳ n converge to those of ℳ. As an application, we propose a finite-state and finite-action truncation technique of the original control model ℳ, which is illustrated by approximating numerically the optimal reward and policy of a controlled population system with catastrophes. We study the corresponding convergence rates.

scholarly journals Discounted Continuous-Time Controlled Markov Chains: Convergence of Control Models

2012 ◽  
Vol 49 (4) ◽  
pp. 1072-1090 ◽  
Author(s):  
Tomás Prieto-Rumeau ◽  
Onésimo Hernández-Lerma
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Convergence Rates ◽  
Control Model ◽  
Population System ◽  
Controlled Markov Chains ◽  
Control Models ◽  
Finite State ◽  
Definition Of ◽  
Optimal Reward

We are interested in continuous-time, denumerable state controlled Markov chains (CMCs), with compact Borel action sets, and possibly unbounded transition and reward rates, under the discounted reward optimality criterion. For such CMCs, we propose a definition of a sequence of control models {ℳn} converging to a given control model ℳ, which ensures that the discount optimal reward and policies of ℳn converge to those of ℳ. As an application, we propose a finite-state and finite-action truncation technique of the original control model ℳ, which is illustrated by approximating numerically the optimal reward and policy of a controlled population system with catastrophes. We study the corresponding convergence rates.

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