1,475 research outputs found
Posterior Consistency via Precision Operators for Bayesian Nonparametric Drift Estimation in SDEs
We study a Bayesian approach to nonparametric estimation of the periodic
drift function of a one-dimensional diffusion from continuous-time data.
Rewriting the likelihood in terms of local time of the process, and specifying
a Gaussian prior with precision operator of differential form, we show that the
posterior is also Gaussian with precision operator also of differential form.
The resulting expressions are explicit and lead to algorithms which are readily
implementable. Using new functional limit theorems for the local time of
diffusions on the circle, we bound the rate at which the posterior contracts
around the true drift function
Adaptive confidence bands for Markov chains and diffusions: Estimating the invariant measure and the drift
As a starting point we prove a functional central limit theorem for
estimators of the invariant measure of a geometrically ergodic Harris-recurrent
Markov chain in a multi-scale space. This allows to construct confidence bands
for the invariant density with optimal (up to undersmoothing)
-diameter by using wavelet projection estimators. In addition our
setting applies to the drift estimation of diffusions observed discretely with
fixed observation distance. We prove a functional central limit theorem for
estimators of the drift function and finally construct adaptive confidence
bands for the drift by using a completely data-driven estimator.Comment: to appear in ESAIM: Probability and Statistic
Parametric and Nonparametric Volatility Measurement
Volatility has been one of the most active areas of research in empirical finance and time series econometrics during the past decade. This chapter provides a unified continuous-time, frictionless, no-arbitrage framework for systematically categorizing the various volatility concepts, measurement procedures, and modeling procedures. We define three different volatility concepts: (i) the notional volatility corresponding to the ex-post sample-path return variability over a fixed time interval, (ii) the ex-ante expected volatility over a fixed time interval, and (iii) the instantaneous volatility corresponding to the strength of the volatility process at a point in time. The parametric procedures rely on explicit functional form assumptions regarding the expected and/or instantaneous volatility. In the discrete-time ARCH class of models, the expectations are formulated in terms of directly observable variables, while the discrete- and continuous-time stochastic volatility models involve latent state variable(s). The nonparametric procedures are generally free from such functional form assumptions and hence afford estimates of notional volatility that are flexible yet consistent (as the sampling frequency of the underlying returns increases). The nonparametric procedures include ARCH filters and smoothers designed to measure the volatility over infinitesimally short horizons, as well as the recently-popularized realized volatility measures for (non-trivial) fixed-length time intervals.
Parametric and Nonparametric Volatility Measurement
Volatility has been one of the most active areas of research in empirical finance and time series econometrics during the past decade. This chapter provides a unified continuous-time, frictionless, no-arbitrage framework for systematically categorizing the various volatility concepts, measurement procedures, and modeling procedures. We define three different volatility concepts: (i) the notional volatility corresponding to the ex-post sample-path return variability over a fixed time interval, (ii) the ex-ante expected volatility over a fixed time interval, and (iii) the instantaneous volatility corresponding to the strength of the volatility process at a point in time. The parametric procedures rely on explicit functional form assumptions regarding the expected and/or instantaneous volatility. In the discrete-time ARCH class of models, the expectations are formulated in terms of directly observable variables, while the discrete- and continuous-time stochastic volatility models involve latent state variable(s). The nonparametric procedures are generally free from such functional form assumptions and hence afford estimates of notional volatility that are flexible yet consistent (as the sampling frequency of the underlying returns increases). The nonparametric procedures include ARCH filters and smoothers designed to measure the volatility over infinitesimally short horizons, as well as the recently-popularized realized volatility measures for (non-trivial) fixed-length time intervals.
Consistency of Bayesian nonparametric inference for discretely observed jump diffusions
We introduce verifiable criteria for weak posterior consistency of
identifiable Bayesian nonparametric inference for jump diffusions with unit
diffusion coefficient and uniformly Lipschitz drift and jump coefficients in
arbitrary dimension. The criteria are expressed in terms of coefficients of the
SDEs describing the process, and do not depend on intractable quantities such
as transition densities. We also show that products of discrete net and
Dirichlet mixture model priors satisfy our conditions, again under an
identifiability assumption. This generalises known results by incorporating
jumps into previous work on unit diffusions with uniformly Lipschitz drift
coefficients.Comment: 20 page
MCMC methods for functions modifying old algorithms to make\ud them faster
Many problems arising in applications result in the need\ud
to probe a probability distribution for functions. Examples include Bayesian nonparametric statistics and conditioned diffusion processes. Standard MCMC algorithms typically become arbitrarily slow under the mesh refinement dictated by nonparametric description of the unknown function. We describe an approach to modifying a whole range of MCMC methods which ensures that their speed of convergence is robust under mesh refinement. In the applications of interest the data is often sparse and the prior specification is an essential part of the overall modeling strategy. The algorithmic approach that we describe is applicable whenever the desired probability measure has density with respect to a Gaussian process or Gaussian random field prior, and to some useful non-Gaussian priors constructed through random truncation. Applications are shown in density estimation, data assimilation in fluid mechanics, subsurface geophysics and image registration. The key design principle is to formulate the MCMC method for functions. This leads to algorithms which can be implemented via minor modification of existing algorithms, yet which show enormous speed-up on a wide range of applied problems
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