37,035 research outputs found
Stability estimating in optimal stopping problem
summary:We consider the optimal stopping problem for a discrete-time Markov process on a Borel state space . It is supposed that an unknown transition probability , , is approximated by the transition probability , , and the stopping rule , optimal for , is applied to the process governed by . We found an upper bound for the difference between the total expected cost, resulting when applying , and the minimal total expected cost. The bound given is a constant times , where is the total variation norm
Optimal dual martingales, their analysis and application to new algorithms for Bermudan products
In this paper we introduce and study the concept of optimal and surely
optimal dual martingales in the context of dual valuation of Bermudan options,
and outline the development of new algorithms in this context. We provide a
characterization theorem, a theorem which gives conditions for a martingale to
be surely optimal, and a stability theorem concerning martingales which are
near to be surely optimal in a sense. Guided by these results we develop a
framework of backward algorithms for constructing such a martingale. In turn
this martingale may then be utilized for computing an upper bound of the
Bermudan product. The methodology is pure dual in the sense that it doesn't
require certain input approximations to the Snell envelope. In an It\^o-L\'evy
environment we outline a particular regression based backward algorithm which
allows for computing dual upper bounds without nested Monte Carlo simulation.
Moreover, as a by-product this algorithm also provides approximations to the
continuation values of the product, which in turn determine a stopping policy.
Hence, we may obtain lower bounds at the same time. In a first numerical study
we demonstrate the backward dual regression algorithm in a Wiener environment
at well known benchmark examples. It turns out that the method is at least
comparable to the one in Belomestny et. al. (2009) regarding accuracy, but
regarding computational robustness there are even several advantages.Comment: This paper is an extended version of Schoenmakers and Huang, "Optimal
dual martingales and their stability; fast evaluation of Bermudan products
via dual backward regression", WIAS Preprint 157
Sequential joint signal detection and signal-to-noise ratio estimation
The sequential analysis of the problem of joint signal detection and
signal-to-noise ratio (SNR) estimation for a linear Gaussian observation model
is considered. The problem is posed as an optimization setup where the goal is
to minimize the number of samples required to achieve the desired (i) type I
and type II error probabilities and (ii) mean squared error performance. This
optimization problem is reduced to a more tractable formulation by transforming
the observed signal and noise sequences to a single sequence of Bernoulli
random variables; joint detection and estimation is then performed on the
Bernoulli sequence. This transformation renders the problem easily solvable,
and results in a computationally simpler sufficient statistic compared to the
one based on the (untransformed) observation sequences. Experimental results
demonstrate the advantages of the proposed method, making it feasible for
applications having strict constraints on data storage and computation.Comment: 5 pages, Proceedings of IEEE International Conference on Acoustics,
Speech, and Signal Processing (ICASSP), 201
Estimating the Spectrum in Computed Tomography Via Kullback–Leibler Divergence Constrained Optimization
Purpose
We study the problem of spectrum estimation from transmission data of a known phantom. The goal is to reconstruct an x‐ray spectrum that can accurately model the x‐ray transmission curves and reflects a realistic shape of the typical energy spectra of the CT system. Methods
Spectrum estimation is posed as an optimization problem with x‐ray spectrum as unknown variables, and a Kullback–Leibler (KL)‐divergence constraint is employed to incorporate prior knowledge of the spectrum and enhance numerical stability of the estimation process. The formulated constrained optimization problem is convex and can be solved efficiently by use of the exponentiated‐gradient (EG) algorithm. We demonstrate the effectiveness of the proposed approach on the simulated and experimental data. The comparison to the expectation–maximization (EM) method is also discussed. Results
In simulations, the proposed algorithm is seen to yield x‐ray spectra that closely match the ground truth and represent the attenuation process of x‐ray photons in materials, both included and not included in the estimation process. In experiments, the calculated transmission curve is in good agreement with the measured transmission curve, and the estimated spectra exhibits physically realistic looking shapes. The results further show the comparable performance between the proposed optimization‐based approach and EM. Conclusions
Our formulation of a constrained optimization provides an interpretable and flexible framework for spectrum estimation. Moreover, a KL‐divergence constraint can include a prior spectrum and appears to capture important features of x‐ray spectrum, allowing accurate and robust estimation of x‐ray spectrum in CT imaging
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