arXiv subjects

Nikolai Dokuchaev

Publications and source records attributed to Nikolai Dokuchaev.

At least 37 records · Page 2Linked to original sources

arXiv2019

On approximation of the distribution for Pearson statistic

The paper considers the classical Goodness of Fit test. It suggests to use the Gamma distribution for the approximation of the distribution of the Pearson statistics with unknown parameters estimated from raw data. The parameters of these Gamma distribution can be estimated from the first two moments of the statistic after averaging over a distribution of the unknown parameter over its range. This allows to simplify calculation of the quantiles for the Pearson statistic, as is shown in some simulation experiments with medium and small sample sizes.

math.ST↗

arXiv2018

First Order BSPDEs in higher dimension for optimal control problems

The paper studies the First Order BSPDEs (Backward Stochastic Partial Differential Equations) suggested earlier for a case of multidimensional state domain with a boundary. These equations represent analogs of Hamilton-Jacobi-Bellman equations and allow to construct the value function for stochastic optimal control problems with unspecified dynamics where the underlying processes do not necessarily satisfy stochastic differential equations of a known kind with a given structure. The problems considered arise in financial modelling.

q-fin.MF↗

arXiv2018

Spectrum degeneracy for functions on branching lines and impact on extrapolation and sampling

The paper studies functions defined on continuous branching lines connected into a system. A notion of spectrum degeneracy for these functions is introduced. This degeneracy is based on the properties of the Fourier transforms for processes representing functions on the branches. It is shown that processes with this spectrum degeneracy are everywhere dense in the set of processes equivalent to functions on the branching lines. Some applications to extrapolation and sampling are considered.

cs.IT↗

arXiv2018

On data recovery with restraints on the spectrum range and the process range

The paper considers recovery of signals from incomplete observations and a problem of determination of the allowed quantity of missed observations, i.e. the problem of determination of the size of the uniqueness sets for a given data recovery procedures. The paper suggests a way to bypass solution of this uniqueness problem via imposing restrictions investigates possibility of data recovery for classes of finite sequences under a special discretization of the process range. It is shown that these sequences can be dense in the space of all sequences and that the uniqueness sets for them can be singletons. Some robustness with respect to rounding of input data can be achieved via including additional observations.

cs.IT↗

arXiv2018

On recovery of signals with single point spectrum degeneracy

The paper study recovery problem for discrete time signals with a finite number of missing values. The paper establishes recoverability of these missing values for signals with Z-transform vanishing with a certain rate at a single point. The transfer functions for the corresponding recovering kernels are presented explicitly. Some robustness of the recovery with respect to data truncation or noise contamination is established.

eess.SP↗

arXiv2018

On a gap between rational annuitization price for producer and price for customer

The paper studies pricing of insurance products focusing on the pricing of annuities under uncertainty. This pricing problem is crucial for financial decision making and was studied intensively, however, many open questions still remain. In particular, there is a so-called "annuity puzzle" related to certain inconsistency of existing financial theory with the empirical observations for the annuities market. The paper suggests a pricing method based on the risk minimization such that both producer and customer seek to minimize the mean square hedging error accepted as a measure of risk. This leads to two different versions of the pricing problem: the selection of the annuity price given the rate of regular payments, and the selection of the rate of payments given the annuity price. It appears that solutions of these two problems are different. This can contribute to explanation for the "annuity puzzle".

econ.GN↗

arXiv2018

On recovering missing values for sequences in a pathwise setting

The paper suggests a frequency criterion of error-free recoverability of a missing value for sequences, i.e. discrete time processes, in a pathwise setting without probabilistic assumptions. The paper establishes error-free recoverability for classes of square-summable sequences with Z-transform vanishing at isolated points with a mild rate; the case of non-summable sequences is not excluded. The transfer functions for recovering algorithm are presented explicitly. Some robustness with respect to noise contamination is established for the suggested recovering algorithm.

cs.IT↗

arXiv2018

Predictors for discrete time processes with energy decay on higher frequencies

The predictability of discrete-time processes is studied in a deterministic setting. A family of one-step-ahead predictors is suggested for processes of which the energy decays at higher frequencies. For such processes, the prediction error can be made arbitrarily small. The predictions can be robust with respect to the noise contamination at higher frequencies.

math.OC↗

arXiv2018

Predictability of sequences and subsequences with spectrum degeneracy at periodically located points

The paper established sufficient conditions of predictability with degeneracy for the spectrum at $M$-periodically located isolated points on the unit circle. It is also shown that $m$-periodic subsequences of these sequences are also predictable if $m$ is a divisor of $M$. The predictability can be achieved for finite horizon with linear predictors defined by convolutions with certain kernels. As an example of applications, it is shown that there exists a class of sequences that is everywhere dense in the class of all square-summable sequences and such that its members can be recovered from their periodic subsequences. This recoverability is associated with certain spectrum degeneracy of a new kind.

cs.IT↗

arXiv2018

On recovery of sequences from subsequences: the case of non-periodic spectrum gaps

The paper investigates recoverability of sequences from their periodic subsequences and offers some modification of the approach suggested in papers arXiv:1605.00414 and arXiv:1803.02233. It is shown that there exists a class of sequences that is everywhere dense in the class of all square-summable sequences and such that its members can be recovered from their periodic subsequences. This recoverability is associated with certain spectrum degeneracy of a new kind.

cs.IT↗

arXiv2018

On causal extrapolation of sequences with applications to forecasting

The paper suggests a method of extrapolation of notion of one-sided semi-infinite sequences representing traces of two-sided band-limited sequences; this features ensure uniqueness of this extrapolation and possibility to use this for forecasting. This lead to a forecasting method for more general sequences without this feature based on minimization of the mean square error between the observed path and a predicable sequence. These procedure involves calculation of this predictable path; the procedure can be interpreted as causal smoothing. The corresponding smoothed sequences allow unique extrapolations to future times that can be interpreted as optimal forecasts.

math.OC↗

arXiv2017

Optimal data recovery and forecasting with dummy long-horizon forecasts

The paper suggests a method of recovering missing values for sequences, including sequences with a multidimensional index, based on optimal approximation by processes featuring spectrum degeneracy. The problem is considered in the pathwise setting, without using probabilistic assumptions on the ensemble. The method requires to solve a closed linear equation connecting the available observations of the underlying process with the values of the approximating process with degenerate spectrum outside the observation range.Some robustness with respect to noise contamination is established for the suggested recovering algorithm. It is suggested to apply this data recovery algorithm to forecasting with a preselected dummy long-horizon forecast that helps to regularize the solution.

cs.IT↗

arXiv2017

On sampling theorem with sparse decimated samples: exploring branching spectrum degeneracy

The paper investigates possibility of recovery of sequences from their decimated subsequences. It is shown that this recoverability is associated with certain spectrum degeneracy of a new kind, and that a sequences of a general kind can be approximated by sequences featuring this degeneracy. This is applied to sparse sampling of continuous time band-limited functions. The paper shows that these functions allow an arbitrarily close approximation by functions that can be recovered from sparse equidistant samples with sampling distance larger than the distance defined by the critical Nyquist rate for the underlying function. This allows to bypass, in a certain sense, the restriction on the sampling rate defined by the Nyquist rate.

cs.IT↗

arXiv2017

A closed equation in time domain for band-limited extensions of one-sided sequences

The paper suggests a method of optimal extension of one-sided semi-infinite sequences of a general type by traces of band-limited sequences in deterministic setting, i.e. without probabilistic assumptions. The method requires to solve a closed linear equation in the time domain connecting the past observations of the underlying process with the future values of the band-limited process. Robustness of the solution with respect to the input errors and data truncation is established in the framework of Tikhonov regularization.

cs.IT↗

arXiv2017

On stochastic integrals with controlled growth of their containing range

This short note suggests special examples of stochastic Ito integrals with controlled growth of their containing range. The integrands for this integrals are presented explicitly. The construction does not involve neither stopping times nor forecasting or calculation of the conditional expectations of a contingent claim

math.PR↗

arXiv2017

On recovering solutions for SPDEs from their averages

We study linear stochastic partial differential equations of parabolic type. We consider a new boundary value problem where a Cauchy condition is replaced by a prescribed average of the solution either over time and probabilistic space for forward SPDEs and over time for backward SPDEs. Well-posedness, existence, uniqueness, and a regularity of the solution for this new problem are obtained. In particular, this can be considered as a possibility to recover a solution of a forward SPDE in a setting where its values at the initial time are unknown, and where the average of the solution over time and probability space is observable, as well as the input processes.

math.PR↗

arXiv2017

A pathwise inference method for the parameters of diffusion terms

We consider inference of the parameters of the diffusion term for Cox-Ingersoll-Ross and similar processes with a power type dependence of the diffusion coefficient from the underlying process. We suggest some original pathwise estimates for this coefficient and for the power index based on an analysis of an auxiliary continuous time complex valued process generated by the underlying real valued process. These estimates do not rely on the distribution of the underlying process and on a particular choice of the drift. Some numerical experiments are used to illustrate the feasibility of the suggested method.

math.PR↗

arXiv2017

On exact and optimal recovering of missing values for sequences

The paper studies recoverability of missing values for sequences in a pathwise setting without probabilistic assumptions. This setting is oriented on a situation where the underlying sequence is considered as a sole sequence rather than a member of an ensemble with known statistical properties. Sufficient conditions of recoverability are obtained; it is shown that sequences are recoverable if there is a certain degree of degeneracy of the Z-transforms. We found that, in some cases, this degree can be measured as the number of the derivatives of Z-transform vanishing at a point. For processes with non-degenerate Z-transform, an optimal recovering based on the projection on a set of recoverable sequences is suggested. Some robustness of the solution with respect to noise contamination and truncation is established.

cs.IT↗