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Nikolai Dokuchaev

Publications and source records attributed to Nikolai Dokuchaev.

At least 19 recordsLinked to original sources
arXiv2024

Sampling Theorem and explicit interpolation formula for non-decaying unbounded signals

The paper establishes an analog Whittaker-Shannon-Kotelnikov sampling theorem for unbounded non-decaying band-limited signals. An explicit interpolation formula is obtained for signals sublinear growth with rate of growth less than 1/2. At any time, the rate of decay for the $k$th coefficients of this formula is $\sim 1/k^2$. In addition, the paper obtains a method for calculating the coefficients of the interpolation formula applicable to signals with arbitrarily high rate of polynomial growth.

math.FA↗
arXiv2024

On predictors and filters for non-decaying unbounded continuous time signals

The paper studies spectral representation and its applications for non-decaying continuous time signals that are not necessarily bounded at $\pm\infty$. The paper introduces notions of transfer functions, spectrum degeneracy, spectrum gaps, and bandlimitness, for these unbounded signals. As an example of applications, explicit formulae are given for transfer functions of low-pass and high-pass filters suitable for these signal. As another example of applications, it is shown that non-decaying unbounded signals with a single point spectrum degeneracy and sublinear rate of growth are predictable. The corresponding transfer functions for the predictors are obtained explicitly.

cs.IT↗
arXiv2024

Approximating intractable short ratemodel distribution with neural network

We propose an algorithm which predicts each subsequent time step relative to the previous timestep of intractable short rate model (when adjusted for drift and overall distribution of previous percentile result) and show that the method achieves superior outcomes to the unbiased estimate both on the trained dataset and different validation data.

stat.ML↗
arXiv2023

Spectral representation of two-sided signals from $\ell_\infty$ and applications to signal processing

The paper studies spectral representation as well as predictability and recoverability problems for non-vanishing discrete time signals from $\ell_\infty$, i.e. for bounded discrete time signals, including signals that do not vanish at $\pm\infty$. The extends the notions of transfer functions, the spectrum gaps, bandlimitness, and filters, on these general type signals. Some frequency conditions of predictability and data recoverability are presented, and some recovery methods and predictors have been suggested.

cs.IT↗
arXiv2023

Near-ideal predictors and causal filters for discrete time signals

The paper presents linear predictors and causal filters for discrete time signals featuring some different kinds of spectrum degeneracy. These predictors and filters are based on approximation of ideal non-causal transfer functions by causal transfer functions represented by polynomials of Z-transform of the unit step signal.

cs.IT↗
arXiv2022

Extrapolation and sampling for processes on spatial graphs

The paper studies processes defined on time domains structured as oriented spatial graphs (or metric graphs, or oriented branched 1-manifolds). This setting can be used, for example, for forecasting models involving branching scenarios. For these processes, a notion of the spectrum degeneracy that takes into account the topology of the graph is introduced. The paper suggests sufficient conditions of uniqueness of extrapolation and recovery from the observations on a single branch. This also implies an analog of sampling theorem for branching processes, i.e., criterions of their recovery from a set of equidistant samples, as well as from a set of equidistant samples from a single branch.

cs.IT↗
arXiv2021

Limited memory predictors based on polynomial approximation of periodic exponents

The paper presents transfer functions for limited memory time-invariant linear integral predictors for continuous time processes such that the corresponding predicting kernels have bounded support. It is shown that processes with exponentially decaying Fourier transforms are predictable with these predictors in some weak sense, meaning that convolution integrals over the future times can be approximated by causal convolutions over past times. For a given predicting horizon, the predictors are based on polynomial approximation of a periodic exponent (complex sinusoid) in a weighted $L_2$-space.

cs.IT↗
arXiv2021

Regularity of complexified hyperbolic equations with integral conditions

This paper considers hyperbolic wave equations with non-local in time conditions involving integrals with respect to time. It is shown that regularity of the solution can be achieved for complexified problem with integral conditions involving harmonic complex exponential weights. The paper establishes existence, uniqueness, and a regularity of the solutions.

math.AP↗
arXiv2020

On the fractional stochastic integration for random non-smooth integrands

The paper suggests a way of stochastic integration of random integrands with respect to fractional Brownian motion with the Hurst parameter H> 1/2. The integral is defined initially on the processes that are "piecewise" predictable on a short horizon. Then the integral is extended on a wide class of square integrable adapted random processes. This class is described via a mild restriction on the growth rate of the conditional mean square error for the forecast on an arbitrarily short horizon given current observations; differentiability of Hölder property of any kind or degree is not required for the integrand. The suggested integration can be interpreted as foresighted integration for integrands featuring corresponding restrictions on the forecasting error. This integration is based on Itô's integration and does not involve Malliavin calculus or Wick products. In addition, it is shown that these stochastic integrals depend continuously on H at H=1/2+0.

math.PR↗
arXiv2020

On recoverability of discrete time signals from sparse observations

The paper investigates recoverability of discrete time signals represented by infinite sequences from incomplte observations. It is shown that there exist wide classes of signals that are everywhere dense in the space of square-summable signals and such that signals from these classes feature robust linear recoverability of their finite traces under very mild restrictions on the location of the observed data. In particular, the case arbitrarily sparse and non-periodic subsequences of observations are not excluded.

cs.IT↗
arXiv2020

Stochastic control problems and HJB equations with excluded parameters of random inputs

This paper introduces a new type of second order stochastic backward Hamilton-Jacobi-Bellman (HJB) equations for optimal stochastic control problems with a currently observable but non-predicable parameter process, in addition to the driving Brownian motion. The main feature of this HJB equation is that it excludes specifications of the parameter process which dynamics can be unspecified or unknown. This allows to reduce the dimension of the state space. The paper considers the case of control dependent diffusion coefficients and fully nonlinear HJB equations under so-called Cordes conditions.

math.OC↗
arXiv2020

On recovering parabolic diffusions from their time-averages

The paper study a possibility to recover a parabolic diffusion from its time-average when the values at the initial time are unknown. This problem can be reformulated as a new boundary value problem where a Cauchy condition is replaced by a prescribed time-average of the solution. It is shown that this new problem is well-posed in certain classes of solutions. The paper establishes existence, uniqueness, and a regularity of the solution for this new problem and its modifications, including problems with singled out terminal values.

math.AP↗
arXiv2020

On linear weak predictability with single point spectrum degeneracy

The paper studies properties of continuous time processes with spectrum degeneracy at a single point where their Fourier transforms vanish with a certain rate. It appears that these processes are linearly predictable in some weak sense, meaning that convolution integrals over future times can be approximated by causal convolutions over past times. The corresponding predicting kernels are time invariant, and they are presented explicitly in the frequency domain via their transfer functions. These predictors are "universal" meaning that they do not require to know details of the spectrum of the underlying processes; the same predictor can be used for the entire class of processes with a single point spectrum degeneracy. The predictors feature some robustness with respect to noise contamination.

cs.IT↗
arXiv2019

Optimal energy storing and selling in continuous time stochastic multi-battery setting

The paper suggests a new stochastic model for energy producing, dispatching, and storing in the multi-battery setting that takes into account the topology of the system of the links between the batteries, the transmission and storage losses, and requirements for special regimes for batteries charging and discharging helping to prolong batteries life. For this model, the problem of optimal energy storing and dispatching is considered and solved using dynamic programming and duality methods.

math.OC↗
arXiv2019

On recovering of solutions of Schrödinger equations from their time averages

The paper study a possibility to recover solutions of Schrödinger equations from its time-averages in the setting where the values at the initial time are unknown. This problem can be reformulated as a new boundary value problem where a Cauchy condition is replaced by a prescribed time-average of the solution. It is shown that this new problem is well-posed in certain classes of solutions. The paper establishes existence, uniqueness, and a regularity of the solution for this new problem.

math-ph↗