arXiv subjects

Nikolai Dokuchaev

Publications and source records attributed to Nikolai Dokuchaev.

At least 55 records · Page 3Linked to original sources

arXiv2016

On predictability of ultra short AR(1) sequences

This paper addresses short term forecast of ultra short AR(1) sequences (4 to 6 terms only) with a single structural break at an unknown time and of unknown sign and magnitude. As prediction of autoregressive processes requires estimated coefficients, the efficiency of which relies on the large sample properties of the estimator, it is a common perception that prediction is practically impossible for such short series with structural break. However, we obtain a heuristic result that some universal predictors represented in the frequency domain allow certain predictability based on these ultra short sequences. The predictors that we use are universal in a sense that they are not oriented on particular types of autoregressions and do not require explicit modelling of structural break. The shorter the sequence, the better the one-step-ahead forecast performance of the smoothed predicting kernel. If the structural break entails a model parameter switch from negative to positive value, the forecast performance of the smoothed predicting kernel is better than that of the linear predictor that utilize AR(1) coefficient estimated from the ultra short sequence without taking the structural break into account regardless whether the innovation terms in the learning sequences are constructed from independent and identically distributed random Gaussian or Gamma variables, scaled pseudo-uniform variables, or first-order auto-correlated Gaussian process.

stat.ME↗

arXiv2016

On detecting and quantification of randomness for one-sided sequences

The paper studies discrete time processes and their predictability and randomness in deterministic pathwise setting, without using probabilistic assumptions on the ensemble. We suggest some approaches to quantification of randomness based on frequency analysis of two-sided and one-sided sequences. In addition, the paper suggests an extension of the notion of bandlimitiness on one-sided sequences and a procedure allowing to represent an one-sided sequence as a sum of left-bandlimited and predictable sequences and a non-reducible noise.

cs.IT↗

arXiv2016

On Nyquist-Shannon Theorem with one-sided half of sampling sequence

The classical sampling Nyquist-Shannon-Kotelnikov theorem states that a band-limited continuous time function can be uniquely recovered without error from a infinite two-sided sampling series taken with a sufficient frequency. This short note shows that the function can be recovered from any one-sided semi-infinite half of any oversampling series, with the same boundary for admissible frequencies as in the classical theorem.

cs.IT↗

arXiv2015

A smooth component of the fractional Brownian motion and optimal portfolio selection

We consider fractional Brownian motion with the Hurst parameters from (1/2,1). We found that the increment of a fractional Brownian motion can be represented as the sum of a two independent Gaussian processes one of which is smooth in the sense that it is differentiable in mean square. We consider fractional Brownian motion and stochastic integrals generated by the Riemann sums. As an example of applications, this results is used to find an optimal pre-programmed strategy in the mean-variance setting for a Bachelier type market model driven by a fractional Brownian motion.

math.PR↗

arXiv2015

On the no-arbitrage market and continuity in the Hurst parameter

We consider a market with fractional Brownian motion with stochastic integrals generated by the Riemann sums. We found that this market is arbitrage free if admissible strategies that are using observations with an arbitrarily small delay. Moreover, we found that this approach eliminates the discontinuity of the stochastic integrals with respect to the Hurst parameter H at H=1/2.

q-fin.MF↗

arXiv2015

Sub-ideal causal smoothing filters for real sequences

The paper considers causal smoothing of the real sequences, i.e.,discrete time processes in a deterministic setting. A family of causal linear time-invariant filters is suggested. These filters approximate the gain decay for some non-causal smoothing filters with transfer functions vanishing at a point of the unit circle and such that they transfer processes into predictable ones. In this sense, the suggested filters are near-ideal; a faster gain decay would lead to the loss of causality. Applications to predicting algorithms are discussed and illustrated by experiments with forecasting of autoregressions with the coefficients that are deemed to be untraceable.

math.OC↗

arXiv2015

On statistical indistinguishability of complete and incomplete discrete time market models

We investigate the possibility of statistical evaluation of the market completeness for discrete time stock market models. It is known that the market completeness is not a robust property: small random deviations of the coefficients convert a complete market model into a incomplete one. The paper shows that market incompleteness is also non-robust. We show that, for any incomplete market from a wide class of discrete time models, there exists a complete market model with arbitrarily close stock prices. This means that incomplete markets are indistinguishable from the complete markets in the terms of the market statistics.

q-fin.MF↗

arXiv2015

Optimal portfolio with unobservable market parameters and certainty equivalence principle

We consider a multi-stock continuous time incomplete market model with random coefficients. We study the investment problem in the class of strategies which do not use direct observations of the appreciation rates of the stocks, but rather use historical stock prices and an a priory given distribution of the appreciation rates. An explicit solution is found for case of power utilities and for a case when the problem can be embedded to a Markovian setting. Some new estimates and filters for the appreciation rates are given.

q-fin.MF↗

arXiv2015

On strong binomial approximation for stochastic processes and applications for financial modelling

This paper considers binomial approximation of continuous time stochastic processes. It is shown that, under some mild integrability conditions, a process can be approximated in mean square sense and in other strong metrics by binomial processes, i.e., by processes with fixed size binary increments at sampling points. Moreover, this approximation can be causal, i.e., at every time it requires only past historical values of the underlying process. In addition, possibility of approximation of solutions of stochastic differential equations by solutions of ordinary equations with binary noise is established. Some consequences for the financial modelling and options pricing models are discussed.

q-fin.CP↗

arXiv2015

Transmission of a continuous signal via one-bit capacity channel

We study the problem of the transmission of currently observed time variable signals via a channel that is capable of sending a single binary signal only for each measurement of the underlying process. For encoding and decoding, we suggest a modification othe adaptive delta modulation algorithm. This modification ensures tracking of time variable signals. We obtained upper estimates for the error for the case of noiseless transmission.

cs.IT↗

arXiv2014

On asymptotic optimality of Merton's myopic portfolio strategies for discrete time market

This paper studies the properties of discrete time stochastic optimal control problems associated with portfolio selection. We investigate if optimal continuous time strategies can be used effectively for a discrete time market after a straightforward discretization. We found that Merton's strategy approximates the performance of the optimal strategy in a discrete time model with the sufficiently small time steps

q-fin.PM↗

arXiv2014

A First-Order BSPDE for Swing Option Pricing: Classical Solutions

In Bender and Dokuchaev (2013), we studied a control problem related to swing option pricing in a general non-Markovian setting. The main result there shows that the value process of this control problem can be uniquely characterized in terms of a first order backward SPDE and a pathwise differential inclusion. In the present paper we additionally assume that the cashflow process of the swing option is left-continuous in expectation (LCE). Under this assumption we show that the value process is continuously differentiable in the space variable that represents the volume which the holder of the option can still exercise until maturity. This gives rise to an existence and uniqueness result for the corresponding backward SPDE in a classical sense. We also explicitly represent the space derivative of the value process in terms of a nonstandard optimal stopping problem over a subset of predictable stopping times. This representation can be applied to derive a dual minimization problem in terms of martingales.

q-fin.PR↗

arXiv2014

On martingale measures and pricing for continuous bond-stock market with stochastic bond

This papers addresses the stock option pricing problem in a continuous time market model where there are two stochastic tradable assets, and one of them is selected as a numéraire. It is shown that the presence of arbitrarily small stochastic deviations in the evolution of the numéraire process causes significant changes in the market properties. In particular, an equivalent martingale measure is not unique for this market, and there are non-replicable claims. The martingale prices and the hedging error can vary significantly and take extreme values, for some extreme choices of the equivalent martingale measures. Some rational choices of the equivalent martingale measures are suggested and discussed, including implied measures calculated from observed bond prices. This allows to calculate the implied market price of risk process.

q-fin.PR↗

arXiv2014

Causal band-limitness and predictability criterions for one-sided sequences

The paper studies frequency characteristics and predictability of real sequences, i.e., discrete time processes in deterministic setting. We consider band-limitness and predictability of one-sided sequences. We establish predictability of some classes of sequences, and we suggest predictors represented by causal convolution sums over past times.

math.OC↗

arXiv2014

Degenerate backward SPDEs in domains: non-local boundary conditions and applications to finance

Backward stochastic partial differential equations of parabolic type in bounded domains are studied in the setting where the coercivity condition is not necessary satisfied and the equation can be degenerate. Some generalized solutions based on the representation theorem are suggested. In addition to problems with a standard Cauchy condition at the terminal time, problems with special non-local boundary conditions are considered. These non-local conditions connect the terminal value of the solution with a functional over the entire past solution. Uniqueness, solvability and regularity results are obtained. Some applications to portfolio selection problem are considered.

math.PR↗

arXiv2014

The structure of optimal portfolio strategies for continuous time markets

The paper studies problem of continuous time optimal portfolio selection for a incom- plete market diffusion model. It is shown that, under some mild conditions, near optimal strategies for investors with different performance criteria can be constructed using a limited number of fixed processes (mutual funds), for a market with a larger number of available risky stocks. In other words, a dimension reduction is achieved via a relaxed version of the Mutual Fund Theorem.

q-fin.PM↗

arXiv2013

On limit periodicity of discrete time stochastic processes

We consider a discrete time dynamic system described by a difference equation with periodic coefficients and with additive stochastic noise. We investigate the possibility of the periodicity for the solution. In particular, we found sufficient conditions for existence of a periodic process such that the solution converges to it, including almost surely convergence.

math.DS↗

arXiv2013

On almost surely periodic and almost periodic solutions of backward SPDEs

We study linear backward stochastic partial differential equations of parabolic type with special boundary conditions in time. The standard Cauchy condition at the terminal time is replaced by a condition that holds almost surely and mixes the random values of the solution at different times, including the terminal time, initial time and continuously distributed times. Uniqueness, solvability and regularity results for the solutions are obtained. In particular, conditions of existence of periodic in time and "almost periodic" solutions are obtained for backward SPDEs.

math.PR↗