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Nikolai Dokuchaev

Publications and source records attributed to Nikolai Dokuchaev.

At least 91 records · Page 5Linked to original sources
arXiv2010

Representation of functionals of Ito processes and their first exit times

The representation theorem is obtained for functionals of non-Markov processes and their first exit times from bounded domains. These functionals are represented via solutions of backward parabolic Ito equations. As an example of applications, analogs of forward Kolmogorov equations are derived for conditional probability density functions of Ito processes being killed on the boundary. In addition, a maximum principle and a contraction property are established for SPDEs in bounded domains.

math.PR↗
arXiv2010

Duality and semi-group property for backward parabolic Ito equations

We study existence, uniqueness, semi-group property, and a priori estimates for solutions for backward parabolic Ito equations in domains with boundary. We study also duality between forward and backward equations. The semi-group for backward equations is established in the form of some anti-causality. The novelty is that the semi-group property involves the diffusion term that is a part of the solution.

math.PR↗
arXiv2009

Mutual Fund Theorem for continuous time markets with random coefficients

We study the optimal investment problem for a continuous time incomplete market model such that the risk-free rate, the appreciation rates and the volatility of the stocks are all random; they are assumed to be independent from the driving Brownian motion, and they are supposed to be currently observable. It is shown that some weakened version of Mutual Fund Theorem holds for this market for general class of utilities; more precisely, it is shown that the supremum of expected utilities can be achieved on a sequence of strategies with a certain distribution of risky assets that does not depend on risk preferences described by different utilities.

q-fin.PM↗
arXiv2009

On identity theorem for real functions

Identity theorem for analytic complex functions says that a function is uniquely defined by its values on a set that contains a density point. The paper presents sufficient conditions for classes of real analytic functions that ensures similar property.

math.FA↗
arXiv2009

Predictability on finite horizon for processes with exponential decrease of energy on higher frequencies

The paper presents sufficient conditions of predictability for continuous time processes in deterministic setting. We found that processes with exponential decay on energy for higher frequencies are predictable in some weak sense on some finite time horizon defined by the rate of decay. Moreover, this predictability can be achieved uniformly over classes of processes. Some explicit formulas for predictors are suggested.

math.OC↗
arXiv2009

Explicit formulas for currents at branching long lines and for maximum of current amplitudes

A system of 'telegrapher's' equations for a number of long lines joined into a network is studied. Explicit formulas for Fourier transforms of current and voltage are derived. These formulas are very suitable for computer application as well as for the analytical study of processes o networks. As an example, the availability of formulas aids the derivation of explicit formulas for maxima of current amplitude over the given class of admissible external influences. These values may be used to indicate the characteristic of network robustness to excess voltage or electromagnetic impulse. The approach is based on the operational solution already proposed by the author for more general partial differential equations on graphs.

physics.class-ph↗
arXiv2008

Predictability of band-limited, high-frequency, and mixed processes in the presence of ideal low-pass filters

Pathwise predictability of continuous time processes is studied in deterministic setting. We discuss uniform prediction in some weak sense with respect to certain classes of inputs. More precisely, we study possibility of approximation of convolution integrals over future time by integrals over past time. We found that all band-limited processes are predictable in this sense, as well as high-frequency processes with zero energy at low frequencies. It follows that a process of mixed type still can be predicted if an ideal low-pass filter exists for this process.

math.OC↗
arXiv2008

Universal estimates for parabolic equations and applications for non-linear and non-local problems

We obtain some "universal" estimates for $L_2$-norm of the solution of a parabolic equation via a weighted version of $H^{-1}$-norm of the free term. More precisely, we found the limit upper estimate that can be achieved by transformation of the equation by adding a constant to the zero order coefficient. The inverse matrix of the higher order coefficients of the parabolic equation is included into the weight for the $H^{-1}$-norm. The constant in the estimate obtained is independent from the choice of the dimension, domain, and the coefficients of the parabolic equation, it is why it can be called an universal estimate. As an example of applications, we found an asymptotic upper estimate for the norm of the solution at initial time. As an another example, we established existence and regularity for non-linear and non-local problems.

math.AP↗
arXiv2008

Cordes conditions and some alternatives for parabolic equations and discontinuous diffusion

The paper considers parabolic equations in non-divergent form with discontinuous coefficients at higher derivatives. Their investigation is most complicated because, in general, in the case of discontinuous coefficients, the uniqueness of a solution for nonlinear parabolic or elliptic equations can fail, and there is no a priory estimate for partial derivatives of a solution. In this paper, existence and regularity results are obtained under some Cordes type restrictions on the coefficients. The results are applied to diffusion processes.

math.AP↗
arXiv2008

Universal estimate of the gradient for parabolic equations

We suggest a modification of the estimate for weighted Sobolev norms of solutions of parabolic equations such that the matrix of the higher order coefficients is included into the weight for the gradient. More precisely, we found the upper limit estimate that can be achieved by variations of the zero order coefficient. As an example of applications, an asymptotic estimate was obtained for the gradient at initial time. The constant in the estimates is the same for all possible choices of the dimension, domain, time horizon, and the coefficients of the parabolic equation. As an another example of application, existence and regularity results are obtained for parabolic equations with time delay for the gradient.

math.AP↗
arXiv2008

Optimal solution of investment problems via linear parabolic equations generated by Kalman filter

We consider optimal investment problems for a diffusion market model with non-observable random drifts that evolve as an Ito's process. Admissible strategies do not use direct observations of the market parameters, but rather use historical stock prices. For a non-linear problem with a general performance criterion, the optimal portfolio strategy is expressed via the solution of a scalar minimization problem and a linear parabolic equation with coefficients generated by the Kalman filter.

q-fin.PM↗
arXiv2007

Regularity of a inverse problem for generic parabolic equations

The paper studies some inverse boundary value problem for simplest parabolic equations such that the homogenuous Cauchy condition is ill posed at initial time. Some regularity of the solution is established for a wide class of boundary value inputs.

math.AP↗
arXiv2007

Parabolic equations with the second order Cauchy conditions on the boundary

The paper studies some ill-posed boundary value problems on semi-plane for parabolic equations with homogenuous Cauchy condition at initial time and with the second order Cauchy condition on the boundary of the semi-plane. A class of inputs that allows some regularity is suggested and described explicitly in frequency domain. This class is everywhere dense in the space of square integrable functions.

math.AP↗
arXiv2003

On $L_1$-distance between first exit times from two regions

First exit times from regions and their dependence on variations of boundaries are discussed for diffusion processes. The paper presents an estimate of $L_1$-distance between exit times from two regions via expectations of exit times.

math.PR↗
arXiv2002

Maximin setting for investment problems and fixed income management with observable but non-predictable parameters

We study optimal investment problem for a diffusion market consisting of a finite number of risky assets (for example, bonds, stocks and options). Risky assets evolution is described by Ito's equation, and the number of risky assets can be larger than the number of driving Brownian motions. We assume that the risk-free rate, the appreciation rates and the volatility of the stocks are all random; they are not necessary adapted to the driving Brownian motion, and their distributions are unknown, but they are supposed to be currently observable. Admissible strategies are based on current observations of the stock prices and the aforementioned parameters. The optimal investment problem is stated as a problem with a maximin performance criterion. This criterion is to ensure that a strategy is found such that the minimum of utility over all distributions of parameters is maximal. Then the maximin problem is solved for a very general case via solution of a linear parabolic equation.

math.PR↗
arXiv2002

Optimal portfolio selection and compression in an incomplete market

We investigate an optimal investment problem with a general performance criterion which, in particular, includes discontinuous functions. Prices are modeled as diffusions and the market is incomplete. We find an explicit solution for the case of limited diversification of the portfolio, i.e. for the portfolio compression problem. By this we mean that an admissible strategies may include no more than m different stocks concurrently, where m may be less than the total number n of available stocks.

math.PR↗
arXiv2002

Pricing rule based on non-arbitrage arguments for random volatility and volatility smile

We consider a generic market model with a single stock and with random volatility. We assume that there is a number of tradable options for that stock with different strike prices. The paper states the problem of finding a pricing rule that gives Black-Scholes price for at-money options and such that the market is arbitrage free for any number of tradable options, even if there are two Brownian motions only: one drives the stock price, the other drives the volatility process. This problem is reduced to solving a parabolic equation.

math.PR↗