Diffusion-type Processes Research Articles

STRATEGIC MANAGEMENT OF THE PORTFOLIO OF FINANCIAL ASSET

Until now, three approaches to asset portfolio management have been used. The first approach is the classic one, based on the "Efficient Market Hypothesis" (EMH). The second and more modern approach is related to the "Fractal Market Hypothesis" (FMH). Modern economic practice is characterized by the presence of structurally unstable markets, included as nodes in the network of the world economy, which functions in real time. The structure of the available financial instruments is heterogeneous and non-Markovian processes arise in them. The third approach is the formation of a dynamic strategy of investment management of the asset portfolio. Due to the complex structure of the modern global financial market, the heterogeneous structure of available financial instruments and traders using different approaches and time horizons, forecasts, as a rule, require a large number of observations, work poorly at the edges of bifurcations, and do not have a computer model that could build forecasts in real time. In these structures, slow diffusion-type processes with a memory phenomenon occur, i.e., non-Markovian processes. Therefore, the formation of a dynamic strategy using modern methods of mathematical and computer modeling is very promising.
 Real financial data is expressed in rational numbers. Computational models were developed on the basis of classical substance analysis. Therefore, p-adic analysis methods are increasingly used in financial mathematical modeling. These methods are used, in particular, in the construction of neural networks, cellular automata, and percolation models. The paper seems to have taken the first step towards building a "synthetic" model of dynamic asset portfolio management. The model has the form of a differential equation in fractional derivatives obtained using the so-called interbasin kinetics method. A general form of the energy market model is also offered, as one of the specific, especially nowadays, markets.

The Study of Three-Dimensional Fluid Flow Regimes on the Basis of the Exact Solution to the Convection Equations

We consider the evaporation/condensation process of diffusion type in two-layer fluid flows under the assumption that both viscous heat-conducting incompressible fluids, a liquid and a gas-vapor mixture, have a common interface and occupy an infinite threedimensional rectangular channel. Based on the exact solution to the convection equations, we study the fluid flows arising in the case of a linearly distributed thermal load on the substrate and thermal insulation of the upper and lateral walls of the channel. It is observed that some complex vortex structures are formed under the action of the longitudinal temperature gradient and transversely directed gravity field. For the ethanol–nitrogen system we present the characteristics of hydrodynamic, thermal, and concentration fields, in particular, the profiles of the evaporative mass flow rates and interface tangential stresses, the topology of flows, and the temperature and vapor concentration distributions.

Ergodic switching control for diffusion-type processes

We consider the control of the discrete component $ n_t $ of a switching Markov process $ x_t = ( z_t, n_t) $ when there is a running cost and an immediate cost $ c(i, j) $ for switching $ n_t $ from $ i $ to $ j $. We study the minimization of the ergodic (or long-term average) total cost. Essentially, this paper treats the case where, for $ n_t=n $ fixed, $ z_t $ is a reflected diffusion or a reflected diffusion with jumps, $ n_t $ being, for fixed $ z $, a continuous-time Markov chain. Using the vanishing discount approach, we extend existing results dealing with the situation where $ n_t $ evolves only by the switching control action and the diffusion is non-degenerate. Moreover, we solve the ergodic problem for a class of diffusions which can be degenerate and for an example with absorbing state.

STRATEGIC BRAND PORTFOLIO MANAGEMENT

In the past twenty years or so, three approaches to brand portfolio management strategies have emerged. The first approach is marketing. This approach is associated with building a corporate brand portfolio. The goal is to increase diversified cash flows by entering new market segments. The second approach is related to the competitive strategy of the enterprise. A false portfolio of intellectual property applications is being created. Competitors are expected to spend resources in retaliation. The third approach is the formation of a dynamic strategy for investment portfolio management. Due to the complex structure of the modern global financial market, the heterogeneous structure of available financial instruments and traders using different approaches and time horizons, forecasts, as a rule, require a large number of observations, work poorly in the vicinity of bifurcations and do not have a computer model that could build forecasts in real time. In such structures, slow diffusion-type processes with the phenomenon of memory arise, that is, non-Markov processes. Moreover, such structures can have fractal properties. In this work, it seems to us, the first step has been taken to build a "synthetic" model of dynamic asset portfolio management. By analyzing the data available in the scientific literature, a mathematical model of strategic brand portfolio management is proposed. In view of the above, the model has the form of a differential equation in fractional derivatives. In connection with the risk analysis, two models of fractional entropy are also considered - fractional Kolmogorov-Sinai entropy and fractional Shannon entropy.

Formation of manganese-enriched austenite at abnormally low temperatures for diffusion type processes at “cascade radiation shaking” of Fe–6.35 at.% Mn alloy with accelerated Ar+ (E = 15 keV) ions

Using the methods of Mössbauer spectroscopy and transmission electron microscopy, it was found that short-term (for 4 s) irradiation of cold-rolled foils of the Fe–6.35 at.% Mn composition with a thickness of 25 μm with Ar+ ions (E = 15 keV) with heating to 300–450 °C causes an α → γ phase transformation with the formation of the repeatedly enriched with manganese austenite (from 23.8 to 38.0 at.%). The estimates made do not allow us to explain these processes by the phenomena of thermal and radiation-enhanced diffusion. As an explanation, the role of nanoscale dynamic effects associated with shaking the medium by powerful post-cascade solitary waves is considered. Such waves, becoming undamped in metastable media and playing, in a certain sense, the role of temperature, cause a short-term giant increase in the mobility of atoms and the transition of these media to a state with minimum free energy. The concentrations of Mn c1 and c2 in the α-and γ-phases, calculated on the basis of decoding the Mössbauer spectra for irradiation temperatures 311, 378 and 449 °C, correspond quite well to the results of extrapolation of the lines of the equilibrium phase diagram to the low temperature region.

On the Integral of Diffusion Process on Interval with Unattainable Edges Boundaries: Semi-Markov Approach

A one-dimension semi-Markov process of diffusion type is considered. A range of values of this process is an open finite interval. It is supposed to have unattainable edges. The integral of this process as a function of time is studied. The invariance principle for this integral is proved.

The law of iterated logarithm for the estimations of diffusion-type processes

This paper mainly discusses the asymptotic behaviours on the lasso-type estimators for diffusion-type processes with a small noise. By constructing the objective function on the estimation, in view of convexity argument, it is proved that the estimator for different values of γ satisfies the iterated logarithm law. The result also presents the exponential convergence principle for the estimator converging to the true value.

On Unattainable Boundary of a Diffusion Process Range: Semi-Markov Approach

One-dimensional homogeneous semi-Markov processes of diffusion type are considered. The transition function of such a process satisfies an ordinary second-order differential equation. It is supposed that the process does not break and has no intervals of constancy. Under these conditions, the Dirichlet problem has a solution on any finite interval. This solution is presented in explicit form in terms of solutions having values 1 and 0 at the boundaries of the interval. A criterion for the left-hand boundary of the interval to be unattainable is derived, and for the corresponding values 0 and 1, a criterion for the right-hand boundary of the interval to be unattainable is derived. This criterion applied to a diffusion process follows from known formulas which are derived by considerably complex methods of the stochastic differential equations theory.

On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes

We obtain closed-form expressions for the value of the joint Laplace transform of the running maximum and minimum of a diffusion-type process stopped at the first time at which the associated drawdown or drawup process hits a constant level before an independent exponential random time. It is assumed that the coefficients of the diffusion-type process are regular functions of the current values of its running maximum and minimum. The proof is based on the solution to the equivalent inhomogeneous ordinary differential boundary-value problem and the application of the normal-reflection conditions for the value function at the edges of the state space of the resulting three-dimensional Markov process. The result is related to the computation of probability characteristics of the take-profit and stop-loss values of a market trader during a given time period.

Diffusion models with accumulated characteristics when operating buildings

Currently, the most common method of describing the evolution in time of rapidly changing processes that occur during the construction and operation of buildings from different influences is stochastic equations, constructed using the Ito`s stochastic differential. For equations of this kind problems of estimating the parameters of their coefficients, building exact and approximate solutions, describing of asymptotic behavior of their solutions are considered. For all importance of such results, they, more often than not, turn out to be unsufficient for developing practical recommendations in the process of using financial instruments. In connection with the above, the question arises of using the available information about the previous evolution of the process for making decisions by analogy with already known situations (the case - study method). Taking into account the practical needs of the construction industry, it is important to develop methods that allow to establish analogies between the behavior of the building of changing the price indicator on the current time interval and some time period from the previous history its evolution. Development of innovative method for comparing the behavior of a random process of diffusion type, associated with the evolution of market indicators, at various time intervals. The methods of the mathematical theory of diffusion, stochastic differential equations, as well as their generalization to the case of locally infinitely divisible processes are used. A generalizations approach to the concept of a diffusion process and the corresponding form of a stochastic differential equation proposed. The concept of the described approach to stochastic processes of this type has been developed for analyzing the possible use of financial instruments in the evolution of market indicators. An innovative technology has been created for teaching the use of financial instruments, such as futures, forwards, options, various swaps and others. Observations on the evolution over time of the price indicator makes it possible to determine and use previously accumulated experience concerning periods of similar evolution. The developed approach allows to classify the behavior of random process of building operation at time intervals.

Adsorption and Capillary Condensation-Induced Imbibition in Nanoporous Media.

Multiphase flow phenomena in nanoporous media are encountered in many science and engineering applications. Shales, for example, possessing complex nanopore networks, have considerable importance as source rocks for unconventional oil and gas production and as low-permeability seals for geologic carbon sequestration or nuclear waste disposal. This study presents a theoretical investigation of the processes controlling adsorption, capillary condensation, and imbibition in such nanoporous media, with a particular focus on understanding the effects of fluid-fluid and fluid-pore wall interaction forces in the interconnected nanopore space. Building on a new theoretical framework, we developed a numerical model for the multiphase nanoporous flow and tested it against water vapor uptake measurements conducted on a shale core sample. The model, which is based on the density functional approach, explicitly includes the relevant interaction forces among fluids and solids while allowing for a continuum representation of the porous medium. The experimental data include gravimetrically measured mass changes in an initially dry core sample exposed to varying levels of relative humidity, starting with a low relative humidity (rh = 0.31) followed by a period of a higher relative humidity (rh = 0.81). During this process, water vapor uptake in the dry core is recorded as a function of time. Our model suggests that, under low rh conditions, the flow within the shale sample is controlled by adsorption- and diffusion-type processes. After increasing the rh to 0.81, the uptake of water vapor becomes more significant, and according to our model, this can be explained by capillary condensation followed by immiscible displacement in the core sample. It appears that strong fluid-pore wall attractive forces cause condensation near the inlet, which then induces water imbibition further into sample.

Hybrid estimators for stochastic differential equations from reduced data

We treat parametric inference for unknown parameters of stochastic differential equations from discrete observations from the viewpoint of computational cost. Following Kamatani et al. (Bull Inf Cybern 48:19–35, 2016) and Kaino and Uchida (Hybrid estimators for ergodic diffusion processes from thinned data, 2018), we present the asymptotic results of the multi-step estimators with the initial Bayes type estimators for both ergodic and non-ergodic diffusion type processes. The initial Bayes type estimators are constructed by means of both the reduced data and the thinned data obtained from the full data. Some examples and simulation results are also given.

ON SOME FUNCTIONALS OF THE FIRST PASSAGE TIMES IN MODELS WITH SWITCHING STOCHASTIC VOLATILITY

We compute some functionals related to the joint generalized Laplace transforms of the first times at which two-dimensional diffusion-type Markov processes exit half strips. It is assumed that the state space components are driven by constantly correlated Brownian motions and the dynamics of the coefficients are described by a continuous-time Markov chain. The method of proof is based on the solutions of the equivalent boundary-value problems for systems of elliptic-type partial differential equations for the associated value functions. The results are illustrated on several two-dimensional continuous mean-reverting or diverting models of switching stochastic volatility.

A correction method for dynamic model calculations using observational data and its application in oceanography

A new data assimilation method for the correction of model calculations is developed and applied. The method is based on the least resistance principle and uses the theory of diffusion-type stochastic processes and stochastic differential equations. Application of the method requires solving a system of linear equations that is derived from this principle. The system can be considered as a generalization of the well-known Kalman scheme taking the model’s dynamics into account. The method is applied to the numerical experiments with the HYbrid Coordinate Ocean Model (HYCOM) and Archiving, Validating, and Interpolating Satellite Ocean (AVISO) data for the Atlantic. The skill of the method is assessed using the results of the experiments. The model’s output is compared with the twin experiments, namely, the model calculations without assimilation, which confirms the consistency and robustness of the proposed method.

Evidential inference for diffusion-type processes

This article analyses diffusion-type processes from a new point-of-view. Consider two statistical hypotheses on a diffusion process. We do not use a classical test to reject or accept one hypothesis using the Neyman–Pearson procedure and do not involve Bayesian approach. As an alternative, we propose using a likelihood paradigm to characterizing the statistical evidence in support of these hypotheses. The method is based on evidential inference introduced and described by Royall [Royall R. Statistical evidence: a likelihood paradigm. London: Chapman and Hall; 1997]. In this paper, we extend the theory of Royall to the case when data are observations from a diffusion-type process instead of iid observations. The empirical distribution of likelihood ratio is used to formulate the probability of strong, misleading and weak evidences. Since the strength of evidence can be affected by the sampling characteristics, we present a simulation study that demonstrates these effects. Also we try to control misleading evidence and reduce them by adjusting these characteristics. As an illustration, we apply the method to the Microsoft stock prices.

Rate-optimal tests for jumps in diffusion processes

Suppose one has a sample of high-frequency intraday discrete observations of a continuous time random process, such as foreign exchange rates and stock prices, and wants to test for the presence of jumps in the process. We show that the power of any test of this hypothesis depends on the frequency of observation. In particular, if the process is observed at intervals of length $$1/n$$ and the instantaneous volatility of the process is given by $$ \sigma _{t}$$ , we show that at best one can detect jumps of height no smaller than $$\sigma _{t}\sqrt{2\log (n)/n}$$ . We present a new test which achieves this rate for diffusion-type processes, and examine its finite-sample properties using simulations.

Asymptotics for functionals of self-normalized residuals of discretely observed stochastic processes

The purpose of this paper is to derive the stochastic expansion of self-normalized-residual functionals stemming from a class of diffusion type processes observed at high frequency, where total observing period may or may not tend to infinity. The result enables us to construct some explicit statistics for goodness of fit tests, consistent against “presence of a jump component” and “diffusion-coefficient misspecification”; then, the acceptance of the null hypothesis may serve as a collateral evidence for using the correctly specified diffusion type model. Especially, our asymptotic result clarifies how to remove the bias caused by plugging in a diffusion-coefficient estimator.

A Kinetic Model for Gas Mixtures Based on a Fokker-Planck Equation

We present a generalized nonlinear Fokker–Planck equation, which describes the dynamics of rarefied monatomic gas mixture flows. The devised kinetic model leads to correct transfer of energy and momentum between gas species and to consistent evolution of molecular stresses and heat fluxes with respect to the generalized Boltzmann equation. Thus, the correct diffusion coefficient together with the mixture viscosity and mixture heat conductivity coefficients are obtained. The strength of the presented model lies on the computational efficiency, which is due to the fact that the resulting stochastic processes are continuous in time. Therefore, unlike in Direct Simulation Monte Carlo (DSMC), here simulated particles do not collide with each other, but move along independent continuous stochastic paths. Another aspect of the new Fokker-Planck model equation is that here the effect of collisions is described via drift and diffusion type processes. Accordingly, a scheme can be derived for which the time step size limitation of the corresponding numerical simulation becomes independent of the Knudsen number. Consequently, this leads to more efficient simulations, especially in low or intermediate Knudsen numbers. Results are presented for helium-argon mixture in a one dimensional geometry. The calculated mixture viscosity is found to be in accordance with experimental data, which reveals the accuracy and relevance of the approach.

Probabilistic whereabouts of the "quantum potential"

We review major appearances of the functional expression ±Δρ1/2/ρ1/2 in the theory of diffusion-type processes and in quantum mechanically supported dynamical scenarios. Attention is paid to various manifestations of "pressure" terms and their meaning.

Subordinated [formula omitted]-stable Ornstein–Uhlenbeck process as a tool for financial data description

The classical financial models are based on the standard Brownian diffusion-type processes. However, in the exhibition of some real market data (like interest or exchange rates) we observe characteristic periods of constant values. Moreover, in the case of financial data, the assumption of normality is often unsatisfied. In such cases the popular Vasiček model, that is a mathematical system describing the evolution of interest rates based on the Ornstein–Uhlenbeck process, seems not to be applicable. Therefore, we propose an alternative approach based on a combination of the popular Ornstein–Uhlenbeck process with a stable distribution and subdiffusion systems that demonstrate such characteristic behavior. The probability density function of the proposed process can be described by a Fokker–Planck type equation and therefore it can be examined as an extension of the basic Ornstein–Uhlenbeck model. In this paper, we propose the parameters’ estimation method and calibrate the subordinated Vasiček model to the interest rate data.