We compute second order backward mean derivatives for diffusion type processes. To do this we first study derivatives of such type for Itoˆ diffusion processes and then go to the non-markovian case of diffusion type processes. Transition from Itoˆ diffusion processes to diffusion type processes is carried out using Malliavin derivatives.
For an abstract linear Sobolev-type equation with a degenerate operator acting on the derivative, the problem of the existence of positive solutions to the initial-final problem is investigated. A distinctive feature of such equations is the ill-posedness of the classical Cauchy problem, caused by the degeneracy of the operator acting on the time derivative. The approach is based on a synthesis of the theory of positive operator semigroups and the theory of degenerate holomorphic resolving semigroups. Sufficient conditions ensuring the existence of a positive solution are established, and the solution to the problem is obtained in explicit form.
T -generalized stochastic bases {Qν τ , 0 ≤ τ ≤ ν ≤ ∞}, where τ and ν are stopping times, were axiomatically introduce. If τ and ν are constant stopping times, we obtain the notion from [1]. We discribe some important properties of T -generalized stochastic bases and of T -generalized martingales. The article explains the rationale for introducing generalized stochastic bases.
We analyze experimental isolines of a free liquid surface in a turbulent wave regime by means of inverse Loewner evolution. The isolines were extracted from diffuse-photography images and divided into chordlength-normalized fragments. For seven experiments containing 525000 contour points and 1050 fragments, the reconstructed driving functions show a nonzero mean drift and a variance growth that is not Brownian. The empirical variance is described by the power law Var(Wt) ∼ t α with α = 1.53±0.09, giving H = 0.763 ± 0.045. The observed isolines are therefore not classical SLE curves. They are more accurately described as Loewner curves with correlated and drifting driving functions.
Screened Poisson equation is a type of elliptic equation that describes stationary systems. Numerical modelling solves it accurately and enables computational analysis. Despite the growth of computing power, numerical analysis of complex mathematical problems still requires development of optimal methods. Aim of this study is to develop asymptotically optimal method for analysis of mixed boundary value problem for screened Poisson equation in three-dimensional domains with complex geometry. The problem in domain with complex geometry is reduced to problem in simpler domain with fictitious domain method, and then discretized by system of linear algebraic equations. Method of iterative extensions is used to solve the system with optimal asymptotic in terms of number of operations. Algorithm of this method is presented in the paper, programmed with C++, and tested. Developed method has linear complexity in terms of number of operations and can be used for effective numerical analysis.
In this paper we study a Leontief-type inclusion with symmetric first-order mean derivative whose matrix pencil is regular and satisfies the rank-degree condition. For this type of inclusion, we prove an analogue of the well-known Filippov lemma.
This paper presents a new asymptotic expansion for the probability density function of normalized sums of independent and identically distributed random variables in the central limit theorem. By establishing sharp bounds for the remainder term, we obtain explicit estimates that avoid the use of Zolotarev’s ideal metric. The theoretical results are validated through a numerical example with an exponential model distribution, demonstrating a significant improvement in accuracy compared to previously known bounds.
The article presents a method for constructing new Fejer pairs from adjoint probability densities using the Khinchin transformation. Their main properties are described, examples of constructing such Fejer pairs for the main positive definite probability densities are given, and relationships between them are found. The article shows the connection between the probability error function and the characteristic function from such a pair.
Motivated by markets in which new entrants challenge established monopolies, we study the problem of determining an optimal switching time for consumers facing competing stochastic price dynamics. In the particular case of the sequences of prices—or premiums—being sub-martingales, an optimal stopping time is described by a simple relation between the increasing processes of the Doob decompositions of these sub-martingales. We present a statistical methodology allowing the application of the results including a Bayesian approach for the case of scarce data. We present a simulation study in the case of average linear growth of the prices increments and we develop two efficient estimation procedures of the parameters of these models allowing accurate estimation of the optimal stopping time. In the particular cases presented, the simulated smallest optimal stopping time coincides with the optimal stopping time studied.
We study the Cauchy problem for a fractional stochastic evolution equation driven by a cylindrical Wiener process, in which the time derivative is understood in the Caputo sense of order q ∈ (0, 1) and the noise enters through a Riemann–Liouville fractional integral. After collecting the required notions on progressively measurable processes, martingales and Hilbert–Schmidt operators, we establish regularity properties of the fractional Itˆo integral and give sufficient conditions for its existence, together with a Burkholder-type inequality. The solution operator families generated by the Mittag-Leffler function are used to formulate the notion of a mild solution and then of a martingale solution. The main result asserts the existence of a martingale solution of the problem under compactness and measurability assumptions on the coefficients.
The paper considers the problem of stabilizing the solutions of the deterministic and stochastic Wentzell equations, which describe the evoluation of a liquid in a circle and on its boundary. The authors address the issue of exponential stability and instability of the deterministic Wentzell equations solutions. They consider different signs of the parameters that describe the medium and the properties of the liquid. The instability gives rise to solving the problem of stabilization using a feedback loop. The obtained results are used in the stochastic Wentzell equations. The Nelson–Gliklikh derivative is considered, and a stochastic process is a solution.
n this paper, we introduced a new parameter, known as reverse Sombor coindex. Also, we obgtained an exact expressions for the reverse Sombor index and reverse Sombor coindex for few standard graphs like paths, cycles, complete graphs, star graphs, wheel graphs, and complete bipartite graphs. Furthermore, several bounds are established with regard to classical graph parameters and wellknown topological descriptors such as the Zagreb indices and forgotten index. Bounds for these descriptors under Cartesian product and composition of graphs are also derived. In addition, the chemical applicability of revrese Sombor index and the reverse Sombor coindex are examined with the help of hetero-atom containing molecules and polycyclic aromatic hydrocarbons (PAHs) and are compared with Sombor index, first Zagreb index, and forgotten index.. The obtained correlation coefficients and coefficients of determination indicate strong linear relationships between the considered descriptors and total 𝜋-electron energy. In particular, the reverse Sombor coindex exhibited the highest predictive performance, yielding correlation coefficients of R=0.9921 for heteroatomcontaining molecules and R=0.9896 for polycyclic aromatic hydrocarbons (PAHs). These results demonstrate the usefulness of reverse degree-based descriptors in chemical graph theory, QSPR/QSAR studies, and molecular structure analysis.
In this paper we focus on the asymptotic analysis of a stochastic HTLV-I infection model using Black-Karasinski process. As a first step, we start by mentioning that there is a unique global solution to the said stochastic model for any initial value. Then we find out an appropriate hypothetical framework leading to the existence of an ergodic stationary distribution. After that, we provide certain sufficient condition for the disease’s extinction.Finaly, we present some numerical simulation example to back up our theoretical outcomes.
The present investigation examines the stagnation point flow of a Casson hybrid nanofluid over a permeable, stretchable surface, accounting for heat sink/source and thermal radiation effects. The flow model incorporates unsteady flow, permeable walls, and Casson fluid rheology, and the hybrid nanofluid is prepared by mixing Al2O3 and Cu nanoparticles in engine oil. A set of nonlinear ordinary differential equations is solved using the Runge-Kutta-Felhberg method. Particular emphasis is placed on the flow behavior in the front and rear stagnation point regions. The results show that, for the front stagnation point flow, the velocity profile increases with higher values of the Casson parameter, whereas, for the rear stagnation point flow, the reverse-flow structure becomes more prominent as the Casson parameter increases. Furthermore, thermal radiation and heat generation significantly increase the temperature distribution and thickness of the thermal boundary layer. The comparative analysis shows that the thermal boundary layer is thicker in the rear stagnation point region than in the front stagnation point region. The outcomes provide valuable data on momentum and heat transfer mechanisms of hybrid nanofluids and can be used for thermal energy systems, polymer extrusion processes, coating technology, lubrication devices, and advanced cooling applications.
We study interval-valued fuzzy weighted graphs whose edge memberships are known only through closed intervals and investigate how this uncertainty propagates to the Laplacian spectrum. For every admissible realization we define the fuzzy weighted Laplacian and prove that the edgeto-Laplacian map is monotone in the Loewner order. Consequently the k-th Laplacian eigenvalue is confined to the exact interval [λk(L−), λk(L+)] generated by the lower and upper endpoint graphs. We derive robust connectivity criteria, midpoint perturbation bounds of Weyl type, Gershgorin-type global enclosures, and explicit formulas for homogeneous interval uncertainty on paths, cycles, stars, and complete graphs. Numerical examples show that exact spectral widths can be substantially smaller than generic perturbation radii. The results place interval-valued fuzzy graph Laplacians into a rigorous spectral-uncertainty framework.
Enhancement of heat transfer efficiency of porous slider systems is of great importance for a wide range of advanced thermal management systems, such as microelectromechanical systems, lubrication technologies, cooling systems, and precision engineering systems. The heat transfer characteristics of nanofluid flow in a porous slider under the combined impact of thermal radiation, heat source/sink and magnetic field is analyzed in the present study. The effect of thermal conductivity is analyzed accurately by comparatively evaluating the Hamilton-Crosser and the Xue thermal conductivity model. Using the appropriate similarity variables the partial differential equations (PDEs) governing the flow and heat transfer processes are transformed to a set of nonlinear ordinary differential equations (ODEs). The subsequent ODEs are solved using the Runge-Kutta-Fehlberg (RKF-45) method to obtain the numerical solutions. Furthermore, Taguchi optimization technique is used to identify the critical parameters affecting the heat transfer rate (H-TR) and to find the optimum operating conditions. The results of the statistical analysis, with coefficient of determination 97.80%, further support the enhanced forecasting ability of the developed optimization model. The maximum H-TR of 1.41595 is obtained at Rd = 2, Q = 0.8, Re = 20, and ϕ = 0.04, where the radiation parameter has the most significant impact (50.43%) on the thermal enhancement and the nanoparticle volume fraction has minimal effect (2.27%).
The stress of a graph is the number of geodesics (shortest paths) that pass through each vertex. A topological index of a chemical structure (graph) is a number that correlates the chemical structure with its chemical reactivity or physical properties. In this paper, we introduce a new topological index for graphs called the second hyper-Gourava stress index, which is defined using the stresses of vertices. Further, we establish several inequalities, prove related results, and compute the second hyper-Gourava stress index for some standard graphs. Similarly, we establish the importance of the second hyper-Gourava stress index in predicting the physicochemical properties of lower alkanes.
Bitcoin price movements are highly volatile and nonlinear over time. Traditional GARCH models are widely used for volatility modeling, but they may not adequately represent nonlinear effects of exogenous variables. In this study, we develop a weighted semiparametric EGARCH model for forecasting Bitcoin volatility by combining the parametric conditional variance from an EGARCH model with a nonlinear crude oil market volatility component estimated using the Nadaraya–Watson kernel estimator. The empirical analysis uses daily Bitcoin and crude oil price data from 2017 to 2026. The forecasting performance is compared with GARCH(1,1), EGARCH(1,1), and semiparametric EGARCH models incorporating crude oil market volatility. The root mean square error, mean absolute error, quasi-likelihood loss, and Diebold–Mariano test are computed to assess the out-of-sample forecast accuracy for various training-testing split designs. Empirical results indicate that the proposed weighted semiparametric EGARCH model achieves higher forecasting accuracy and robustness for forecasting Bitcoin volatility.
