Drift Term Research Articles

Stochastic gradient descent in continuous time for drift identification in multiscale diffusions

We consider the setting of multiscale overdamped Langevin stochastic differential equations, and study the problem of learning the drift function of the homogenized dynamics from continuous-time observations of the multiscale system. We decompose the drift term in a truncated series of basis functions, and employ the stochastic gradient descent in continuous time to infer the coefficients of the expansion. Due to the incompatibility between the multiscale data and the homogenized model, the estimator alone is not able to reconstruct the exact drift. We therefore propose to filter the original trajectory through appropriate kernels and include filtered data in the stochastic differential equation for the estimator, which indeed solves the misspecification issue. Several numerical experiments highlight the accuracy of our approach. Moreover, we show theoretically in a simplified framework the asymptotic unbiasedness of our estimator in the limit of infinite data and when the multiscale parameter describing the fastest scale vanishes.

Long-time behavior of solutions for time-periodic reaction-diffusion equations and applications

This article concerns the asymptotic behavior of solutions for time-periodic reaction-diffusion equations with a drift term in one dimensional space. Assuming that drift term is decaying, we analyze the effect of the decaying rate of the drift term on the propagation speed of solutions. Also we show some applications of our results in high-dimensional domains. For more information see https://ejde.math.txstate.edu/Volumes/2024/69/abstr.html

Stabilization for degenerate equations with drift and small singular term

We consider a degenerate/singular wave equation in lone dimension, with drift and in presence of a leading operator that is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.

An elliptic problem in dimension N with a varying drift term bounded in L N

The present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in L N ( Ω ), with N the dimension of the space. It is known that there exists a unique solution for each of these problems in the Sobolev space H 0 1 ( Ω ). However, because the operators are not coercive, there is no uniform estimate of the solutions in this space. We use some estimates in (J. Differential Equations 258 (2015) 2290–2314), and a regularization obtained by adding a small nonlinear first order term, to pass to the limit in these problems.

Efficiency Investigation of Langevin Monte Carlo Ray Tracing

The main computationally expensive task of realistic computer graphics is the calculation of global illumination. Currently, most of the lighting simulation methods are based on various types of Monte Carlo ray tracing. One of them, the Langevin Monte Carlo ray tracing, generates samples using the time series of a system of the Langevin dynamics. The method seems to be very promising for calculating the global illumination. However, it remains poorly studied, while its analysis could significantly speed up the calculations without losing the quality of the result. In our work, we analyzed the most computationally expensive operations of this method and also conducted the computational experiments demonstrating the contribution of a particular operation to the convergence speed. One of our main conclusions is that the computationally expensive drift term can be dropped because it does not improve convergence. Another important conclution is that the preconditioning matrix makes the greatest contribution to the improvement of convergence. At the same time, calculation of this matrix is not so expensive, because it does not require calculating the gradient of the potential. The results of our study allow to significantly speed up the method.

Langevin dynamics for a heavy particle immersed within a flow of light particles

A micro-hydrodynamics model based on elastic collisions of light point solvent particles with a heavy solute particle is investigated in the setting where the light particles have velocity distribution corresponding to a background flow. Considering a range of stationary background flows and distributions for the solvent particle velocities, the macroscopic Langevin-type description of the behaviour of the heavy particle is derived in the form of a generalized Ornstein–Uhlenbeck process. At leading order, the drift term in this process depends upon both the geometric structure of the background flow and the size of the heavy particle, while both drift and diffusion terms scale with moments of the light particle velocity distribution. Computational methods for simulating the micro-hydrodynamics model are then designed to confirm the theoretical results. To enable long-time calculations, simulations are performed in a frame co-moving with the heavy particle. Efficient methods for sampling the position and velocity distributions of incoming solvent particles at the boundaries of the co-moving frame are derived for a range of distributions of solvent particles. The simulations show good agreement with the theoretical results.

Moderate deviations for a stochastic Schrödinger equation with linear drift

Moderate deviation principle is achieved by the weak convergence approach for a stochastic Schrödinger type equation with linear drift term and noise driven by a [Formula: see text]-Wiener process. The central limit theorem is also shown for the equation to further analyze its asymptotic behavior.

Revised and Generalized Results of Averaging Principles for the Fractional Case

The averaging principle involves approximating the original system with a simpler system whose behavior can be analyzed more easily. Recently, numerous scholars have begun exploring averaging principles for fractional stochastic differential equations. However, many previous studies incorrectly defined the standard form of these equations by placing ε in front of the drift term and ε in front of the diffusion term. This mistake results in incorrect estimates of the convergence rate. In this research work, we explain the correct process for determining the standard form for the fractional case, and we also generalize the result of the averaging principle and the existence and uniqueness of solutions to fractional stochastic delay differential equations in two significant ways. First, we establish the result in Lp space, generalizing the case of p=2. Second, we establish the result using the Caputo–Katugampola operator, which generalizes the results of the Caputo and Caputo–Hadamard derivatives.

On weak and strong solutions of time inhomogeneous Itô’s equations with VMO diffusion and Morrey drift

We prove the existence of weak solutions of Itô’s stochastic time dependent equations with irregular diffusion and drift terms of Morrey spaces. Weak uniqueness (generally conditional) and a conjecture pertaining to strong solutions are also discussed. Our results are new even if the drift term vanishes.

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We investigate the interior regularity to nonlinear subelliptic systems in divergence form with drift term for the case of superquadratic controllable structure conditions in the Heisenberg group. On the basis of a generalization of the A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{A}$\\end{document}-harmonic approximation technique, C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$C^{1}$\\end{document}-regularity is established for horizontal gradients of vector-valued solutions to the subelliptic systems with drift term. Specially, our result is optimal in the sense that in the case of Hölder continuous coefficients we directly attain the optimal Hölder exponent for the horizontal gradients of weak solutions on the regular set.

Gradient flows of generalized relative entropy and functional inequalities on graphs

We derive the discrete forms of the porous medium equation and fast diffusion equation with drift terms on a finite graph as gradient flows of the m-relative entropy, with the global existence of solutions. We show the Łojasiewicz inequality to prove the convergence of solutions and some important functional inequalities. More specifically, the Łojasiewicz inequality holds near the stationary solution with exponent 12, which leads to the exactly exponential convergence rate with a finite trajectory length. It can also be applied to show the Talagrand-type inequality and the log-Sobolev-type inequality directly.

Measuring ESG risk premia with contingent claims

We propose a contingent claims approach for estimating ESG risk premia from market information and market participants' decisions. To this end, we infer the asset value dynamics via the structural model of Merton [1974, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.” Journal of Finance 29: 449–470.] for a large panel of S&P 500 firms using an estimation algorithm that utilizes the information embedded in stock market prices, CDS spreads, and default probabilities. We find a statistically significant relationship between the ESG score and the volatility and drift terms of the asset value process, suggesting that ESG factors are structurally connected to the value of the firm. We establish a mapping between ESG scores and the cost of equity and debt as implied by firm's contingent claims, and derive estimates of the ESG risk premium across different ESG and leverage profiles. In addition, we break down the ESG risk premia by industry, and demonstrate how practitioners can adjust the weighed average cost of capital of ESG laggard firms for valuation and decision making purposes.

System transformation and model-free value iteration algorithms for continuous-time linear quadratic stochastic optimal control problems

In this paper, we investigate a continuous-time linear quadratic stochastic optimal control (LQSOC) problem in an infinite horizon, where diffusion and drift terms of the corresponding stochastic system depend on both state and control variables. In light of the stochastic control theory, this LQSOC problem is reduced to solving a generalised algebraic Riccati equation (GARE). With the help of an existing model-based value iteration (VI) algorithm, we propose two data-driven VI algorithms to solve the GARE. The first one relies on transforming the stochastic system into a deterministic control system first and then solving the LQSOC problem by the data of the deterministic system. Consequently, this algorithm does not need the information of two system coefficients and has a lower algorithm complexity. The second algorithm directly uses the data generated by the stochastic system, and thus it circumvents the requirement of all system coefficients. We also provide convergence proofs of these two data-driven algorithms and validate these algorithms through two simulation examples.

First SOLPS-ITER predictions of the impact of cross-field drifts on divertor and scrape-off layer conditions in double-null configuration of STEP

Introduction of cross-field drifts in SOLPS-ITER simulations of connected double-null plasmas in STEP with the ion ∇B×B drift towards the upper divertors was found to enhance the detachment of the inner divertors with decreased target densities and ion and heat fluxes, while simultaneously complicating the access to detachment in the outer lower divertor by increasing the target temperature and heat loads to levels above the engineering limits. The ∇B×B drift was observed to significantly affect the radial heat transport between the core and the scrape-off layer (SOL), altering the poloidal temperature and pressure profiles and, consequently, the poloidal conductive and convective heat transport in the SOL. As a result, up-down asymmetries of 52:48 and 58:42 biased towards the outer lower and upper inner divertors, respectively, were observed to arise in the unmitigated power entering the divertor regions, breaking the up-down symmetry seen in simulations without the drift terms and contradicting with earlier experimental observations on the low-field side. Moreover, the upstream electron density was found to decrease noticeably in the core and separatrix regions following the activation of the drifts due to an increased share of the neutrals arriving from D2 injections near the upper and lower X-points ionizing already in the private flux regions.

Modeling bitcoin price by using Euler-Maruyama method

<p>In this study, we use Euler-Maruyama method to simulate Bitcoin prices dynamics. We investigate a year-long movement of Bitcoin prices. Daily closing prices were collected over a period starting from May 27th, 2023 and ending on May 27th, 2024. This data provides a comprehensive picture of how Bitcoin behaved on a daily basis throughout that specific year. The Euler-Maruyama method is used as numerical method for solving stochastic differential equations (SDEs). The method involves discretizing the SDE into an iterative process to obtain a simulated price trajectory. The drift term was estimated by the average daily return of Bitcoin prices over the study period. The volatility term was estimated by the standard deviation of daily Bitcoin returns. A Monte Carlo simulation was performed to generate a range of possible price trajectories using the Euler-Maruyama method. The result shows that the Euler-Maruyama method was able to capture the short-term trend of Bitcoin prices effectively. However, the method can also be used to capture the long-term trend of Bitcoin prices.</p>

Stochastic Modeling of Human Civilizationʹs Future under Competing AI Influences

This study explores the potential trajectories of human civilization influenced by the development and competition of advanced artificial intelligence (AI) systems. Using a system of stochastic partial differential equations (SPDEs), we model the probability density function representing the state of civilization in a phase space defined by prosperity and knowledge. The model incorporates diffusion, growth, saturation, and drift terms, alongside stochastic noise to reflect the uncertainties and random fluctuations in AI impacts. The simulation results are visualized in a probability density graph, revealing the likelihood of various outcomes ranging from extinction and regression, to prosperity and technological advancement. The analysis highlights the balanced prospects of human future, with significant probabilities in both positive and negative directions. Positive scenarios suggest potential for increased prosperity and knowledge, emphasizing the importance of effective AI management and international cooperation. Conversely, the notable risks of regression and extinction underline the need for strategic interventions to mitigate adverse impacts. Our findings stress the stochastic nature of future developments and the critical role of adaptive and flexible policies in steering human civilization towards favorable outcomes. This study provides a simple yet comprehensive framework for understanding the complex dynamics at play and underscores the importance of proactive strategies in the age of AI

On the solvability of some systems of integro-differential equations with and without a drift

We prove the existence of solutions for some integro-differential systems containing equations with and without drift terms in the H 2 spaces by virtue of the fixed point technique when the elliptic equations contain second order differential operators with and without Fredholm property, on the whole real line or on a finite interval with periodic boundary conditions. Let us emphasise that the study of the system case is more complicated than of the scalar situation and requires to overcome more cumbersome technicalities.

On neural networks application in integral sliding mode control

In this paper, the combined use of deep neural networks (DNNs) and integral sliding mode (ISM) control is investigated, giving rise to novel DNN-ISM control strategies for nonlinear systems affine in the control law, characterized by fully unknown dynamics and subject to disturbances. In particular, two feed-forward DNNs are used to approximate the drift term and the control effectiveness matrix of the nonlinear systems model. The DNNs weights are tuned online relying on adaptation laws derived from Lyapunov’s stability analysis. The DNNs with their weights tuning mechanism are exploited as building blocks in the design of an ISM control scheme to solve trajectory tracking problems. The proposed framework is then extended to the case in which the system dynamics is subject to time-varying state constraints, producing a second scheme, still of DNN-ISM type. Both control schemes are analyzed, providing interesting theoretical guarantees. Possible solutions to reduce, in practical implementations, the chattering effect, as well as the initial estimation error of the DNNs are also discussed. The proposed DNNs-based control schemes are finally assessed in simulation relying on a realistic model of an anthropomorphic 7-axes robot manipulator.

Tipping points of evolving epidemiological networks: Machine learning-assisted, data-driven effective modeling.

We study the tipping point collective dynamics of an adaptive susceptible-infected-susceptible (SIS) epidemiological network in a data-driven, machine learning-assisted manner. We identify a parameter-dependent effective stochastic differential equation (eSDE) in terms of physically meaningful coarse mean-field variables through a deep-learning ResNet architecture inspired by numerical stochastic integrators. We construct an approximate effective bifurcation diagram based on the identified drift term of the eSDE and contrast it with the mean-field SIS model bifurcation diagram. We observe a subcritical Hopf bifurcation in the evolving network's effective SIS dynamics that causes the tipping point behavior; this takes the form of large amplitude collective oscillations that spontaneously-yet rarely-arise from the neighborhood of a (noisy) stationary state. We study the statistics of these rare events both through repeated brute force simulations and by using established mathematical/computational tools exploiting the right-hand side of the identified SDE. We demonstrate that such a collective SDE can also be identified (and the rare event computations also performed) in terms of data-driven coarse observables, obtained here via manifold learning techniques, in particular, Diffusion Maps. The workflow of our study is straightforwardly applicable to other complex dynamic problems exhibiting tipping point dynamics.

Optimal investment for asset–liability management with delay and partial information under Ornstein–Uhlenbeck process

In this paper, we investigate the optimal investment strategy of asset liability management (ALM) with bounded memory and partial information. Suppose that investors invest their assets in a financial market consisting of a risk-free bond and a risk-free stock, while also taking on liabilities, in which the value of liabilities and the price of risky assets satisfy the Ornstein–Uhlenbeck (O–U) processes whose drift terms are unobserved. By constructing a dynamic portfolio of risk-free bonds, risky stocks and liabilities, a stochastic delay differential equation is obtained to depict the surplus process of investor. The ALM problem is formulated as finding the best strategy to maximize the terminal utility of the sum of terminal surplus and some historical wealth under partial information, and the corresponding full information case is also studied as a supplement. For both cases of partial information and full information, we apply the dynamic programming method to derive HJB equations, verification theorems, and closed-form solutions of optimal strategies and value functions. Moreover the relationship between optimal strategy and value function under full information and partial information is also given. Finally, numerical examples are carried out to illustrate the influence of some important parameters on the obtained results.