Drift Function Research Articles

Path-dependent infinite-dimensional SDE with non-regular drift: An existence result

We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither small or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy. Our result strongly improves the previous ones obtained for free dynamics with a small perturbative drift. The originality of our method leads in the use of the specific entropy as a tightness tool and on a description of such stochastic differential equation as solution of a variational problem on the path space.

Joint maximum a posteriori state path and parameter estimation in stochastic differential equations

In this article, we introduce the joint maximum a posteriori state path and parameter estimator (JME) for continuous-time systems described by stochastic differential equations (SDEs). This estimator can be applied to nonlinear systems with discrete-time (sampled) measurements with a wide range of measurement distributions. We also show that the minimum-energy state path and parameter estimator (MEE) obtains the joint maximum a posteriori noise path, initial conditions, and parameters. These estimators are demonstrated in simulated experiments, in which they are compared to the prediction error method (PEM) using the unscented Kalman filter and smoother. The experiments show that the MEE is biased for the damping parameters of the drift function. Furthermore, for robust estimation in the presence of outliers, the JME attains lower state estimation errors than the PEM.

Theoretical Study of the Conditions of Maximum Manifestation of the Error Due to Inhomogeneity of Thermocouple Legs

The method of theoretical analysis of temperature ranges for the maximum manifestation of the error due to acquired thermoelectric inhomogeneity of thermocouple legs is proposed in this paper. The drift function of the reference function of a type K thermocouples in a ceramic insulation, that consisted of 1.2 mm diameter thermoelements after their exposure to 800 \(^{\circ }\)C for 10 000 h in an oxidizing atmosphere (air), is analyzed. The method takes into account various operating conditions to determine the optimal conditions for studying inhomogeneous thermocouples. The method can be applied for other types of thermocouples when taking into account their specific characteristics and the conditions that they have been exposed to.

Zero-sum continuous-time Markov pure jump game over a fixed duration

This paper considers a two-person zero-sum continuous-time Markov pure jump game in Borel state and action spaces over a fixed finite horizon. The main assumption on the model is the existence of a drift function, which bounds the reward rate. Under some regularity conditions, we show that the game has a value, and both of the players have their optimal policies.

Chance and necessity in the genome evolution of endosymbiotic bacteria of insects

An open question in evolutionary biology is how does the selection–drift balance determine the fates of biological interactions. We searched for signatures of selection and drift in genomes of five endosymbiotic bacterial groups known to evolve under strong genetic drift. Although most genes in endosymbiotic bacteria showed evidence of relaxed purifying selection, many genes in these bacteria exhibited stronger selective constraints than their orthologs in free-living bacterial relatives. Remarkably, most of these highly constrained genes had no role in the host–symbiont interactions but were involved in either buffering the deleterious consequences of drift or other host-unrelated functions, suggesting that they have either acquired new roles or their role became more central in endosymbiotic bacteria. Experimental evolution of Escherichia coli under strong genetic drift revealed remarkable similarities in the mutational spectrum, genome reduction patterns and gene losses to endosymbiotic bacteria of insects. Interestingly, the transcriptome of the experimentally evolved lines showed a generalized deregulation of the genome that affected genes encoding proteins involved in mutational buffering, regulation and amino acid biosynthesis, patterns identical to those found in endosymbiotic bacteria. Our results indicate that drift has shaped endosymbiotic associations through a change in the functional landscape of bacterial genes and that the host had only a small role in such a shift.

Cardiovascular adjustments to heat stress during prolonged exercise.

Prolonged effort in hot environments is increasing popular in extreme sports. The number of long distance athletes taking part in races in extremely warm condition, as in deserts, is larger than in the past and also the race distances are increased thus. The demanding for the heart function are higher than those challenged in the past only by few athletes and in easier conditions. This puts news challenges on the professional involved in these activities and there is a need to better understand how the cardiocirculatory system responses in these extreme environmental conditions. This review paper is focused on the mechanism of cardiovascular drift function during prolonged exercise in heat conditions. Many aspects are treated in the literature. The emerging topics treated in this paper are the sympathetic and vagal nervous activity, heart rate functioning during exercise, dehydration and hyperthermia, the relationships between heart rate and stroke volume oxygen uptake and blood flow dynamic, circulatory responses, left ventricular functions in athletes after strenuous prolonged exercise, and heart rate variability. The literature on the topic appear to be vast and address many important factors of interest for the performance development occurred in hot condition and necessary to be known for the preservation of the health of the athletes.

Strong Feller Property of Sticky Reflected Distorted Brownian Motion

Using Girsanov transformations we construct from sticky reflected Brownian motion on $$[0,\infty )$$ a conservative diffusion on $$E:=[0,\infty )^n$$ , $$n \in \mathbb {N}$$ , and prove that its transition semigroup possesses the strong Feller property for a specified general class of drift functions. By identifying the Dirichlet form of the constructed process we characterize it as sticky reflected distorted Brownian motion. In particular, the relations of the underlying analytic Dirichlet form methods to the probabilistic methods of random time changes and Girsanov transformations are presented. Our studies of the mathematical model are motivated by its applications to the dynamical wetting model with $$\delta $$ -pinning and repulsion.

Solvable Diffusion Models with Linear and Mean-Reverting Nonlinear Drifts

We present new extensions to a method for constructing families of solvable one-dimensional time-homogeneous diffusions whose transition densities and other related quantities are obtainable in analytically closed form. Our approach is based on a dual application of the so-called diffusion canonical transformation method that combines a smooth monotonic differentiable map with nonzero derivative (diffeomorphism) and a measure change via a Doob-$h$ transform. This gives rise to new multiparameter solvable diffusions that are divided into two main classes; the first is specified by having affine (linear) drift with various resulting nonlinear diffusion coefficient functions, while the second class allows for several specifications of a (generally nonlinear) diffusion coefficient with resulting nonlinear drift function. The first class of models, having linear drift and nonlinear (state-dependent) volatility functions, is useful for pricing equity and foreign exchange (FX) options in finance, while the second class of diffusions contains new models that are mean-reverting and are applicable to pricing interest-rate and other path-dependent derivatives such as volatility index (VIX) options. As specific examples of the first class of affine drift models, we present explicit results for two new families of models that arise from the squared Bessel process (the Bessel family) and the Ornstein--Uhlenbeck diffusion (the OU family). For the second class of nonlinear drift models, we give examples of solvable subfamilies called the Bessel family of mean-reverting diffusions and derive closed-form integral formulas for conditional expectations of certain functionals. In particular, we derive a new closed-form analytical formula for the Laplace transform with respect to the strike price of a standard call VIX option. We then succeed in Laplace inverting the expression to obtain numerically exact VIX call option prices with realistic implied volatilities with respect to strike and maturity. Moreover, we accurately calibrate the OU family of models to FX option market data, exhibiting pronounced implied volatility smiles across several strikes and maturities.

Application of evolution-based uncertainty design on gear

The evolution of mechanical parameters, a factor affecting the mechanical reliability, has gathered more attention nowadays. However, studies on time varying uncertainty can hardly be found. A new method based on evolution-based uncertainty design (EBUD) is applied to the design of gear in this paper. Considering the wear evolution over the lifetime, a tooth wear’s time-varying uncertainty model based on the continuous-time model and Ito lemma is established. Drift and volatility functions dependent on the drift rate and volatility rate of rotational speed and torque are used to express the time-varying uncertainty of tooth thickness. The method can predict the reliability and provide an instruction in reliability improving, maintenance and repair of the gear system.

Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes

This paper focuses on two variants of the Milstein scheme, namely the split-step backward Milstein method and a newly proposed projected Milstein scheme, applied to stochastic differential equations which satisfy a global monotonicity condition. In particular, our assumptions include equations with super-linearly growing drift and diffusion coefficient functions and we show that both schemes are mean-square convergent of order 1. Our analysis of the error of convergence with respect to the mean-square norm relies on the notion of stochastic C-stability and B-consistency, which was set up and applied to Euler-type schemes in [Beyn, Isaak, Kruse, J. Sci. Comp., 2015]. As a direct consequence we also obtain strong order 1 convergence results for the split-step backward Euler method and the projected Euler-Maruyama scheme in the case of stochastic differential equations with additive noise. Our theoretical results are illustrated in a series of numerical experiments.

Least squares estimators for stochastic differential equations driven by small Lévy noises

We study parameter estimation for discretely observed stochastic differential equations driven by small Lévy noises. We do not impose Lipschitz condition on the dispersion coefficient function σ and any moment condition on the driving Lévy process, which greatly enhances the applicability of our results to many practical models. Under certain regularity conditions on the drift and dispersion functions, we obtain consistency and rate of convergence of the least squares estimator (LSE) of parameter when ε→0 and n→∞ simultaneously. We present some simulation study on a two-factor financial model driven by stable noises.

Variational estimation of the drift for stochastic differential equations from the empirical density

We present a method for the nonparametric estimation of the drift function of certain types of stochastic differential equations from the empirical density. It is based on a variational formulation of the Fokker–Planck equation. The minimization of an empirical estimate of the variational functional using kernel based regularization can be performed in closed form. We demonstrate the performance of the method on second order, Langevin-type equations and show how the method can be generalized to other noise models.

A simple approach for a spatial terrestrial exposure assessment of the insecticide fenoxycarb, based on a high-resolution landscape analysis.

BACKGROUNDThe objective was to refine the standard regulatory exposure scenario used in plant protection product authorisations by developing a more realistic landscape‐related GIS‐based exposure assessment for terrestrial non‐target arthropods. We quantified the proportion of adjacent off‐target area in agricultural landscapes potentially exposed to insecticide drift from applications of the active substance fenoxycarb. High‐resolution imagery, landscape classification and subsequent stepwise analysis of a whole landscape using drift and interception functions were applied to selected areas in representative fruit‐producing regions in Germany.RESULTSEven under worst‐case assumptions regarding treated area, use rate and drift, less than 12% of the non‐agricultural habitat area would potentially be exposed to fenoxycarb drift above regulatory acceptable concentrations. Additionally, if the filtering effect of tall vegetation were taken into account, this number would decrease to 6.6%. Further refinements to landscape elements and application conditions indicate that less than 5% of the habitat area might be exposed above regulatory acceptable concentrations, meaning that 95% of the non‐agricultural habitat area will be unimpacted (i.e. no unacceptable effects) and can serve as refuge for recolonisation.CONCLUSIONApproaches and tools are proposed for standardisable and transparent refinements in regulatory risk assessments on the landscape level. © 2016 The Authors. Pest Management Science published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry.

Nonparametric estimation of a scalar diffusion model from discrete time data: a survey

In view of rapid developments on nonparametric estimation of the drift and volatility functions in scalar diffusion models in financial econometrics, from discrete-time observations, we provide, in this paper, a survey of its state-of-the-art with new insights into current practices, as well as elaborating on our own recent contributions. In particular, in presenting the main principles of estimation for both stationary and nonstationary cases, we show the possibility to estimate nonparametrically the drift and volatility functions without distinguishing these two frameworks.

Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition

In this paper the numerical approximation of stochastic differential equations satisfying a global monotonicity condition is studied. The strong rate of convergence with respect to the mean square norm is determined to be $\frac{1}{2}$ for the two-step BDF-Maruyama scheme and for the backward Euler-Maruyama method. In particular, this is the first paper which proves a strong convergence rate for a multi-step method applied to equations with possibly superlinearly growing drift and diffusion coefficient functions. We also present numerical experiments for the $\tfrac32$-volatility model from finance, which verify our results in practice and indicate that the BDF2-Maruyama method offers advantages over Euler-type methods if the stochastic differential equation is stiff or driven by a noise with small intensity.

Exact sampling of diffusions with a discontinuity in the drift

AbstractWe introduce exact methods for the simulation of sample paths of one-dimensional diffusions with a discontinuity in the drift function. Our procedures require the simulation of finite-dimensional candidate draws from probability laws related to those of Brownian motion and its local time, and are based on the principle of retrospective rejection sampling. A simple illustration is provided.

Pre-averaged kernel estimators for the drift function of a diffusion process in the presence of microstructure noise

We consider estimation of the drift function of a stationary diffusion process when we observe high-frequency data with microstructure noise over a long time interval. We propose to estimate the drift function at a point by a Nadaraya–Watson estimator that uses observations that have been pre-averaged to reduce the noise. We give conditions under which our estimator is consistent and asympotically normal. Its rate and asymptotic bias and variance are the same as those without microstructure noise. To use our method in data analysis, we propose a data-based cross-validation method to determine the bandwidth in the Nadaraya–Watson estimator. Via simulation, we study several methods of bandwidth choices, and compare our estimator to several existing estimators. In terms of mean squared error, our new estimator outperforms existing estimators.

Pathwise solvability of stochastic integral equations with generalized drift and non-smooth dispersion functions

We study one-dimensional stochastic integral equations with non-smooth dispersion coefficients, and with drift components that are not restricted to be absolutely continuous with respect to Lebesgue measure. In the spirit of Lamperti, Doss and Sussmann, we relate solutions of such equations to solutions of certain ordinary integral equations, indexed by a generic element of the underlying probability space. This relation allows us to solve the stochastic integral equations in a pathwise sense.

Drift correction of the dissolved signal in single particle ICPMS.

A method is presented where drift, the random fluctuation of the signal intensity, is compensatedfor based on the estimation of the drift function by a moving average. It was shown using single particle ICPMS (spICPMS) measurements of 10 and 60 nm Au NPs that drift reduces accuracy of spICPMS analysis at the calibration stage and during calculations of the particle size distribution (PSD), but that the present method can again correct the average signal intensity as well as the signal distribution of particle-containing samples skewed by drift. Moreover, deconvolution, a method that models signal distributions of dissolved signals, fails in some cases when using standards and samples affected by drift, but the present method was shown to improve accuracy again. Relatively high particle signals have to be removed prior to drift correction in this procedure, which was done using a 3 × sigma method, and the signals are treated separately and added again. The method can also correct for flicker noise that increases when signal intensity is increased because of drift. The accuracy was improved in many cases when flicker correction was used, but when accurate results were obtained despite drift, the correction procedures did not reduce accuracy. The procedure may be useful to extract results from experimental runs that would otherwise have to be run again. Graphical Abstract A method is presented where a spICP-MS signal affected by drift (left) is corrected (right) by adjusting the local (moving) averages (green) and standard deviations (purple) to the respective values at a reference time (red). In combination with removing particle events (blue) in the case of calibration standards, this method is shown to obtain particle size distributions where that would otherwise be impossible, even when the deconvolution method is used to discriminate dissolved and particle signals.

Exploring quasifission characteristics forS34+Th232formingSg266

Background: Fission fragments from heavy ion collisions with actinide nuclei show mass-asymmetric and mass-symmetric components. The relative probabilities of these two components vary rapidly with beam energy with respect to the capture barrier, indicating a strong dependence on the alignment of the deformed nucleus with the partner in the collisions.Purpose: To study the characteristics of the mass-asymmetric quasifission component by reproducing the experimental mass-angle distributions to investigate mass evolution and sticking times.Methods: Fission fragment mass-angle distributions were measured for the $^{34}\text{S}+^{232}\text{Th}$ reaction. Simulations to match the measurements were made by using a classical phenomenological approach. Mass ratio distributions and angular distributions of the mass-asymmetric quasifission component were simultaneously fit to constrain the free parameters used in the simulation.Results: The mass-asymmetric quasifission component---predominantly originating from tip (axial) collisions with the prolate deformed $^{232}\text{Th}$---is found to be peaked near $A=200$ at all energies and center-of-mass angles. A Monte Carlo model using the standard mass equilibration time constant of $5.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}21}$ s predicts more symmetric mass splits. Three different hypotheses assuming (i) a mass halt at $A=200$, (ii) a slower mass equilibration time, or (iii) a Fermi-type mass drift function reproduced the main experimental features.Conclusions: In tip collisions for the $^{34}\text{S}+^{232}\text{Th}$ reaction, mass-asymmetric fission with $A\ensuremath{\sim}200$ is the dominant outcome. The average sticking time is found to be $\ensuremath{\sim}7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}21}$ s, independent of the scenario used for mass evolution.