Unknown Drift Research Articles

Practical Reference-Frame-Independent Measurement-Device-Independent Quantum Key Distribution Systems Against the Worst Relative Rotation of Reference Frames**Supported by the National Key Research and Development Program of China under Grant Nos. 2018YFA0306400 and 2017YFA0304100, the National Natural Science Foundation of China under Grant Nos. 61475197, 61590932, 11774180, and 61705110, the Natural Science Foundation of the Jiangsu Higher Education Institutions under Grant Nos. 15KJA120002

Reference-frame-independent measurement-device-independent QKD (RFI-MDI-QKD), immune to both the slow unknown drift of reference frames and detector side channel attacks, can generate information-theoretic secure keys. Despite its natural resistance to the slow drift of reference frames, the performance of practical RFI-MDI-QKD systems descends with the increasing drift of reference frames. In this paper, we demonstrate the worst relative rotation of reference frames for practical RFI-MDI-QKD systems, and investigate their performance against the worst-case scenario, both of which are unclear to date. Simulation results show that, practical RFI-MDI-QKD systems can achieve quite good performance even against the worst-case scenario, which clearly demonstrates that it is possible to implement practical MDI-QKD systems with freely drifting reference frames.

Asset Liquidation Under Drift Uncertainty and Regime-Switching Volatility

Optimal liquidation of an asset with unknown constant drift and stochastic regime-switching volatility is studied. The uncertainty about the drift is represented by an arbitrary probability distribution; the stochastic volatility is modelled by m-state Markov chain. Using filtering theory, an equivalent reformulation of the original problem as a four-dimensional optimal stopping problem is found and then analysed by constructing approximating sequences of three-dimensional optimal stopping problems. An optimal liquidation strategy and various structural properties of the problem are determined. Analysis of the two-point prior case is presented in detail, building on which, an outline of the extension to the general prior case is given.

Data-Based Reinforcement Learning for Nonzero-Sum Games With Unknown Drift Dynamics.

This paper is concerned about the nonlinear optimization problem of nonzero-sum (NZS) games with unknown drift dynamics. The data-based integral reinforcement learning (IRL) method is proposed to approximate the Nash equilibrium of NZS games iteratively. Furthermore, we prove that the data-based IRL method is equivalent to the model-based policy iteration algorithm, which guarantees the convergence of the proposed method. For the implementation purpose, a single-critic neural network structure for the NZS games is given. To enhance the application capability of the data-based IRL method, we design the updating laws of critic weights based on the offline and online iterative learning methods, respectively. Note that the experience replay technique is introduced in the online iterative learning, which can improve the convergence rate of critic weights during the learning process. The uniform ultimate boundedness of the critic weights are guaranteed using the Lyapunov method. Finally, the numerical results demonstrate the effectiveness of the data-based IRL algorithm for nonlinear NZS games with unknown drift dynamics.

The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing When the Horizon Is Infinite

Assume that we want to shell an asset with unknown drift but known that the drift is a two value random variable, and the initial distribution can be estimated. As time goes by, this distribution is updated and base on the probability of the drift takes the small one gives us the stopping rule. Research results show that the optimal strategy to sell the asset is if the initial probability that the drift receives a small value greater than a certain threshold then liquidates the asset immediately, otherwise the asset holder will wait until the probability of the drift receives a small value passing a certain threshold, it is the optimal time to liquidate the asset.

Uncoupled PID Control of Multi-Agent Nonlinear Uncertain Stochastic Systems

This paper proposes a PID (proportional-integral-derivative) control design method to solve the regulation problem for a class of coupled multi-agent nonlinear uncertain stochastic systems, where each agent only has access to its own regulation error without communicating with others. A three-dimensional manifold will be constructed based on the information about the Lipschitz constants of both the unknown nonlinear drift and diffusion terms, such that the three parameters of each agent’s PID controller can be chosen arbitrarily from this common manifold. It will be shown that such an uncoupled PID design method can globally stabilize the whole nonlinear uncertain stochastic multi-agent system with the regulation error of each agent approaching to zero asymptotically.

Adaptive nonparametric drift estimation of an integrated jump diffusion process

In the present article, we investigate nonparametric estimation of the unknown drift function b in an integrated Lévy driven jump diffusion model. Our aim will be to estimate the drift on a compact set based on a high-frequency data sample. Instead of observing the jump diffusion process V itself, we observe a discrete and high-frequent sample of the integrated process Xt := ∫0t Vsds Based on the available observations of Xt, we will construct an adaptive penalized least-squares estimate in order to compute an adaptive estimator of the corresponding drift function b. Under appropriate assumptions, we will bound the L2-risk of our proposed estimator. Moreover, we study the behavior of the proposed estimator in various Monte Carlo simulation setups.

A two-step estimation of diffusion processes using noisy observations

ABSTRACTThis paper considers the estimation of unknown drift and diffusion functions of a one-dimensional diffusion process when the observation is a discrete sampling of with an additive noise, at times , . In order to reduce the noise effect, a two-step estimation method is proposed based on the joint use of the pre-averaging technique and kernel smoothing. Under some suitable conditions, the proposed estimators are consistent and asymptotically normal. A simulation study and a real data application are given to evaluate the finite sample performance of the proposed method in comparison with alternative methods.

Local asymptotic mixed normality of approximate maximum likelihood estimator of drift parameters in diffusion model

We assume that the diffusion $X$ satisfies a stochastic differential equation of the form: $dX_t=\mu(X_t, \theta)dt+\sigma_0\nu(X_t)dW_t$, with unknown drift parameter $\theta$ and known diffusion coefficient parameter $\sigma_0$. We prove that approximate maximum likelihood estimator of drift parameter $\bar{;\theta};_n$ obtained from discrete observations $(X_{;i\Delta_n};, 0\leq i\leq n)$ along fixed time interval $[0, T]$, and when $\Delta_n =\frac{;T};{;n};$ tends to zero, is locally asymptotic mixed normal, with covariance matrix which depends on MLE $\hat{;\theta};$ obtained from continuous observations $(X_t, 0\leq t\leq T)$ along fixed time interval $[0, T]$, and on path $(X_t, 0\leq t\leq T)$.

Fisher information and Cramér-Rao bound for unknown systematic errors

In order to understand the lower bound of achievable measurement uncertainties, the Cramér-Rao inequality is known to be an utmost useful tool. However, the calculation of the Cramér-Rao bound requires a known probability density function that describes the occurring stochastic process. For this reason, the Cramér-Rao bound is applied for determining the lower limit of the measurement uncertainty due to random errors. According to the international guide to the expression of uncertainty in measurement (GUM), unknown systematic errors shall be treated as random errors. This approach is adopted here to enhance the applicability of the Cramér-Rao bound for unknown systematic errors. As a key result, the concept of Fisher information and the Cramér-Rao bound is shown to be applicable also to unknown systematic errors, which is demonstrated for several examples. An unknown offset, an unknown linear drift and successive unknown linear drifts are investigated in detail as systematic errors. Each derived corresponding Fisher information shows a characteristic behavior with respect to the measurement time. In contrast to random errors with a constant variance, the Fisher information can decrease for unknown systematic errors and, thus, the Cramér-Rao bound can increase with an increasing measurement time. For the typically existing case of simultaneously occurring random and unknown systematic errors, an optimal measurement time exists for which the achievable measurement uncertainty becomes minimal. In summary, the examples demonstrate how to determine the Fisher information and the Cramér-Rao bound for unknown systematic errors.

Threshold Estimation for Stochastic Processes with Small Noise

AbstractConsider a process satisfying a stochastic differential equation with unknown drift parameter, and suppose that discrete observations are given. It is known that a simple least squares estimator (LSE) can be consistent but numerically unstable in the sense of large standard deviations under finite samples when the noise process has jumps. We propose a filter to cut large shocks from data and construct the same LSE from data selected by the filter. The proposed estimator can be asymptotically equivalent to the usual LSE, whose asymptotic distribution strongly depends on the noise process. However, in numerical study, it looked asymptotically normal in an example where filter was chosen suitably, and the noise was a Lévy process. We will try to justify this phenomenon mathematically, under certain restricted assumptions.

Distribution of the mean reversion estimator in the Ornstein–Uhlenbeck process

ABSTRACTWe derive the exact distribution of the maximum likelihood estimator of the mean reversion parameter (κ) in the Ornstein–Uhlenbeck process using numerical integration through analytical evaluation of a joint characteristic function. Different scenarios are considered: known or unknown drift term, fixed or random start-up value, and zero or positive κ. Monte Carlo results demonstrate the remarkably reliable performance of our exact approach across all the scenarios. In comparison, misleading results may arise under the asymptotic distributions, including the advocated infill asymptotic distribution, which performs poorly in the tails when there is no intercept in the regression and the starting value of the process is nonzero.

Event-Triggered Optimal Control for Partially Unknown Constrained-Input Systems via Adaptive Dynamic Programming

Event-triggered control has been an effective tool in dealing with problems with finite communication and computation resources. In this paper, we design an event-triggered control for nonlinear constrained-input continuous-time systems based on the optimal policy. Constraints on controls are handled using a bounded function. To learn the optimal solution with partially unknown dynamics, an online adaptive dynamic programming algorithm is proposed. The identifier network, the critic network, and the actor network are employed to approximate the unknown drift dynamics, the optimal value, and the optimal policy, respectively. The identifier is tuned based on online data, which further trains the critic and actor at triggering instants. A concurrent learning technique repeatedly uses past data to train the critic. Stability of the closed-loop system, and convergence of neural networks to the optimal solutions are proved by Lyapunov analysis. In the end, the algorithm is applied to the overhead crane system to observe the performance. The event-triggered optimal controller with constraints stabilizes the system and consumes much less sampling times.

Hypothesis testing of the drift parameter sign for fractional Ornstein–Uhlenbeck process

We consider the fractional Ornstein–Uhlenbeck process with an unknown drift parameter and known Hurst parameter $H$. We propose a new method to test the hypothesis of the sign of the parameter and prove the consistency of the test. Contrary to the previous works, our approach is applicable for all $H\in(0,1)$.

An Algorithm for In-Flight Spectral Calibration of Imaging Spectrometers

Accurate spectral calibration of satellite and airborne spectrometers is essential for remote sensing applications that rely on accurate knowledge of center wavelength (CW) positions and slit function parameters (SFP). We present a new in-flight spectral calibration algorithm that retrieves CWs and SFPs across a wide spectral range by fitting a high-resolution solar spectrum and atmospheric absorbers to in-flight radiance spectra. Using a maximum a posteriori optimal estimation approach, the quality of the fit can be improved with a priori information. The algorithm was tested with synthetic spectra and applied to data from the APEX imaging spectrometer over the spectral range of 385–870 nm. CWs were retrieved with high accuracy (uncertainty <0.05 spectral pixels) from Fraunhofer lines below 550 nm and atmospheric absorbers above 650 nm. This enabled a detailed characterization of APEX’s across-track spectral smile and a previously unknown along-track drift. The FWHMs of the slit function were also retrieved with good accuracy (<10% uncertainty) for synthetic spectra, while some obvious misfits appear for the APEX spectra that are likely related to radiometric calibration issues. In conclusion, our algorithm significantly improves the in-flight spectral calibration of APEX and similar spectrometers, making them better suited for the retrieval of atmospheric and surface variables relying on accurate calibration.

Asymptotic growth of trajectories of multifractional Brownian motion, with statistical applications to drift parameter estimation

We construct the least-square estimator for the unknown drift parameter in the multifractional Ornstein–Uhlenbeck model and establish its strong consistency in the non-ergodic case. The proofs are based on the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of multifractional Brownian motion (mBm) and of some other functionals of mBm, including increments and fractional derivatives. As the auxiliary results having independent interest, we produce the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of the general Gaussian process and some functionals of it, in terms of the covariance function of its increments.

Opportunistic Time-of-Arrival localization in fully asynchronous wireless networks

We present a novel Time-of-Arrival localization scheme for fully asynchronous systems, where each node clock is affected by unknown bias (temporal offset) and drift (frequency offset). The proposed method relies exclusively on the measurement of reception times, and does not require knowledge nor control of the transmission times. Therefore, it can exploit opportunistically signals and packets that are already available over-the-air for communication purposes. We formalize the problem and provide a low-complexity resolution algorithm based on pseudo-ranges. Simulation results are provided to validate the resolution approach and to assess the sensitivity of the localization accuracy to the main scenario parameters.

Pathwise stability of likelihood estimators for diffusions via rough paths

We consider the classical estimation problem of an unknown drift parameter within classes of nondegenerate diffusion processes. Using rough path theory (in the sense of T. Lyons), we analyze the Maximum Likelihood Estimator (MLE) with regard to its pathwise stability properties as well as robustness toward misspecification in volatility and even the very nature of the noise. Two numerical examples demonstrate the practical relevance of our results.

On drift parameter estimation for reflected fractional Ornstein–Uhlenbeck processes

We consider a reflected Ornstein–Uhlenbeck process X driven by a fractional Brownian motion with Hurst parameter . Our goal is to estimate an unknown drift parameter on the basis of continuous observation of the state process. We establish Girsanov theorem for the process X, derive the standard maximum likelihood estimator of the drift parameter , and prove its strong consistency and asymptotic normality. As an improved estimator, we obtain the explicit formulas for the sequential standard maximum likelihood estimator and its mean squared error by assuming the process is observed until a certain information reaches a specified precision level. The estimator is shown to be unbiased, uniformly normally distributed, and efficient in the mean square error sense.

Model-Based Reinforcement Learning for Infinite-Horizon Approximate Optimal Tracking.

This brief paper provides an approximate online adaptive solution to the infinite-horizon optimal tracking problem for control-affine continuous-time nonlinear systems with unknown drift dynamics. To relax the persistence of excitation condition, model-based reinforcement learning is implemented using a concurrent-learning-based system identifier to simulate experience by evaluating the Bellman error over unexplored areas of the state space. Tracking of the desired trajectory and convergence of the developed policy to a neighborhood of the optimal policy are established via Lyapunov-based stability analysis. Simulation results demonstrate the effectiveness of the developed technique.

Optimal Liquidation of an Asset under Drift Uncertainty

We study a problem of finding an optimal stopping strategy to liquidate an asset with unknown drift. Taking a Bayesian approach, we model the initial beliefs of an individual about the drift by all ...