A note on wavelet density deconvolution for weakly dependent data

Stochastic volatility (SV) models take a very important role in financial market, while there still exist some difficulties in estimating parameters in SV models. In this paper, a unified parameter estimation algorithm is proposed to estimate continuous-time SV model. Parameters in equity prices and volatilities stochastic processes are estimated separately after orthogonalization of Brownian motion in SV models. A closed-form of Maximum Likelihood Estimation (MLE) and moment estimation results for four parameters in SV models are deducted. Five typical SV models are simulated to reveal the characteristics in empirical volatilities data. Parameter estimation results are presented to prove the convergence of estimation algorithm. We also compare our algorithm with Least Square (LS) estimations and numerical solutions of all the parameters in with MLE method, respectively. LS can only work in a specific form of SV models and the convergence speed is limited. Numerical solutions of the likelihood function with four parameters are unstable and can not converge to true values of all the parameters. Thus our algorithm works more efficiently than other methods.

We consider variance-optimal hedging in general continuous-time affine stochastic volatility models. The optimal hedge and the associated hedging error are determined semiexplicitly in the case that the stock price follows a martingale. The integral representation of the solution opens the door to efficient numerical computation. The setup includes models with jumps in the stock price and in the activity process. It also allows for correlation between volatility and stock price movements. Concrete parametric models will be illustrated in a forthcoming paper.

In this paper, we analyze the robustness and sensitivity of various continuous-time rough Volterra stochastic volatility models in relation to the process of market calibration. Model robustness is examined from two perspectives: the sensitivity of option price estimates and the sensitivity of parameter estimates to changes in the option data structure. The following sensitivity analysis consists of statistical tests to determine whether a given studied model is sensitive to changes in the option data structure based on the distribution of parameter estimates. Empirical study is performed on a data set consisting of Apple Inc. equity options traded on four different days in April and May 2015. In particular, the results for RFSV, rBergomi and alpha RFSV models are provided and compared to the results for Heston, Bates, and AFSVJD models.

We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our ‘COGARCH’ (continuous-time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.

We consider a continuous-time stochastic volatility model. The model contains a stationary volatility process, the density of which, at a fixed instant in time, we aim to estimate. We assume that we observe the process at discrete instants in time. The sampling times will be equidistant with vanishing distance. A Fourier-type deconvolution kernel density estimator based on the logarithm of the squared processes is proposed to estimate the volatility density. An expansion of the bias and a bound on the variance are derived.

Estimation of continuous-time stochastic volatility models with jumps using high-frequency data

Attainability of European Path-Independent Claims in Incomplete Markets

Covid-19 epidemic dramatically relaunched the importance of mathematical modelling in supporting governments decisions to slow down the disease propagation. On the other hand, it remains a challenging task for mathematical modelling. The interplay between different models could be a key element in the modelling strategies. Here we propose a continuous space-time non-linear probabilistic model from which we can derive many of the existing models both deterministic and stochastic as for example SI, SIR, SIR stochastic, continuous-time stochastic models, discrete stochastic models, Fisher–Kolmogorov model. A partial analogy with the statistical interpretation of quantum mechanics provides an interpretation of the model. Epidemic forecasting is one of its possible applications; in principle, the model can be used in order to locate those regions of space where the infection probability is going to increase. The connection between non-linear probabilistic and non-linear deterministic models is analyzed. In particular, it is shown that the Fisher–Kolmogorov equation is connected to linear probabilistic models. On the other hand, a generalized version of the Fisher–Kolmogorov equation is derived from the non-linear probabilistic model and is shown to be characterized by a non-homogeneous time-dependent diffusion coefficient (anomalous diffusion) which encodes information about the non-linearity of the probabilistic model.

Computational modeling plays an important role in a gamut of research fields. In affect research, continuous-time stochastic models are becoming increasingly popular. Recently, a non-linear, continuous-time, stochastic model has been introduced for affect dynamics, called the Affective Ising Model (AIM). The drawback of non-linear models like the AIM is that they generally come with serious computational challenges for parameter estimation and related statistical analyses. The likelihood function of the AIM does not have a closed form expression. Consequently, simulation based or numerical methods have to be considered in order to evaluate the likelihood function. Additionally, the likelihood function can have multiple local minima. Consequently, a global optimization heuristic is required and such heuristics generally require a large number of likelihood function evaluations. In this paper, a Julia software package is introduced that is dedicated to fitting the AIM. The package includes an implementation of a numeric algorithm for fast computations of the likelihood function, which can be run both on graphics processing units (GPU) and central processing units (CPU). The numerical method introduced in this paper is compared to the more traditional Euler-Maruyama method for solving stochastic differential equations. Furthermore, the estimation software is tested by means of a recovery study and estimation times are reported for benchmarks that were run on several computing devices (two different GPUs and three different CPUs). According to these results, a single parameter estimation can be obtained in less than thirty seconds using a mainstream NVIDIA GPU.

Decision letter for "Red noise in continuous-time stochastic modelling"

This paper presents a new and efficient method for the construction of optimal designs for regression models with dependent error processes. In contrast to most of the work in this field, which starts with a model for a finite number of observations and considers the asymptotic properties of estimators and designs as the sample size converges to infinity, our approach is based on a continuous time model. We use results from stochastic analysis to identify the best linear unbiased estimator (BLUE) in this model. Based on the BLUE, we construct an efficient linear estimator and corresponding optimal designs in the model for finite sample size by minimizing the mean squared error between the optimal solution in the continuous time model and its discrete approximation with respect to the weights (of the linear estimator) and the optimal design points, in particular in the multiparameter case. In contrast to previous work on the subject, the resulting estimators and corresponding optimal designs are very efficient and easy to implement. This means that they are practically not distinguishable from the weighted least squares estimator and the corresponding optimal designs, which have to be found numerically by nonconvex discrete optimization. The advantages of the new approach are illustrated in several numerical examples.

Hyperbolic Normal Stochastic Volatility Model

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We investigate a linear estimation problem under model uncertainties using a competitive algorithm framework under mean square error (MSE) criteria. Here, the performance of a linear estimator is defined relative to the performance of the linear minimum MSE estimator tuned to the underlying unknown system model. We then find the linear estimator that minimizes this relative performance measure, i.e., the regret, for the worst possible system model. Two definitions of regret are given: first as a difference of MSEs and second as a ratio of MSEs. We demonstrate that finding the linear estimators that minimize these regret definitions can be cast as a Semidefinite Programming (SDP) problem and provide numerical examples. </para>

The performance of time-domain channel estimation deteriorates due to the presence of Gaussian mixture model (GMM) noise, which results in high mean squared error (MSE) as a challenging issue. The performance of the estimator further decreases when the complexity of the estimator is high due to the high convergence rate. In this paper, an optimized channel estimation method is proposed with low complexity and high accuracy in the GMM environment. In this channel estimation, an improved Gauss-Seidel iterative method is utilized with a minimum number of iterations. The convergence rate of the Gauss-Seidel method is improved by estimating an appropriate initial guess value when no guard bands are used in the orthogonal frequency-division multiplexing (OFDM) symbol. Simulation results provide an acceptable MSE for GMM environments, up to the probability of 5% impulsive noise component. This paper also presents the design and implementation of the proposed estimator in the NEXYS-2 FPGA platform that provides resources allocation, reconfigurability, schematic, and the timing diagram for detailed insight.

Although it is only during the last decade that continuous-time models have been extensively used in applied econometric work, the development of statistical methods applicable to such models commenced over 40 years ago. The first significant contribution to the problem of estimating the parameters of continuous-time stochastic models from discrete data was made by the British statistician Bartlett (1946) only three years after the pioneer contribution of Haavelmo (1943) on simultaneous-equations models. Moreover, by this time the fundamental mathematical theory of continuous-time stochastic models was already well-developed, major contributions having been made by some of the leading mathematicians of the twentieth century, including Einstein, Wiener and Kolmogorov.KeywordsContinuous TimeStochastic Differential EquationDiscrete DataAsymptotic BiasEconometric TheoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.