On non-negative modeling with CARMA processes

Continuous‐time autoregressive moving average (CARMA) processes have recently been used widely in the modelling of non‐uniformly spaced data and as a tool for dealing with high‐frequency data of the form ,n = 0, 1, 2,…, where Δ is small and positive. Such data occur in many fields of application, particularly in finance and in the study of turbulence. This article is concerned with the characteristics of the process , when Δ is small and the underlying continuous‐time process is a specified CARMA process.

The multiple disorder problem consists of finding a sequence of stopping times which are as close as possible to the (unknown) times of ’disorder’ when the distribution of an observed process changes its probability characteristics. We present a formulation and solution of the multiple disorder problem for a Wiener and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial optimal switching problems to the corresponding coupled optimal stopping problems and solving the equivalent coupled free-boundary problems by means of the smooth- and continuous-fit conditions.

This paper aims to propose efficient mathematical model of variance swap to study the effect of stochastic volatility in different time-scales on the option pricing. Two types of stochastic volatility, including Ornstein-Uhlenbeck (OU) process and Cox-Ingersoll-Ross (CIR) process are considered. Analytical solution of CIR model is presented. For the OU process, a numerical algorithm based on the finite element approach is established for solution of the model.

Dynamical behavior and numerical simulation of a stochastic eco-epidemiological model with Ornstein–Uhlenbeck process

In this paper we consider two classes of reflected Ornstein–Uhlenbeck (OU) processes: the reflected OU process with jumps and the Markov-modulated reflected OU process. We prove that their stationary distributions exist. Furthermore, for the jump reflected OU process, the Laplace transform (LT) of the stationary distribution is given. As for the Markov-modulated reflected OU process, we derive an equation satisfied by the LT of the stationary distribution.

Occupation times of Lévy-driven Ornstein–Uhlenbeck processes with two-sided exponential jumps and applications

In this paper we give easy to verify conditions for the strong consistency of the maximum likelihood estimator (MLE) in the case when data is sampled from a parametric family of selfdecomposable distributions. The difficulty arises from the fact that standard conditions for the consistency of the MLE are based on the pdf, which, for most selfdecomposable distributions, is not available in a closed form. Instead, our conditions are based on properties of the Levy triplet (i.e. the Levy measure, the Gaussian part, and the shift) of the distribution. Further, we extend out results to certain selfdecomposable stochastic processes, and, in particular, we give conditions in the case when the data is sampled from a Levy or an Ornstein–Uhlenbeck process.

In this paper, a stochastic differential equation based on Ornstein---Uhlenbeck (OU) process has been employed to image enhancement and edge detection processes. The simulation results show that the method is efficient to enhance the color image. In addition, the performance of OU method on edge detection is acceptable in both gray and color images.

The worldwide data for COVID-19 for active, infected individuals in multiple waves show that traditional epidemic models with constant parameters are not able to capture this kind of disease behavior. We solved this major open mathematical problem in this report. We first consider the disease transmission rate for the stochastic SIRVI epidemic model, which satisfies the mean-reverting Ornstein–Uhlenbeck (OU) process, and we propose a new stochastic SIRVI model. We then showed the existence and uniqueness of the global solution and obtained sufficient conditions for the persistent mean and exponential extinction of infectious disease, which have not been given before. In the second part, we derive a nonlinear system of differential equations for the time-dependent transmission rate from the deterministic SIRVI model and present an algorithm to compute the time-dependent transmission rate directly from the given active, infected individuals’ data. We then show that the time-dependent transmission obtained from and perturbed by the Ornstein–Uhlenbeck process could be represented after using a smoothing technique using a finite linear combination of a Gaussian radial basis function, which was obtained from our algorithm. This novel computer-assisted proof provides a theoretical basis for other epidemic models and epidemic waves. Finally, some numerical solutions of the stochastic SIRVI model are presented using COVID-19 data from Saudi Arabia and Austria.

Sparse inference of the drift of a high-dimensional Ornstein–Uhlenbeck process

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

Recognizing the crucial impacts of dispersal and noise intensity in ecosystems, this article explores a two-species stochastic competitive model with a Holling Type-II functional response, in which the intrinsic growth rates are driven by the Ornstein–Uhlenbeck process. Firstly, we demonstrate the existence and uniqueness of the global solution to the model, as well as confirming the boundedness of the moment. Secondly, we proceed to derive sufficient conditions to guarantee the asymptotic stability of the model’s positive equilibrium point and acquire the value of constant b that will affect this property. This indicates that the weaker the noise intensity, the closer the stochastic model approaches the positive equilibrium of the corresponding deterministic model in the mean sense. Furthermore, we build the model by introducing a proper Lyapunov function and provide sufficient conditions under which a stationary distribution exists. Finally, through several numerical simulations, we yield results indicating that weaker noise can ensure the existence and uniqueness of a stationary distribution. Furthermore, this article extends the existing ones.

In this paper we present closed form solutions of some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems, where the normal-reflection and smooth-fit conditions may break down and the latter then replaced by the continuous-fit condition. We show that, under certain relationships on the parameters of the model, the optimal stopping boundary can be uniquely determined as a component of the solution of a two-dimensional system of nonlinear ordinary differential equations. The obtained results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.

In this paper, we consider a Sparre Andersen risk model perturbed by a spectrally negative Lévy process (SNLP). Assuming that the interclaim times follow a Coxian distribution, we show that the Laplace transforms and defective renewal equations for the Gerber–Shiu functions can be obtained by employing the roots of a generalized Lundberg equation. When the SNLP is a combination of a Brownian motion and a compound Poisson process with exponential jumps, explicit expressions and asymptotic formulas for the Gerber–Shiu functions are obtained for exponential claim size distribution and heavy-tailed claim size distribution, respectively.

The noise processes associated with dynamical systems have limited bandwidth having finite, non-zero correlation time enabling us to model them as the Ornstein–Uhlenbeck process. The confluence of the deterministic system dynamics with these noisy forcing functions may trigger a series of unwarranted complex, peculiar phenomenon or stability issues in the system under different initial conditions and noise intensities. Thus, revisiting the system dynamics in the Ornstein–Uhlenbeck setting using the estimation and filtration algorithms seems an accurate and imperative choice. The Ito calculus, a very powerful classical tool, is used to evaluate the performance of stochastic differential equations. However, the Ito calculus cannot be directly applied to the systems under the influence of the Ornstein–Uhlenbeck process and we come across what is more fondly called as the curse of dimensionality, the augmented state vector approach. This paper develops a system theoretic-estimation algorithm for a switched-mode DC–DC power converter driven by the Ornstein–Uhlenbeck process. This is a novel attempt in this research direction. The algorithm is tested under a various data set and initial conditions to validate the findings.