Qu, Dassios, and Zhao (2021) suggested an exact simulation method for tempered stable Ornstein–Uhlenbeck processes, but their algorithms contain some errors. This short note aims to correct their algorithms and conduct some numerical experiments.

Explicit bounds are given for the Kolmogorov and Wasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some $\sigma$ -algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the $\sigma$ -algebra. The proof is based on Stein’s method. As an application, we consider the Yule–Ornstein–Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.

This paper investigates a financial market where stock returns depend on an unobservable Gaussian mean reverting drift process. Information on the drift is obtained from returns and randomly arriving discrete-time expert opinions. Drift estimates are based on Kalman filter techniques. We study the asymptotic behavior of the filter for high-frequency experts with variances that grow linearly with the arrival intensity. The derived limit theorems state that the information provided by discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process. These diffusion approximations are extremely helpful for deriving simplified approximate solutions of utility maximization problems.

We present Lyapunov-type conditions for non-strong ergodicity of Markov processes. Some concrete models are discussed, including diffusion processes on Riemannian manifolds and Ornstein–Uhlenbeck processes driven by symmetric $\alpha$-stable processes. In particular, we show that any process of d-dimensional Ornstein–Uhlenbeck type driven by $\alpha$-stable noise is not strongly ergodic for every $\alpha\in (0,2]$.

Let X be an Ornstein–Uhlenbeck process driven by a Brownian motion. We propose an expression for the joint density / distribution function of the process and its running supremum. This law is expressed as an expansion involving parabolic cylinder functions. Numerically, we obtain this law faster with our expression than with a Monte Carlo method. Numerical applications illustrate the interest of this result.

We introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. Using potential theory and perturbation theory, we are able to deduce closed-form truncated probability densities, as asymptotics or approximations to the original first passage time densities, for single-side level crossing problems. The framework is applicable to diffusion processes with continuous drift functions; in particular, for bounded drift functions, we show that the perturbation series converges. In the present paper, we demonstrate examples of applying our framework to the Ornstein–Uhlenbeck, Bessel, exponential-Shiryaev, and hypergeometric diffusion processes (the latter two being previously studied by Dassios and Li (2018) and Borodin (2009), respectively). The purpose of this paper is to provide a fast and accurate approach to estimating first passage time densities of various diffusion processes.

We investigate the long-time behavior of the Ornstein–Uhlenbeck process driven by Lévy noise with regime switching. We provide explicit criteria on the transience and recurrence of this process. Contrasted with the Ornstein–Uhlenbeck process driven simply by Brownian motion, whose stationary distribution must be light-tailed, both the jumps caused by the Lévy noise and the regime switching described by a Markov chain can derive the heavy-tailed property of the stationary distribution. The different role played by the Lévy measure and the regime-switching process is clearly characterized.

Pricing variance swaps have become a popular subject recently, and most research of this type come under Heston’s two-factor model. This paper is an extension of some recent research which used the dimension-reduction technique based on the Heston model. A new closed-form pricing formula focusing on a log-return variance swap is presented here, under the assumption that the underlying asset prices can be described by a mean-reverting Gaussian volatility model (Ornstein–Uhlenbeck process). Numerical tests in two respects using the Monte Carlo (MC) simulation are included. Moreover, we discuss a procedure of solving a quadratic differential equation with one variable. Our method can avoid the previously encountered limitations, but requires more time for calculation than other recent analytical discrete models.

In this paper we consider an Ornstein–Uhlenbeck (OU) process (M(t))t≥0 whose parameters are determined by an external Markov process (X(t))t≥0 on a finite state space {1, . . ., d}; this process is usually referred to as Markov-modulated Ornstein–Uhlenbeck. We use stochastic integration theory to determine explicit expressions for the mean and variance of M(t). Then we establish a system of partial differential equations (PDEs) for the Laplace transform of M(t) and the state X(t) of the background process, jointly for time epochs t = t1, . . ., tK. Then we use this PDE to set up a recursion that yields all moments of M(t) and its stationary counterpart; we also find an expression for the covariance between M(t) and M(t + u). We then establish a functional central limit theorem for M(t) for the situation that certain parameters of the underlying OU processes are scaled, in combination with the modulating Markov process being accelerated; interestingly, specific scalings lead to drastically different limiting processes. We conclude the paper by considering the situation of a single Markov process modulating multiple OU processes.

We study the asymptotic behaviors of estimators of the parameters in an Ornstein–Uhlenbeck process with linear drift, such as the law of the iterated logarithm (LIL) and Berry–Esseen bounds. As an application of the Berry–Esseen bounds, the precise rates in the LIL for the estimators are obtained.

Many prominent continuous-time stochastic volatility models exhibit certain functional relationships between price jumps and volatility jumps. We show that stochastic volatility models like the Ornstein–Uhlenbeck and other continuous-time CARMA models as well as continuous-time GARCH and EGARCH models all exhibit such functional relations. We investigate the asymptotic behaviour of certain functionals of price and volatility processes for discrete observations of the price process on a grid, which are relevant for estimation and testing problems.

Based on the explicit coupling property, the ergodicity and the exponential ergodicity of Lévy-driven Ornstein–Uhlenbeck processes are established.

A Markov chain X with finite state space {0,…,N} and tridiagonal transition matrix is considered, where transitions from i to i-1 occur with probability (i/N)(1-p(i/N)) and transitions from i to i+1 occur with probability (1-i/N)p(i/N). Here p:[0,1]→[0,1] is a given function. It is shown that if p is continuous with p(x)≤p(1) for all x∈[0,1] then, for each N, a dual process Y to X (with respect to a specific duality function) exists if and only if 1-p is completely monotone with p(0)=0. A probabilistic interpretation of Y in terms of an ancestral process of a mixed multitype Moran model with a random number of types is presented. It is shown that, under weak conditions on p, the process Y, properly time and space scaled, converges to an Ornstein–Uhlenbeck process as N tends to ∞. The asymptotics of the stationary distribution of Y is studied as N tends to ∞. Examples are presented involving selection mechanisms. results.

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.

This paper introduces a rather general class of stationary continuous-time processes with long memory by randomizing the time-scale of short-memory processes. In particular, by randomizing the time-scale of continuous-time autoregressive and moving-average processes, many power-law decay and slow decay correlation functions are obtained.

We analyse a class of diffusion models that (i) allows an explicit expression for the likelihood function of discrete time observation, (ii) allows the possibility of heavy-tailed observations, and (iii) allows an analysis of the tails of the increments. The class simply consists of transformed Ornstein–Uhlenbeck processes and is of relevance for heavy-tailed time series. We also treat the question of the existence of an equivalent martingale measure for the class of models considered.

In this paper, transient characteristics related to excursions of the occupation process of M/M/∞ queues are studied, when the excursion level is large and close to the mean offered load. We show that the classical diffusion approximation by an Ornstein–Uhlenbeck (OU) process captures well the average values of the transient variables considered, while the asymptotic distributions of these variables depart from those corresponding to the OU process. They exhibit, however, equivalent tail behaviour at infinity and numerical evidence shows that they are amazingly close to each other over the whole half-line.

This paper introduces exchangeable bisexual models with fixed population sizes and non-overlapping generations. In each generation there are N pairs of individuals consisting of a female and a male. The N pairs of a generation produce N daughters and N sons altogether, and these 2N children form the N pairs of the next generation at random.

First the extinction of the lines of descendants of a fixed number of pairs is studied, when the population size becomes large. Under suitable conditions this structure can be approximately treated in the framework of a Galton-Watson process. In particular it is shown for the Wright-Fisher model that the discrepancy between the extinction probabilities in the model and in the approximating Galton-Watson process is of order N.

Next, the process of the number of ancestor-pairs of all pairs of a generation is analysed. Under suitable conditions this process, properly normed, has a weak limit as N becomes large. For the Wright-Fisher model this limit is an Ornstein–Uhlenbeck process (restricted to a discrete time-set). The corresponding stationary distributions of the backward processes converge to the normal distribution, as expected.

Here a response of a non-linear oscillator of the Liénard type with a large parameter α ≥ 0 is formulated as a solution of a two-dimensional stochastic differential equation with mean-field of the McKean type. This solution is governed by a special form of the Fokker–Planck equation such as the Smoluchowski–Kramers equation, which is an equation of motion for distribution functions in position and velocity space describing the Brownian motion of particles in an external field. By a change of time and displacement we find that the velocity process converges to a one-dimensional Ornstein–Uhlenbeck process as α →∞.

Making use of the integral equations given in [1], [2] and [3], the asymptotic behaviour of the first-passage time (FPT) p.d.f.'s through certain time-varying boundaries, including periodic boundaries, is determined for a class of one-dimensional diffusion processes with steady-state density. Sufficient conditions are given for the cases both of single and of pairs of asymptotically constant and asymptotically periodic boundaries, under which the FPT densities asymptotically exhibit an exponential behaviour. Explicit expressions are then worked out for the processes that can be obtained from the Ornstein–Uhlenbeck process by spatial transformations. Some new asymptotic results for the FPT density of the Wiener process are finally proved, together with a few miscellaneous results.