Fractional Ornstein-Uhlenbeck Research Articles

On some local asymptotic properties of sequences with a random index

«Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия» — научно-теоретический рецензируемый журнал, публикующий исследования в области математики, механики и астрономии, а также математического моделирования.

Rough stochastic elasticity of variance and option pricing

This study is concerned with the elasticity of variance for risky assets. We show that the elasticity of variance for S&P500 exhibits short-range correlations. By using asymptotic and martingale methods, we obtain a semi-analytical expression for the option price in the two-scale regime where the constant elasticity of variance is perturbed by a smooth and bounded function of a rapid fractional Ornstein–Uhlenbeck process with Hurst exponent within (0,12). The associated implied volatility is presented and discussed. As a result, the scope of Markov stochastic elasticity of variance model is extended to a non-Markov case.

Volatility estimation in fractional Ornstein-Uhlenbeck models

In this article, we study the asymptotic behavior of the realized quadratic variation of a process where u is a β-Hölder continuous process with and where and BH is a fractional Brownian motion with Hurst index By exploiting the concentration phenomena, we prove almost sure convergence of the quadratic variation, that holds uniformly in time and on the full range As an application, we construct strongly consistent estimator for the integrated volatility parameter in a model driven by

Least squares estimator of Ornstein-Uhlenbeck processes driven by fractional Lévy processes with periodic mean

We deal with the least squares estimator for the drift parameters of an Ornstein-Uhlenbeck process with periodic mean function driven by fractional Levy process. For this estimator, we obtain consistency and the asymptotic distribution. Compared with fractional Ornstein-Uhlenbeck and Ornstein-Uhlenbeck driven by Levy process, they can be regarded both as a Levy generalization of fractional Brownian motion and a fractional generalization of Levy process.

Fractional Ornstein-Uhlenbeck Process with Stochastic Forcing, and its Applications

We consider a fractional Ornstein-Uhlenbeck process involving a stochastic forcing term in the drift, as a solution of a linear stochastic differential equation driven by a fractional Brownian motion. For such process we specify mean and covariance functions, concentrating on their asymptotic behavior. This gives us a sort of short- or long-range dependence, under specified hypotheses on the covariance of the forcing process. Applications of this process in neuronal modeling are discussed, providing an example of a stochastic forcing term as a linear combination of Heaviside functions with random center. Simulation algorithms for the sample path of this process are finally given.

On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes

We investigate the main statistical parameters of the integral over time of the fractional Brownian motion and of a kind of pseudo-fractional Gaussian process, obtained as a classical Gauss–Markov process from Doob representation by replacing Brownian motion with fractional Brownian motion. Possible applications in the context of neuronal models are highlighted. A fractional Ornstein–Uhlenbeck process is considered and relations with the integral of the pseudo-fractional Gaussian process are provided.

Volatility tail risk under fractionality

Abstract We study the probabilistic properties of the fractional Ornstein–Uhlenbeck process, which is a relevant framework for volatility modeling in continuous time. First, we compute an expression for its variance for any value of the Hurst parameter, H ∈ (0, 1). Second, we derive the density of the process and we calculate the probability of its supremum to be above a given threshold. We provide a number of illustrations based on fractional stochastic volatility models, such as those of Comte and Renault (1998), Bayer et al. (2016) and Gatheral et al. (2018). Finally, the empirical analysis, based on the realized variance series of S&P500, shows the usefulness of these theoretical results for risk management purposes, especially when a characterization of the volatility tail risk is needed.

Standard and fractional Ornstein-Uhlenbeck process on a growing domain

We study normal diffusive and subdiffusive processes in a harmonic potential (Ornstein-Uhlenbeck process) on a uniformly growing or contracting domain. Our starting point is a recently derived fractional Fokker-Planck equation, which covers both the case of Brownian diffusion and the case of a subdiffusive continuous-time random walk (CTRW). We find a high sensitivity of the random walk properties to the details of the domain growth rate, which gives rise to a variety of regimes with extremely different behaviors. At the origin of this rich phenomenology is the fact that the walkers still move while they wait to jump, since they are dragged by the deterministic drift arising from the domain growth. Thus, the increasingly long waiting times associated with the aging of the subdiffusive CTRW imply that, in the time interval between two consecutive jumps, the walkers might travel over much longer distances than in the normal diffusive case. This gives rise to seemingly counterintuitive effects. For example, on a static domain, both Brownian diffusion and subdiffusive CTRWs yield a stationary particle distribution with finite width when a harmonic potential is at play, thus indicating a confinement of the diffusing particle. However, for a sufficiently fast growing or contracting domain, this qualitative behavior breaks down, and differences between the Brownian case and the subdiffusive case are found. In the case of Brownian particles, a sufficiently fast exponential domain growth is needed to break the confinement induced by the harmonic force; in contrast, for subdiffusive particles such a breakdown may already take place for a sufficiently fast power-law domain growth. Our analytic and numerical results for both types of diffusion are fully confirmed by random walk simulations.

Self-normalized asymptotic properties for the parameter estimation in fractional Ornstein–Uhlenbeck process

In this paper, we consider the self-normalized asymptotic properties of the parameter estimators in the fractional Ornstein–Uhlenbeck process. The deviation inequalities, Cramér-type moderate deviations and Berry–Esseen bounds are obtained. The main methods include the deviation inequalities and moderate deviations for multiple Wiener–Itô integrals [P. Major, Tail behavior of multiple integrals and U-statistics, Probab. Surv. 2 (2005) 448–505; On a multivariate version of Bernsteins inequality, Electron. J. Probab. 12 (2007) 966–988; M. Schulte and C. Thäle, Cumulants on Wiener chaos: Moderate deviations and the fourth moment theorem, J. Funct. Anal. 270 (2016) 2223–2248], as well as the Delta methods in large deviations [F. Q. Gao and X. Q. Zhao, Delta method in large deviations and moderate deviations for estimators, Ann. Statist. 39 (2011) 1211–1240]. For applications, we propose two test statistics which can be used to construct confidence intervals and rejection regions in the hypothesis testing for the drift coefficient. It is shown that the Type II errors tend to be zero exponential when using the proposed test statistics.

Nonparametric estimation in fractional SDE

This paper deals with the consistency and a rate of convergence for a Nadaraya-Watson estimator of the drift function of a stochastic differential equation driven by an additive fractional noise. The results of this paper are obtained via both some long-time behavior properties of Hairer and some properties of the Skorokhod integral with respect to the fractional Brownian motion. These results are illustrated on the fractional Ornstein-Uhlenbeck process.

Fractional iterated Ornstein-Uhlenbeck Processes

In this work we present a Gaussian process that arise from the iteration of p fractional Ornstein-Uhlenbeck processes generated by the same fractional Brownian motion. This iteration results, when the values of lambdas are pairwise differents, in a particular linear combination of those processes. Although for $H>1/2$ each term of the linear combination is a long memory processes, we prove that it results in a short memory processes. We include applications to real data that show improvement in predictive performance compared with different ARMA models.

Exponential stability of SDEs driven by fBm with Markovian switching

In this paper, we focus on the exponential stability of stochastic differential equations driven by fractional Brownian motion (fBm) with Hurst parameter $ H\in(1/2, 1) $. Based on the generalized Itô formula and representation of the fBm, some sufficient conditions for exponential stability of a class of SDEs with additive fractional noise are given. Besides, we present a criterion on the exponential stability for the fractional Ornstein-Uhlenbeck process with Markov switching. A numerical example is provided to illustrate our results.

Laws of large numbers for supercritical branching Gaussian processes

A general class of non-Markov, supercritical Gaussian branching particle systems is introduced and its long-time asymptotics is studied. Both weak and strong laws of large numbers are developed with the limit object being characterized in terms of particle motion/mutation. Long memory processes, like branching fractional Brownian motion and fractional Ornstein–Uhlenbeck processes with large Hurst parameters, as well as rough processes, like fractional processes with smaller Hurst parameter, are included as important examples. General branching with second moments is allowed and moment measure techniques are utilized.

Optimal portfolio under fractional stochastic environment

AbstractRough stochastic volatility models have attracted a lot of attention recently, in particular for the linear option pricing problem. In this paper, starting with power utilities, we propose to use a martingale distortion representation of the optimal value function for the nonlinear asset allocation problem in a (non‐Markovian) fractional stochastic environment (for all values of the Hurst index ). We rigorously establish a first‐order approximation of the optimal value, when the return and volatility of the underlying asset are functions of a stationary slowly varying fractional Ornstein–Uhlenbeck process. We prove that this approximation can be also generated by a fixed zeroth‐ order trading strategy providing an explicit strategy which is asymptotically optimal in all admissible controls. Furthermore, we extend the discussion to general utility functions, and obtain the asymptotic optimality of this fixed strategy in a specific family of admissible strategies.

Ergodicity and Drift Parameter Estimation for Infinite-Dimensional Fractional Ornstein–Uhlenbeck Process of the Second Kind

We introduce the Hilbert-valued fractional Ornstein–Uhlenbeck of the second kind as the mild solution of a stochastic evolution equation with fractional-type Gaussian noise. We study the stationarity and the ergodicity for this infinite-dimensional process. Finally, via Malliavin calculus, we also analyze the least squares estimator of the drift parameter of the fractional Ornstein–Uhlenbeck of the second kind.

LAN property for stochastic differential equations with additive fractional noise and continuous time observation

We consider a stochastic differential equation with additive fractional noise with Hurst parameter H>1∕2, and a non-linear drift depending on an unknown parameter. We show the Local Asymptotic Normality property (LAN) of this parametric model with rate τ as τ→∞, when the solution is observed continuously on the time interval [0,τ]. The proof uses ergodic properties of the equation and a Girsanov-type transform. We analyze the particular case of the fractional Ornstein–Uhlenbeck process and show that the Maximum Likelihood Estimator is asymptotically efficient in the sense of the Minimax Theorem.

Option pricing under fast‐varying long‐memory stochastic volatility

AbstractRecent empirical studies suggest that the volatility of an underlying price process may have correlations that decay slowly under certain market conditions. In this paper, the volatility is modeled as a stationary process with long‐range correlation properties in order to capture such a situation, and we consider European option pricing. This means that the volatility process is neither a Markov process nor a martingale. However, by exploiting the fact that the price process is still a semimartingale and accordingly using the martingale method, we can obtain an analytical expression for the option price in the regime where the volatility process is fast mean reverting. The volatility process is modeled as a smooth and bounded function of a fractional Ornstein–Uhlenbeck process. We give the expression for the implied volatility, which has a fractional term structure.

Option Pricing with Fractional Stochastic Volatility and Discontinuous Payoff Function of Polynomial Growth

We consider the pricing problem related to payoffs of polynomial growth that can have discontinuities of the 1st kind. The asset price dynamic is modeled within the Black-Scholes framework characterized by a stochastic volatility term driven by a fractional Ornstein-Uhlenbeck process. In order to solve the aforementioned problem, we consider three approaches. The first one consists in a suitable transformation of the initial value of the asset price, in order to eliminate possible discontinuities. Then we discretize both the Wiener process and the fractional Brownian motion and estimate the rate of convergence of the related discretized price to its real value whose closed-form analytical expression is usually difficult to obtain. The second approach consists in considering the conditional expectation with respect to the entire trajectory of the fractional Brownian motion (fBm). Here we derive a presentation for the option price which involves only an integral functional depending on the fBm trajectory, and then discretize the fBm and estimate the rate of convergence of the associated numerical scheme. In both cases the rate of convergence is the same and equals n−rH, where n is the partition size, H is the Hurst index of the fBm, and r is the Holder exponent of the volatility function. The third method consists in calculating the density of the integral functional depending on the trajectory of the fBm via Malliavin calculus and providing the option price in terms of the associated probability density.

Parameter estimation for discretely observed non-ergodic fractional Ornstein–Uhlenbeck processes of the second kind

We use the least squares type estimation to estimate the drift parameter $\theta>0$ of a non-ergodic fractional Ornstein–Uhlenbeck process of the second kind defined as $dX_{t}=\theta X_{t}\,dt+dY_{t}^{(1)},X_{0}=0$, $t\geq0$, where $Y_{t}^{(1)}=\int_{0}^{t}e^{-s}\,dB_{a_{s}}$ with $a_{t}=He^{\frac{t}{H}}$, and $\{B_{t},t\geq0\}$ is a fractional Brownian motion of Hurst parameter $H\in(\frac{1}{2},1)$. We assume that the process $\{X_{t},t\geq0\}$ is observed at discrete time instants $t_{1}=\Delta_{n},\ldots,t_{n}=n\Delta_{n}$. We construct two estimators $\hat{\theta}_{n}$ and $\check{\theta}_{n}$ of $\theta$ which are strongly consistent and we prove that these estimators are $\sqrt{n\Delta_{n}}$-consistent, in the sense that the sequences $\sqrt{n\Delta_{n}}(\hat{\theta}_{n}-\theta)$ and $\sqrt{n\Delta_{n}}(\check{\theta}_{n}-\theta)$ are tight.

Fractional Ornstein–Uhlenbeck noise

Fractional Ornstein–Uhlenbeck noise is considered and investigated. The fractional Ornstein–Uhlenbeck noise may be linked with a supercapacitor driven by the white noise, and its correlation function for the stationary state shows monotonic and oscillatory decays. In the case of the oscillatory behavior the correlation function presents behaviors similar to those of the harmonic noise (harmonic oscillator driven by the white noise). For application, the Langevin equation with the harmonic potential driven by the fractional Ornstein–Uhlenbeck noise is considered; the first two moments and mean energy are investigated.