Related papers: Gradient Estimate for Ornstein-Uhlenbeck Jump Proc…
The score function for the diffusion process, also known as the gradient of the log-density, is a basic concept to characterize the probability flow with important applications in the score-based diffusion generative modelling and the…
Observing prices of European put and call options, we calibrate exponential L\'evy models nonparametrically. We discuss the efficient implementation of the spectral estimation procedures for L\'evy models of finite jump activity as well as…
This paper studies the identification of the L\'{e}vy jump measure of a discretely-sampled semimartingale. We define successive Blumenthal-Getoor indices of jump activity, and show that the leading index can always be identified, but that…
In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on an unknown parameter $\theta$. We suppose that the process is discretely observed at the instants (t n i)i=0,...,n with $\Delta$n = sup…
Assuming that a reflected Ornstein-Uhlenbeck state process is observed at discrete time instants, we propose generalized moment estimators to estimate all drift and diffusion parameters via the celebrated ergodic theorem. With the sampling…
We consider a class of L\'evy-type processes derived via a Doob-transform from L\'evy processes conditioned by a control function called potential. These processes have position-dependent and generally unbounded components, with stationary…
L\'{e}vy flight models whose jumps have infinite moments are mathematically used to describe the superdiffusion in complex systems. Exponentially tempering the Levy measure of L\'{e}vy flights leads to the tempered stable L\'{e}vy processes…
In this paper we study the problem of statistical inference for a continuous-time moving average L\'evy process of the form $$Z_{t} = \int_{\mathbb{R}}\mathcal{K}(t-s)\, dL_{s},\quad t\in\mathbb{R}$$ with a deterministic kernel (\K\) and a…
Let the Ornstein-Uhlenbeck process $(X_t)_{t\ge0}$ driven by a fractional Brownian motion $B^{H }$, described by $dX_t = -\theta X_t dt + \sigma dB_t^{H }$ be observed at discrete time instants $t_k=kh$, $k=0, 1, 2, \cdots, 2n+2 $. We…
We present here a simple method for computing the large deviation of long time average for stochastic jump processes. We show that the computation of the rate function can be reduced to that of a partial differential equation governing the…
In this paper, we establish a moderate deviation principle for stochastic models of two-dimensional second grade fluids driven by L\'evy noise. We will adopt the weak convergence approach. Because of the appearance of jumps, this result is…
Let $X$ be a L\'evy process with absolutely continuous L\'evy measure $\nu$. Small time polynomial expansions of order $n$ in $t$ are obtained for the tails $P(X_{t}\geq{}y)$ of the process, assuming smoothness conditions on the L\'evy…
A new multi-factor short rate model is presented which is bounded from below by a real-valued function of time. The mean-reverting short rate process is modeled by a sum of pure-jump Ornstein--Uhlenbeck processes such that the related bond…
Efficient estimation of a non-Gaussian stable Levy process with drift and symmetric jumps observed at high frequency is considered. For this statistical experiment, the local asymptotic normality of the likelihood is proved with a…
This article study the class of distributions obtained by subordinating L\'evy processes and L\'evy bases. To do this we derive properties of a suitable mapping obtained via L\'evy mixing. We show that our results can be used to solve the…
In this article, we employ a collection of stochastic differential equations with drift and diffusion coefficients approximated by neural networks to predict the trend of chaotic time series which has big jump properties. Our contributions…
We consider the problem of modelling restricted interactions between continuously-observed time series as given by a known static graph (or network) structure. For this purpose, we define a parametric multivariate Graph Ornstein-Uhlenbeck…
In this paper, an integration by parts formula was derived for jump processes on Hilbert spaces. Using this formula, we investigated derivative formula and exponential ergodicity for nonlinear SPDEs driven by purely jump processes.
We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients…
Nonparametric methods for the estimation of the Levy density of a Levy process are developed. Estimators that can be written in terms of the ``jumps'' of the process are introduced, and so are discrete-data based approximations. A model…