Related papers: Fractional L\'{e}vy-driven Ornstein--Uhlenbeck pro…

In this paper we introduce a model, the stochastic fractional delay differential equation (SFDDE), which is based on the linear stochastic delay differential equation and produces stationary processes with hyperbolically decaying…

We present an $L_{p}$-theory ($p\geq 2$) for time-fractional stochastic partial differential equations driven by L\'evy processes of the type $$ \partial^{\alpha}_{t}u=\sum_{i,j=1}^d a^{ij}u_{x^{i}x^{j}}…

In this paper, we get some convergence rates in total variation distance in approximating discretized paths of L{\'e}vy driven stochastic differential equations, assuming that the driving process is locally stable. The particular case of…

Fractional Ornstein-Uhlenbeck process of the second kind $(\text{fOU}_{2})$ is solution of the Langevin equation $\mathrm{d}X_t = -\theta X_t\,\mathrm{d}t+\mathrm{d}Y_t^{(1)}, \ \theta >0$ with Gaussian driving noise $ Y_t^{(1)} := \int^t_0…

In this paper, we study reflected generalized backward doubly stochastic differential equations driven by Teugels martingales associated with L\'evy process (RGBDSDELs, in short) with one continuous barrier. Under uniformly Lipschitz…

We develop efficient methods for simulating processes of Ornstein-Uhlenbeck type related to the class of $p$-tempered $\alpha$-stable ($\ts$) distributions. Our results hold for both the univariate and multivariate cases and we consider…

In this paper, we develop and analyze numerical methods for high dimensional Fokker-Planck equations by leveraging generative models from deep learning. Our starting point is a formulation of the Fokker-Planck equation as a system of…

Using a method of eigenfunction expansion, a stochastic equation is developed for the generalized Schr{\"o}dinger equation with random fluctuations. The wave field $ {\psi} $ is expanded in terms of eigenfunctions: $ {\psi} = \sum_{n} a_{n}…

It is well known that certain fractional diffusion equations can be solved by the densities of stable L\'evy motions. In this paper we use the classical semigroup approach for L\'evy processes to define semi-fractional derivatives, which…

We consider a Volterra convolution equation in $\mathbb{R}^d$ perturbed with an additive fractional Brownian motion of Riemann-Liouville type with Hurst parameter $H\in (0,1)$. We show that its solution solves a stochastic partial…

Using generalized Blumenthal--Getoor indices, we obtain criteria for the finiteness of the $p$-variation of L\'evy-type processes. This class of stochastic processes includes solutions of Skorokhod-type stochastic differential equations…

A general method is proposed which allows one to estimate drift and diffusion coefficients of a stochastic process governed by a Langevin equation. It extends a previously devised approach [R. Friedrich et al., Physics Letters A 271, 217…

We study a system of Forward-Backward Stochastic Differential Equations (FBSDEs) with time-delayed generators. The forward process includes a reflection component expressed via a Stieltjes integral, while the backward process takes the form…

This paper provides a probabilistic approach to solve linear equations involving Caputo and Riemann-Liouville type derivatives. Using the probabilistic interpretation of these operators as the generators of interrupted Feller processes, we…

We introduce a new class of integrators for stiff ODEs as well as SDEs. These integrators are (i) {\it Multiscale}: they are based on flow averaging and so do not fully resolve the fast variables and have a computational cost determined by…

We consider stochastic differential equations (SDEs) driven by a fractional Brownian motion with a drift coefficient that is allowed to be arbitrarily close to criticality in a scaling sense. We develop a comprehensive solution theory that…

We introduce an extended version of the fractional Ornstein-Uhlenbeck (FOU) process where the integrand is replaced by the exponential of an independent L\'evy process. We call the process the generalized fractional Ornstein-Uhlenbeck…

The study of non-stationary processes whose local form has controlled properties is a fruitful and important area of research, both in theory and applications. We present here a construction of multifractional multistable processes, based…

In this article, we consider slow-fast McKean-Vlasov stochastic differential equations driven by Brownian motions and fractional Brownian motions. We give a definition of the large deviation principle (LDP) on the product space related to…

We demonstrate that two Ornstein--Uhlenbeck processes, that is, solutions to certain stochastic differential equations that are driven by a L\'evy process L have equivalent laws as long as the eigenvalues of the covariance operator…