Related papers: Nonparametric Estimation in SDE Models Involving a…

We study pathwise approximation of scalar stochastic differential equations at a single point. We provide the exact rate of convergence of the minimal errors that can be achieved by arbitrary numerical methods that are based (in a…

Non-uniform sampling arises when an experimenter does not have full control over the sampling characteristics of the process under investigation. Moreover, it is introduced intentionally in algorithms such as Bayesian optimization and…

Considering a linear parabolic stochastic partial differential equation driven by impulsive space time noise, dX_t+AX_t dt= Q^{1/2}dZ_t, X_0=x_0\in H, t\in [0,T], we approximate the distribution of X_T. (Z_t)_{t\in[0,T]} is an impulsive…

We present a versatile framework to study strong existence and uniqueness for stochastic differential equations (SDEs) in Hilbert spaces with irregular drift. We consider an SDE in a separable Hilbert space $H$ \begin{equation*} dX_t= (A…

In this paper we study weak solutions for the following type of stochastic differential equation \[ dX_{t}=dW_{t}+b(t, X_{t})dt, \quad t\ge s, \quad X_{s}=x, \] where $b: [0,\infty) \times \mathbb{R}^{d} \to \mathbb{R}^{d}$ is a measurable…

We consider the Stochastic Differential Equation $X_t = X_0 + \int_0^t b(s,X_s) ds + B_t$, in $\mathbb{R}^d$. We give an example of a drift $b$ such that there does not exist a weak solution, but there exists a solution for almost every…

In this paper we focus on the so called identification problem for a backward SDE driven by a continuous local martingale and a possibly non quasi-left-continuous random measure. Supposing that a solution (Y, Z, U) of a backward SDE is such…

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…

We present a Bayesian non-parametric way of inferring stochastic differential equations for both regression tasks and continuous-time dynamical modelling. The work has high emphasis on the stochastic part of the differential equation, also…

Strongly consistent and asymptotically normal estimators of the Hurst parameter of solutions of stochastic differential equations are proposed. The estimators are based on discrete observations of the underlying processes.

We study nonparametric estimation in dynamical systems described by ordinary differential equations (ODEs). Specifically, we focus on estimating the unknown function $f \colon \mathbb{R}^d \to \mathbb{R}^d$ that governs the system dynamics…

In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms \[\mathbb{R}^d\ni x\quad\longmapsto\quad\phi_{s,t}(x)\in \mathbb{R}^d,\qquad s,t\in\mathbb{R}\] for a stochastic differential equation (SDE) of the…

We consider a non-linear parabolic partial differential equation (PDE) on $\mathbb R^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity…

We consider the one-dimensional squared Bessel process given by the stochastic differential equation (SDE) \begin{align*} dX_t = 1\,dt + 2\sqrt{X_t}\,dW_t, \quad X_0=x_0, \quad t\in[0,1], \end{align*} and study strong (pathwise)…

This paper establishes a Transition Path Theory (TPT) for L\'{e}vy-type processes, addressing a critical gap in the study of the transition mechanism between meta-stabile states in non-Gaussian stochastic systems. A key contribution is the…

We study SDE $$ d X_t = b(X_t) \, dt + A(X_{t-}) \, d Z_t, \quad X_{0} = x \in \mathbb{R}^d, \quad t \geq 0 $$ where $Z=(Z^1, \dots, Z^d)^T$, with $Z^i, i=1,\dots, d$ being independent one-dimensional symmetric jump L\'evy processes, not…

The numerical analysis of stochastic parabolic partial differential equations of the form $$ du + A(u) = f \,dt + g \, dW, $$ is surveyed, where $A$ is a partial operator and $W$ a Brownian motion. This manuscript unifies much of the theory…

We consider a general one-dimensional overdamped diffusion model described by the It\^{o} stochastic differential equation (SDE) ${dX_t=\mu(X_t,t)dt+\sigma(X_t,t)dW_t}$, where $W_t$ is the standard Wiener process. We obtain a specific…

We prove that the weak version of the SPDE problem \begin{align*} dV_{t}(x) & = [-\mu V_{t}'(x) + \frac{1}{2} (\sigma_{M}^{2} + \sigma_{I}^{2})V_{t}"(x)]dt - \sigma_{M} V_{t}'(x)dW^{M}_{t}, \quad x > 0, \\ V_{t}(0) &= 0 \end{align*} with a…

We consider stochastic PDEs \[dY_t = L(Y_t)\, dt + A(Y_t).\, dB_t, t > 0\] and associated PDEs \[du_t = L u_t\, dt, t > 0\] with regular initial conditions. Here, $L$ and $A$ are certain partial differential operators involving…