Related papers: Exact Moment Estimation of Stochastic Differential…

Overparameterized stochastic differential equation (SDE) models have achieved remarkable success in various complex environments, such as PDE-constrained optimization, stochastic control and reinforcement learning, financial engineering,…

It is critical yet challenging for deep learning models to properly characterize uncertainty that is pervasive in real-world environments. Although a lot of efforts have been made, such as heteroscedastic neural networks (HNNs), little work…

Recently in [M. Hairer, M. Hutzenthaler, and A. Jentzen, Ann. Probab. 43, 2 (2015), 468--527] and [A. Jentzen, T. M\"uller-Gronbach, and L. Yaroslavtseva, Commun. Math. Sci. 14, 6 (2016), 1477--1500] stochastic differential equations (SDEs)…

We consider the "moment vanishing problem" for a general class of piecewise-analytic functions which satisfy on each continuity interval a linear ODE with polynomial coefficients. This problem, which essentially asks how many zero first…

We develop a method to approximate the moments of a discrete-time stochastic polynomial system. Our method is built upon Carleman linearization with truncation. Specifically, we take a stochastic polynomial system with finitely many states…

To study the nonlinear properties of complex natural phenomena, the evolution of the quantity of interest can be often represented by systems of coupled nonlinear stochastic differential equations (SDEs). These SDEs typically contain…

Stochasticity plays a key role in many biological systems, necessitating the calibration of stochastic mathematical models to interpret associated data. For model parameters to be estimated reliably, it is typically the case that they must…

We present an explicit method for simulating stochastic differential equations (SDEs) that have variable diffusion coefficients and satisfy the detailed balance condition with respect to a known equilibrium density. In Tupper and Yang…

In this paper, we propose a data-driven framework for model discovery of stochastic differential equations (SDEs) from a single trajectory, without requiring the ergodicity or stationary assumption on the underlying continuous process. By…

We introduce and analyze a novel class of inverse problems for stochastic dynamics: Given the ergodic invariant measure of a stochastic process governed by a nonlinear stochastic ordinary or partial differential equation (SODE or SPDE), we…

We solve the Skorokhod embedding problem for a class of stochastic processes satisfying an inhomogeneous stochastic differential equation (SDE) of the form $d A_t =\mu (t, A_t) d t + \sigma(t, A_t) d W_t$. We provide sufficient conditions…

Stochastic differential equations (SDEs) or diffusions are continuous-valued continuous-time stochastic processes widely used in the applied and mathematical sciences. Simulating paths from these processes is usually an intractable problem,…

In this work, we determine the full expression for the global truncation error of hyperbolic partial differential equations (PDEs). In particular, we use theoretical analysis and symbolic algebra to find exact expressions for the…

Distributed stochastic optimization algorithms can simultaneously process large-scale datasets, significantly accelerating model training. However, their effectiveness is often hindered by the sparsity of distributed networks and data…

In the recent article [Hairer, M., Hutzenthaler, M., Jentzen, A., Loss of regularity for Kolmogorov equations, Ann. Probab. 43 (2015), no. 2, 468--527] it has been shown that there exist stochastic differential equations (SDEs) with…

Stochastic differential equations (SDEs) are a staple of mathematical modelling of temporal dynamics. However, a fundamental limitation has been that such models have typically been relatively inflexible, which recent work introducing…

We study stochastic differential equations (SDEs) whose drift and diffusion coefficients are path-dependent and controlled. We construct a value process on the canonical path space, considered simultaneously under a family of singular…

This paper studies stabilities of stochastic differential equation (SDE) driven by time-changed L\'evy noise in both probability and moment sense. This provides more flexibility in modeling schemes in application areas including physics,…

The moments of spatial probabilistic systems are often given by an infinite hierarchy of coupled differential equations. Moment closure methods are used to approximate a subset of low order moments by terminating the hierarchy at some order…

Physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution.…