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In this article, we examine a stochastic partial differential equation (SPDE) driven by a symmetric $\alpha$-stable (S$\alpha$S) L\'evy noise, that is multiplied by a linear function $\sigma(u)=u$ of the solution. The solution is…
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…
A new asymptotic expansion scheme for backward SDEs (BSDEs) is proposed.The perturbation parameter is introduced just to scale the forward stochastic variables within a BSDE. In contrast to the standard small-diffusion asymptotic expansion…
Stochastic differential equations (SDEs) are of utmost importance in various scientific and industrial areas. They are the natural description of dynamical processes whose precise equations of motion are either not known or too expensive to…
This paper presents a novel mathematical framework for representing uncertainty in large deformation diffeomorphic image registration. The Bayesian posterior distribution over the deformations aligning a moving and a fixed image is…
Multiscale stochastic dynamical systems have been widely adopted to a variety of scientific and engineering problems due to their capability of depicting complex phenomena in many real world applications. This work is devoted to…
Understanding the training dynamics of deep learning models is perhaps a necessary step toward demystifying the effectiveness of these models. In particular, how do data from different classes gradually become separable in their feature…
We identify effective stochastic differential equations (SDE) for coarse observables of fine-grained particle- or agent-based simulations; these SDE then provide useful coarse surrogate models of the fine scale dynamics. We approximate the…
Large sectors of the recent optimization literature focused in the last decade on the development of optimal stochastic first order schemes for constrained convex models under progressively relaxed assumptions. Stochastic proximal point is…
We investigate the problem of joint statistical estimation of several parameters for a stochastic differential equation driven by an additive fractional Brownian motion. Based on discrete-time observations of the model, we construct an…
We consider a linear stochastic differential equation with stochastic drift and multiplicative noise. We study the problem of approximating its solution with the process that solves the equation where the possibly stochastic drift is…
Stochastic neurons and hard non-linearities can be useful for a number of reasons in deep learning models, but in many cases they pose a challenging problem: how to estimate the gradient of a loss function with respect to the input of such…
The Latent Stochastic Differential Equation (SDE) is a powerful tool for time series and sequence modeling. However, training Latent SDEs typically relies on adjoint sensitivity methods, which depend on simulation and backpropagation…
We propose and analyse a new type of fully discrete surface finite element approximation of a class of linear parabolic stochastic evolution equations with additive noise. Our discretization uses a surface finite element approximation of…
We present DeepFDM, a differentiable finite-difference framework for learning spatially varying coefficients in time-dependent partial differential equations (PDEs). By embedding a classical forward-Euler discretization into a convolutional…
In this study we explore a new simulation scheme for partial differential equations known as Information Field Dynamics (IFD). Information field dynamics attempts to improve on existing simulation schemes by incorporating Bayesian field…
The concept of the path-dependent partial differential equation (PPDE) was first introduced in the context of path-dependent derivatives in financial markets. Its semilinear form was later identified as a non-Markovian backward stochastic…
The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations…
Backward stochastic differential equations (BSDEs) appear in numeruous applications. Classical approximation methods suffer from the curse of dimensionality and deep learning-based approximation methods are not known to converge to the BSDE…
We present convergence results in expectation for stochastic subspace correction schemes and their accelerated versions to solve symmetric positive-definite variational problems, and discuss their potential for achieving fault tolerance in…