Related papers: Jump-Diffusions in Hilbert Spaces: Existence, Stab…
In this paper we study jump-diffusion stochastic differential equations (SDEs) with a discontinuous drift coefficient and a possibly degenerate diffusion coefficient. Such SDEs appear in applications such as optimal control problems in…
We propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central…
A novel approach to Riemann--Hilbert problems of particular class is introduced. The approach is applicable to problems in which the multiplicative jump is set on a half-line. Such problems are linked to some Wiener--Hopf problems motivated…
This paper proposes a novel low-rank approximation to the multivariate State-Space Model. The Stochastic Partial Differential Equation (SPDE) approach is applied component-wise to the independent-in-time Mat\'ern Gaussian innovation term in…
We derive the numerical schemes for the strong order integration of the set of the stochastic differential equations (SDEs) corresponding to the non-stationary Parker transport equation (PTE). PTE is 5-dimensional (3 spatial coordinates,…
We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable travelling wave solutions to the deterministic system retain their orbital stability if the…
The purpose of this paper is to establish the well-posedness of the stochastic Stefan problem on moving hypersurfaces. Through a specially designed transformation, it turns out we need to solve stochastic partial differential equations on a…
We consider the steady Swift - Hohenberg partial differential equation. It is a one-parameter family of PDE on the plane, modeling for example Rayleigh - B\'enard convection. For values of the parameter near its critical value, we look for…
Under nondegeneracy assumptions on the diffusion coefficients, we establish the derivative formulae of Bismut-Elworthy-Li's type for forward-backward stochastic differential equations with respect to Poisson random measure using the lent…
We study path-dependent SDEs in Hilbert spaces. By using methods based on contractions in Banach spaces, we prove existence and uniqueness of mild solutions, continuity of mild solutions with respect to perturbations of all the data of the…
The existence of stationary distributions to distribution dependent stochastic differential equations are investigated by using the ergodicity of the associated decoupled equation and the Schauder fixed point theorem. By using Zvonkin's…
We construct a moduli space of formally integrable and involutive ideal sheaves arising from systems of partial differential equations (PDEs) in the algebro-geometric setting, by introducing the $\mathcal{D}$-Hilbert and $\mathcal{D}$-Quot…
We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We…
This paper extends deterministic notions of Strong Stability Preservation (SSP) to the stochastic setting, enabling nonlinearly stable numerical solutions to stochastic differential equations (SDEs) and stochastic partial differential…
We propose a new, unified approach to solving jump-diffusion partial integro-differential equations (PIDEs) that often appear in mathematical finance. Our method consists of the following steps. First, a second-order operator splitting on…
We analyze the concepts of analytically weak solutions of stochastic differential equations (SDEs) in Hilbert spaces with time-dependent unbounded operators and give conditions for existence and uniqueness of such solutions. Our studies are…
Stochastic differential equations in Hilbert space as random nonlinear modified Schroedinger equations have achieved great attention in recent years; of particular interest is the long time behavior of their solutions. In this note we…
We establish well-posedness for a class of systems of SDEs with non-Lipschitz coefficients in the diffusion and jump terms and with two sources of interdependence: a monotone function of all the components in the drift of each SDE and the…
We present a novel approach for solving steady-state stochastic partial differential equations (PDEs) with high-dimensional random parameter space. The proposed approach combines spatial domain decomposition with basis adaptation for each…
We study the dependence of mild solutions to linear stochastic evolution equations on Hilbert space driven by Wiener noise, with drift having linear part of the type $A+\varepsilon G$, on the parameter $\varepsilon$. In particular, we study…