In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretisation on an unbounded domain is convergent of first order in time and second order in the spatial grid size, and that the discretisation is stable with respect to boundary data. Numerica...

This paper develops a fractional stochastic partial differential equation (SPDE) to model the evolution of random tangent vector field on unit sphere. The SPDE is governed by diffusion operator Lévy-type behaviour spatial solution, derivative in time depict intermittency its temporal and driven vector-valued Brownian motion sphere characterize long-range dependence. solution presented form Kar...

Modulationor Amplitude-Equations are a universal tool to approximate solutions of complicated systems given by partial or stochastic partial differential equations (SPDEs) near a change of stability, when there is no center manifold theory available. Relying on the natural separation of time-scales at the bifurcation the solution of the original equation is well approximated by the bifurcating ...

With the fragmentation of electronic markets, exchanges are now competing in order to attract trading activity on their platform. Consequently, they developed several regulatory tools control liquidity provision / consumption pool. In this paper, we study problem an exchange using incentives increase market liquidity. We model limit book as solution a stochastic partial differential equation (S...

We deal with the Sobolev space theory for stochastic partial differential equation (SPDE) driven by Wiener processes $$\begin{aligned} \partial _{t}^{\alpha }u=\left( \phi (\varDelta ) u +f(u) \right) + _t^\beta \sum _{k=1}^\infty \int _0^t g^k(u)\,dw_s^k, \quad t>0, \,x\in {\mathbb {R}}^d \end{aligned}$$ as well SPDE space-time white noise ^{\alpha }_{t}u=\phi )u f(u) ^{\beta -1}_{t}h(u) {\dot...

The solutions of parabolic and hyperbolic stochastic partial differential equations (SPDEs) driven by an infinite dimensional Brownian motion, which is a martingale, are in general not semi-martingales any more and therefore do not satisfy an Itô formula like the solutions of finite dimensional stochastic differential equations (SODEs). In particular, it is not possible to derive stochastic Tay...

We consider conditional McKean–Vlasov stochastic differential equations (SDEs), as the ones arising in large-system limit of mean field games and particle systems with interactions when common noise is present. The time-marginals solutions to these SDEs are governed by non-linear partial (SPDEs) second order, whereas their laws satisfy Fokker–Planck on space probability measures. Our paper esta...

in this paper, a high-order and conditionally stable stochastic difference scheme is proposed for the numerical solution of $rm ithat{o}$ stochastic advection diffusion equation with one dimensional white noise process. we applied a finite difference approximation of fourth-order for discretizing space spatial derivative of this equation. the main properties of deterministic difference schemes,...

A non-stationary Gaussian random field model is developed based on a combination of the stochastic partial differential equation (SPDE) approach and classical deformation method. With method, stationary defined domain which deformed so that becomes non-stationary. We show if Matérn as solution to fractional SPDE, resulting can be represented another SPDE domain. By defining in this way, computa...

We provide existence and uniqueness of global (and local) mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear growth (or local boundedness, resp.) conditions. The socalled “method of the moving frame” allows us to reduce the SPDE problems to SDE problems.