Abstract. We study the asymptotic behavior of solutions to stochastic evolution equations with monotone drift and multiplicative Poisson noise in the variational setting, thus covering a large class of (fully) nonlinear partial differential equations perturbed by jump noise. In particular, we provide sufficient conditions for the existence, ergodicity, and uniqueness of invariant measures. Furt...

the numerical methods are of great importance for approximating the solutions of nonlinear ordinary or partial differential equations, especially when the nonlinear differential equation under consideration faces difficulties in obtaining its exact solution. in this latter case, we usually resort to one of the efficient numerical methods. in this paper, the chebyshev collocation method is sugge...

A numerical approach for the approximation of inertial manifolds of stochastic evolutionary equations with multiplicative noise is presented and illustrated. After splitting the stochastic evolutionary equations into a backward and a forward part, a numerical scheme is devised for solving this backward-forward stochastic system, and an ensemble of graphs representing the inertial manifold is co...

In common finance literature, Black-Scholes partial differential equation of option pricing is usually derived with no-arbitrage principle. Considering an asset market, Merton applied the Hamilton-Jacobi-Bellman techniques of his continuous-time consumption-portfolio problem, deriving general equilibrium relationships among the securities in the asset market. In special case where the interest ...

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 ...

In this talk, we will introduce high accurate numerical schemes for solving forward backward stochastic differential equations (FBSDEs) with jumps. In these schemes, the simplest Euler scheme with only one jump is used to solve the forward stochastic differential equation (SDE), and multistep schemes is used to solve the backward stochastic differential equation (BSDE) with high convergence rat...

We extend Walsh’s theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulatio...

The dynamics of a kink driven by noise is analyzed using the two collective variables of the Rice ansatz: position and width. Starting from a stochastic partial differential equation, with the phi(4) potential in the overdamped limit, the pair of stochastic differential equations for the collective variables are derived without approximation other than the ansatz itself. From the steady state p...

Perturbations to Markov chains and Markov processes are considered. The unperturbed problem is assumed to be geometrically ergodic in the sense usually established through the use of Foster–Lyapunov drift conditions. The perturbations are assumed to be uniform, in a weak sense, on bounded time intervals. The long-time behavior of the perturbed chain is studied. Applications are given to numeric...

In this article spatial and temporal regularity of the solution process of a stochastic partial differential equation (SPDE) of evolutionary type with nonlinear multiplicative trace class noise is analyzed.