We consider parametric estimation for a parabolic linear second order stochastic partial differential equation (SPDE) from high frequency data which are observed in time and space. By using thinned obtained the data, adaptive estimators of coefficient parameters including volatility parameter proposed. Moreover, we give some examples simulation results SPDE model based on data.

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...

We deal with parametric estimation for a parabolic linear second-order stochastic partial differential equation (SPDE) small dispersion parameter based on high-frequency data which are observed in time and space. Using the thinned respect to space obtained from data, minimum contrast estimators of two coefficient parameters SPDE proposed. With these we construct an approximation coordinate proc...

Linear-implicit versions of strong Taylor numerical schemes for finite dimensional It6 stochastic differential equations (SDEs) are shown to have the same order as the original scheme. The combined truncation and global discretization error of an 7 strong linear-implicit Taylor scheme with time-step A applied to the N dimensional It6-Galerkin SDE for a class of parabolic stochastic partial diff...

The coefficient function of the leading differential operator is estimated from observations a linear stochastic partial equation (SPDE). estimation based on continuous time which are localised in space. For asymptotic regime with fixed horizon and spatial resolution tending to zero, we provide rate-optimal estimators establish scaling limits deterministic PDE SPDE growing domains. robust lower...

Existence and uniqueness for semilinear stochastic evolution equations with additive noise by means of finite-dimensional Galerkin approximations is established and the convergence rate of the Galerkin approximations to the solution of the stochastic evolution equation is estimated. These abstract results are applied to several examples of stochastic partial differential equations (SPDEs) of ev...

The numerical solution of stochastic partial differential equations (SPDEs) is at a stage of development roughly similar to that of stochastic ordinary differential equations (SODEs) in the 1970s, when stochastic Taylor schemes based on an iterated application of the Itô formula were introduced and used to derive higher order numerical schemes. An Itô formula in the generality needed for Taylor...

Diffusion processes intended to model the continuous state space limit of birth–death processes, chemical reactions, and other discrete particle systems often involve multiplicative noise where the diffusion vanishes near one (or more) of the state space boundaries. Standard direct numerical simulation schemes for the associated stochastic differential equations run the risk of “overshooting”, ...

Kardar-Parisi-Zhang (KPZ) equation is a quasilinear stochastic partial differential equation(SPDE) driven by a space-time white noise. In recent years there have been several works directed towards giving a rigorous meaning to a solution of this equation. Bertini, Cancrini and Giacomin [2, 3] have proposed a notion of a solution through a limiting procedure and a certain renormalization of the ...

Neural network dynamics are governed by the interaction of spiking neurons. Stochastic aspects of single-neuron dynamics propagate up to the network level and shape the dynamical and informational properties of the population. Mean-field models of population activity disregard the finite-size stochastic fluctuations of network dynamics and thus offer a deterministic description of the system. H...