Luciano Tubaro

Expansion of the global error for numerical schemes solving stochastic differential equations

Given the solution (Xt ) of a Stochastic Differential System, two situat,ions are considered: computat,ion of Ef(Xt ) by a Monte–Carlo method and, in the ergodic case, integration of a function f w.r.t. the invariant probability law of (Xt ) by simulating a simple t,rajectory. For each case it is proved the expansion of the global approximat,ion error—for a class of discret,isat,ion schemes and of funct,ions f—in powers of the discretisation step size, extending in the fist case a result of Gragg for deterministic O.D.E. Some nn~nerical examples are shown to illust,rate the applicat,ion of extrapolation methods, justified by the foregoing expansion, in order to improve the approximation accuracy

Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space

We consider the stochastic reflection problem associated with a selfadjoint operator A and a cylindrical Wiener process on a convex set K with nonempty interior and regular boundary E in a Hilbert space H. We prove the existence and uniqueness of a smooth solution for the corresponding elliptic infinite-dimensional Kolmogorov equation with Neumann boundary condition on E.

Some Results about Dissipativity of Kolmogorov Operators

Given a Hilbert space H with a Borel probability measure ν, we prove the m-dissipativity in L1(H, ν) of a Kolmogorov operator K that is a perturbation, not necessarily of gradient type, of an Ornstein-Uhlenbeck operator.

Self-adjointness of some infinite-dimensional elliptic operators and application to stochastic quantization

Abstract We consider an operator K˚ϕ = Lϕ−: in a Hilbert space H, where L is an Ornstein–Uhlenbeck operator, U∈W1,4(H, μ) and μ is the invariant measure associated with L. We show that K˚ is essentially self-adjoint in the space L2(H, ν) where ν is the “Gibbs” measure ν(dx) = Z−:1e−:2U(x)dx. An application to Stochastic quantization is given.

Fully nonlinear stochastic partial differential equations

The authors study a class of fully nonlinear stochastic partial differential equations by the reduction to a family of deterministic fully nonlinear equations using the stochastic characteristic method.

Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space II

Here A :D(A) ⊂H →H is a self-adjoint operator, K = {x ∈H :g(x) ≤ 1}, where g :H→ R is convex and of class C∞, NK(x) is the normal cone to K at x and W (t) is a cylindrical Wiener process in H (see Hypothesis 1.1 for more precise assumptions). Obviously the expression in (1.1) is formal and its precise meaning should be defined. When H is finite-dimensional a solution to (1.1) is a pair of continuous adapted processes (X,η) such that X is K-valued, η is of bounded variation with dη concentrated on the set of times where X(t) ∈ Σ (the boundary of K) and

SURFACE MEASURES IN INFINITE DIMENSION

We construct surface measures associated to Gaussian measures in separable Banach spaces, and we prove several properties including an integration by parts formula.

A Stochastic Parabolic Equation with Nonlinear Flux on the Boundary Driven by a Gaussian Noise

This paper concerns a class of stochastic parabolic equations with nonlinear boundary conditions and boundary noise, which is either a Wiener process or a fractional Brownian motion with Hurst parameter

Asymptotic Behavior of a Class of Nonlinear Stochastic Heat Equations with Memory Effects

\mathscr{H} > 1/2

The Stochastic Reflection Problem in Hilbert Spaces

. Boundary degeneracy of the solution is already known in the literature; we show that in our framework we can overcome this difficulty and treat a nonlinear perturbation term on the boundary. The boundary nonlinear term is either Lipschitz continuous or a maximal monotone mapping.