The goal of this contribution is to introduce some approaches to uncertainty modeling in a way accessible to non-specialists. Elements of the Monte Carlo method, polynomial chaos method, Dempster-Shafer approach, fuzzy set theory, and the worst (case) scenario method are presented.

This paper discusses analytical and numerical issues related to elliptic equations with random coefficients which are generally nonlinear functions of white noise. Singularity issues are avoided by using the Itô-Skorohod calculus to interpret the interactions between the coefficients and the solution. The solution is constructed by means of the Wiener Chaos (Cameron-Martin) expansions. The existence and uniqueness of the solutions are established under rather weak assumptions, the main of which...

We study ergodic properties of stochastic geometric wave equations on a particular model with the target being the 2D sphere while considering only solutions which are independent of the space variable. This simplification leads to a degenerate stochastic equation in the tangent bundle of the 2D sphere. Studying this equation, we prove existence and non-uniqueness of invariant probability measures for the original problem and obtain also results on attractivity towards an invariant measure. We also...

We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form ${\left|x\right|}^{\alpha }$, $\alpha \in [1/2,1)$. In that case, we study the rate of convergence of a symmetrized version of the Euler scheme. This symmetrized version is easy to simulate on a computer. We prove its strong convergence and obtain the same rate of convergence as when the coefficients are Lipschitz.

We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form |x|α, α ∈ [1/2,1). In that case, we study the rate of convergence of a symmetrized version of the Euler scheme. This symmetrized version is easy to simulate on a computer. We prove its strong convergence and obtain the same rate of convergence as when the coefficients are Lipschitz.

This paper is concerned with the problem of simulation of ${\left(X_t\right)}_{0\le t\le T}$, the solution of a stochastic differential equation constrained by some boundary conditions in a smooth domain $D$: namely, we consider the case where the boundary $\partial D$ is killing, or where it is instantaneously reflecting in an oblique direction. Given $N$ discretization times equally spaced on the interval $[0,T]$, we propose new discretization schemes: they are fully implementable and provide a weak error of order $N^{-1}$ under some conditions. The construction...

This paper is concerned with the problem of simulation of (Xt)0≤t≤T, the solution of a stochastic differential equation constrained by some boundary conditions in a smooth domain D: namely, we consider the case where the boundary ∂D is killing, or where it is instantaneously reflecting in an oblique direction. Given N discretization times equally spaced on the interval [0,T], we propose new discretization schemes: they are fully implementable and provide a weak error of order N-1 under some conditions....

We study convergence in law for the Euler and Euler-Peano schemes for stochastic differential equations reflecting on the boundary of a general convex domain. We assume that the coefficients are measurable and continuous almost everywhere with respect to the Lebesgue measure. The proofs are based on new estimates of Krylov's type for the approximations considered.

In this note we propose an exact simulation algorithm for the solution of (1)$$\mathrm{d}{X}_t=\mathrm{d}{W}_t+\overline{b}\left(X_t\right)\mathrm{d}t,X_0=x,$$d X t = d W t + b̅ ( X t ) d t,   X 0 = x, where $\overline{b}$b̅is a smooth real function except at point 0 where $\overline{b}(0+)\ne \overline{b}(0-)$b̅(0 + ) ≠ b̅(0 −) . The main idea is to sample an exact skeleton of Xusing an algorithm deduced from the convergence of the solutions of the skew perturbed equation (2)$$\mathrm{d}{X}_t^{\beta }=\mathrm{d}{W}_t+\overline{b}\left(X_t^{\beta }\right)\mathrm{d}t+\beta \mathrm{d}{L}_t^0\left(X^{\beta }\right),X_0=x$$d X t β = d W t + b̅ ( X t β ) d t + β d L t 0 ( X β ) ,   X 0 = x towardsX solution of (1) as β ≠ 0 tends to 0. In this note, we show that this convergence...

We study a free energy computation procedure, introduced in [Darve and Pohorille, J. Chem. Phys.115 (2001) 9169–9183; Hénin and Chipot, J. Chem. Phys.121 (2004) 2904–2914], which relies on the long-time behavior of a nonlinear stochastic differential equation. This nonlinearity comes from a conditional expectation computed with respect to one coordinate of the solution. The long-time convergence of the solutions to this equation has been proved in [Lelièvre et al., Nonlinearity21 (2008) 1155–1181],...