On stochastic differential equations in a configuration space.
On stochastic differential equations with locally unbounded drift
We study the regularizing effect of the noise on differential equations with irregular coefficients. We present existence and uniqueness theorems for stochastic differential equations with locally unbounded drift.
On Stochastic Differential Equations with Reflecting Boundary Condition in Convex Domains
Let D be an open convex set in and let F be a Lipschitz operator defined on the space of adapted càdlàg processes. We show that for any adapted process H and any semimartingale Z there exists a unique strong solution of the following stochastic differential equation (SDE) with reflection on the boundary of D: , t ∈ ℝ⁺. Our proofs are based on new a priori estimates for solutions of the deterministic Skorokhod problem.
On stochastic Euler equation in .
On surrogate learning for linear stability assessment of Navier-Stokes equations with stochastic viscosity
We study linear stability of solutions to the Navier-Stokes equations with stochastic viscosity. Specifically, we assume that the viscosity is given in the form of a stochastic expansion. Stability analysis requires a solution of the steady-state Navier-Stokes equation and then leads to a generalized eigenvalue problem, from which we wish to characterize the real part of the rightmost eigenvalue. While this can be achieved by Monte Carlo simulation, due to its computational cost we study three surrogates...
On the approximate solutions of the Stratonovitch equation.
On the Azéma martingales
On the consistency of a least squares identification procedure
On the constructions of the skew Brownian motion.
On the continuity of random operators.
On the control of the difference between two Brownian motions: a dynamic copula approach
We propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. Our approach is based on the copula between the Brownian motion and its reflection. We show that the class of admissible copulae for the Brownian motions are not limited to the class of Gaussian copulae and that it also contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions to have higher value in the right...
On the control of the difference between two Brownian motions: an application to energy markets modeling
We derive a model based on the structure of dependence between a Brownian motion and its reflection according to a barrier. The structure of dependence presents two states of correlation: one of comonotonicity with a positive correlation and one of countermonotonicity with a negative correlation. This model of dependence between two Brownian motions B1 and B2 allows for the value of [...] to be higher than 1/2 when x is close to 0, which is not the case when the dependence is modeled by a constant...
On the convergence of bilinear and quadratic forms in independent random variables
On the convergence of generalized polynomial chaos expansions
A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial chaos expansions to the correct limit and complement...
On the convergence of generalized polynomial chaos expansions
A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial...
On the convergence of the stochastic Galerkin method for random elliptic partial differential equations
In this article we consider elliptic partial differential equations with random coefficients and/or random forcing terms. In the current treatment of such problems by stochastic Galerkin methods it is standard to assume that the random diffusion coefficient is bounded by positive deterministic constants or modeled as lognormal random field. In contrast, we make the significantly weaker assumption that the non-negative random coefficients can be bounded strictly away from zero and infinity by random...
On the convergence of the wavelet-Galerkin method for nonlinear filtering
The aim of the paper is to examine the wavelet-Galerkin method for the solution of filtering equations. We use a wavelet biorthogonal basis with compact support for approximations of the solution. Then we compute the Zakai equation for our filtering problem and consider the implicit Euler scheme in time and the Galerkin scheme in space for the solution of the Zakai equation. We give theorems on convergence and its rate. The method is numerically much more efficient than the classical Galerkin method....
On the density for the solution of a Burgers-type SPDE
On the density of some Wiener functionals: an application of Malliavin calculus.
Using a representation as an infinite linear combination of chi-square independent random variables, it is shown that some Wiener functionals, appearing in empirical characteristic process asymptotic theory, have densities which are tempered in the properly infinite case and exponentially decaying in the finite case.
On the difference between the product and the convolution product of distribution functions.