
Wellposedness and tamed Euler schemes for McKeanVlasov equations driven by Lévy noise
We prove the wellposedness of solutions to McKeanVlasov stochastic dif...
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Sampling of Stochastic Differential Equations using the KarhunenLoève Expansion and Matrix Functions
We consider linearizations of stochastic differential equations with add...
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Improving the approximation of the first and second order statistics of the response process to the random Legendre differential equation
In this paper, we deal with uncertainty quantification for the random Le...
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Explicit solution of divideandconquer dividing by a half recurrences with polynomial independent term
Divideandconquer dividing by a half recurrences, of the form x_n =a· x...
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On mathematical aspects of evolution of dislocation density in metallic materials
This paper deals with the solution of delay differential equations descr...
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Nonlinear Gaussian smoothing with Taylor moment expansion
This letter is concerned with solving continuousdiscrete Gaussian smoot...
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Taylor Moment Expansion for ContinuousDiscrete Gaussian Filtering and Smoothing
The paper is concerned with nonlinear Gaussian filtering and smoothing ...
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Polynomial propagation of moments in stochastic differential equations
We address the problem of approximating the moments of the solution, X(t), of an Itô stochastic differential equation (SDE) with drift and a diffusion terms over a timegrid t_0, t_1, …, t_n. In particular, we assume an explicit numerical scheme for the generation of sample paths X̂(t_0), …, X̂(t_n), … and then obtain recursive equations that yield any desired noncentral moment of X̂(t_n) as a function of the initial condition X_0. The core of the methodology is the decomposition of the numerical solution into a "central part" and an "effective noise" term. The central term is computed deterministically from the ordinary differential equation (ODE) that results from eliminating the diffusion term in the SDE, while the effective noise accounts for the stochastic deviation from the numerical solution of the ODE. For simplicity, we describe algorithms based on an EulerMaruyama integrator, but other explicit numerical schemes can be exploited in the same way. We also apply the moment approximations to construct estimates of the 1dimensional marginal probability density functions of X̂(t_n) based on a GramCharlier expansion. Both for the approximation of moments and 1dimensional densities, we describe how to handle the cases in which the initial condition is fixed (i.e., X_0 = x_0 for some known x_0) or random. In the latter case, we resort to polynomial chaos expansion (PCE) schemes to approximate the target moments. The methodology has been inspired by the PCE and differential algebra (DA) methods used for uncertainty propagation in astrodynamics problems. Hence, we illustrate its application for the quantification of uncertainty in a 2dimensional Keplerian orbit perturbed by a Wiener noise process.
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