Weak approximation of stochastic differential delay equations

Adaptive Weak Approximation of Stochastic Differential Equations

Adaptive time-stepping methods based on the Monte Carlo Euler method for weak approximation of Itô stochastic differential equations are developed. The main result is new expansions of the computational error, with computable leading-order term in a posteriori form, based on stochastic flows and discrete dual backward problems. The expansions lead to efficient and accurate computation of error ...

Weak discrete time approximation of stochastic differential equations with time delay

A Robust Weak Taylor Approximation Scheme for Solutions of Jump-Diffusion Stochastic Delay Differential Equations

and Applied Analysis 3 Now we present some lemmas to be used later for the proof of the convergence theorem. Consider a right continuous process YΔ l = {YΔ (t), t ∈ [−γ, T]}. YΔ l is called a discrete-time numerical approximationwithmaximum step size Δ l , if it is obtained by using a time discretization t Δ l , and the random variable YΔ l tn is F tn -measurable for n ∈ {1, . . . , N}. Further...

Weak approximation of stochastic partial differential equations: the nonlinear case

We study the error of the Euler scheme applied to a stochastic partial differential equation. We prove that as it is often the case, the weak order of convergence is twice the strong order. A key ingredient in our proof is Malliavin calculus which enables us to get rid of the irregular terms of the error. We apply our method to the case a semilinear stochastic heat equation driven by a space-ti...

Convergence Rates for Adaptive Weak Approximation of Stochastic Differential Equations

Convergence rates of adaptive algorithms for weak approximations of Itô stochastic differential equations are proved for the Monte Carlo Euler method. Two algorithms based either on optimal stochastic time steps or optimal deterministic time steps are studied. The analysis of their computational complexity combines the error expansions with a posteriori leading order term introduced in Szepessy...