Weak discrete time approximation of stochastic differential equations with time delay

Weak discrete time approximation of stochastic differential equations with time delay

Strong Discrete Time Approximation of Stochastic Di erential Equations with Time Delay

The paper introduces an approach for the derivation of discrete time approximations for solutions of stochastic di erential equations with time delay. The suggested approximations converge in a strong sense. Furthermore, explicit solutions for linear stochastic delay equations are given.

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 ...

Estimating the Delay Time in Affine Stochastic Delay Differential Equations

We consider linear differential equations with bounded time delay driven by additive white noise. Our aim is the estimation of the maximal delay time from observations of one realisation of the solution process X under nonparametric drift assumptions. In the stationarity case the covariance function has a jump in the third derivative according to the location of the delay time. Based on this re...

Discrete approximation of stochastic differential equations

It is shown how stochastic Itô-Taylor schemes for stochastic ordinary differential equations can be embedded into standard concepts of consistency, stability and convergence. An appropriate choice of function spaces and norms, in particular a stochastic generalization of Spijker’s norm (1968), leads to two-sided estimates for the strong error of convergence under the usual assumptions.