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Discrete-time approximations of stochastic delay equations: the Milstein scheme. (English) Zbl 1062.60065

The authors consider stochastic delay differential equations of the type \[ \begin{split} x(t)=\eta(0) + \int_0^t g(s,x(s-s_{1,1}),\ldots,x(s-s_{1,k_1}))\,dW(s) \\+ \int_0^t h(s,x(s-s_{2,1}),\dots,x(s-s_{2,k_2}))\,ds\end{split}\tag{1} \] for \(t\geq 0\), with initial condition \(x(t)=\eta(t)\) for \(t \in [-\tau,0]\), \(\tau = \max\{s_{1,1},\dots,s_{1,k_1}, s_{2,1}, \dots, s_{2,k_2}\}\). The driving process \(W\) is a \(d\)-dimensional standard Brownian motion.
The aim of the paper is to develop the Milstein scheme for the above class of equations and to analyse its strong convergence properties. In this case the Milstein scheme also contains multiple stochastic integrals of the form \(\int_a^b \int_{a-s_{i,j}}^{t-s_{i,j}}\,dW(u)\,dW(t)\), \(1\leq j \leq k_i\), \(i=1,2\). The convergence analysis requires the application of an appropriate Itô-formula for functions of the form of \(g\) and \(h\) in (1). It turns out that one needs Malliavin calculus to develop an Itô-formula for this case. The reason (briefly and heuristically) is the following: the standard approach to develop an Itô-formula for a function \(\varphi(x(s))\) is to apply the deterministic Taylor-formula to \(\varphi\) and then to analyse the resulting terms. In the case of functions of the form \(\varphi(x(s),x(s-\tau))\) the application of the Taylor-formula yields terms of the form \(\frac{\partial}{\partial x_2} \varphi(x(s),x(s-\tau))dW(s-\tau)\) and their analysis requires results from anticipating calculus.
The article under review provides an appropriate Itô-formula and results on weak differentiability of solutions of (1). The latter are needed for estimates of the remainder terms in the convergence proof of the Milstein scheme. The main result of the article is the proof of strong convergence of order 1 of the Milstein scheme for the approximation of solutions of (1).

MSC:

60H20 Stochastic integral equations
60H07 Stochastic calculus of variations and the Malliavin calculus
34K50 Stochastic functional-differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
37H10 Generation, random and stochastic difference and differential equations
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