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Adaptive weak approximation of stochastic differential equations. (English) Zbl 1024.60028
The authors consider a $$d$$-dimensional system of Itô stochastic differential equations, driven by $$l_0$$ independent Wiener processes: $dX_k(t) = a_k(t,X(t)) dt + \sum_{l=1}^{l_0}b_k^{l}(t,X(t)) dW{l}(t), \quad k= 1,2,\dots, d,\;t > 0. \tag{1}$ They develop and analyse two adaptive time-stepping strategies used with the Euler-method to approximate the quantity $$E[g(X(T))]$$ for a given function $$g$$. The basis for their results is the work by D. Talay and L. Tubaro [Stochastic Anal. Appl. 8, 483-509 (1990; Zbl 0718.60058)], where an expansion of the error $E(g(X(T))-g({\overline X}(T)))\tag{2}$ is established, yielding an a priori estimate involving unknown quantities, such as the exact solution. In the article under review a similar expansion of the error (2) is developed, but here the leading error term of the expansion includes only known or computable quantities. The authors use stochastic flows and dual functions as their main tools to obtain their error expansion. The time-stepping strategies that the authors propose are a) a deterministic one, i.e. the time-steps get adapted, but the mesh is then fixed for all trajectories of the solution process, and b) a stochastic one, i.e. the time-steps get adapted for different trajectories of the solution process. Numerical experiments illustrate the theoretical results.

MSC:
 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 65C30 Numerical solutions to stochastic differential and integral equations 65C05 Monte Carlo methods
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