Convergence rates for adaptive weak approximation of stochastic differential equations
2005 (English)In: Stochastic Analysis and Applications, ISSN 0736-2994, E-ISSN 1532-9356, Vol. 23, no 3, 511-558 p.Article in journal (Refereed) Published
Convergence rates of adaptive algorithms for weak approximations of Ito stochastic differential equations are proved for the Monte Carlo Euler method. Two algorithms based either oil 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 et al. [Szepessy, A.. R. Tempone, and G. Zouraris. 2001. Comm. Pare Appl. Math. 54:1169-1214] and ail extension of the convergence results for adaptive algorithms approximating deterministic ordinary differential equations, derived in Moon et al. [Moon, K.-S., A. Szepessy, R. Tempone, and G. Zouraris. 2003. Numer. Malh. 93:99-129]. The main step in the extension is the proof of the almost sure convergence of the error density. Both adaptive alogrithms are proven to stop with asymptotically optimal number of steps up to a problem independent factor defined in the algorithm. Numerical examples illustrate the behavior of the adaptive algorithms, motivating when stochastic and deterministic adaptive time steps are more efficient than constant time steps and when adaptive stochastic steps are more efficient than adaptive deterministic steps.
Place, publisher, year, edition, pages
2005. Vol. 23, no 3, 511-558 p.
adaptive mesh refinement algorithm, almost sure convergence, computational complexity, Monte Carlo method, stochastic differential equations, finite-element methods, error
IdentifiersURN: urn:nbn:se:kth:diva-14798DOI: 10.1081/sap-200056678ISI: 000229613900005ScopusID: 2-s2.0-20444378209OAI: oai:DiVA.org:kth-14798DiVA: diva2:332839
QC 201005252010-08-052010-08-052010-08-25Bibliographically approved