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Convergence rates for adaptive weak approximation of stochastic differential equations
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
2005 (Engelska)Ingår i: Stochastic Analysis and Applications, ISSN 0736-2994, E-ISSN 1532-9356, Vol. 23, nr 3, s. 511-558Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

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.

Ort, förlag, år, upplaga, sidor
2005. Vol. 23, nr 3, s. 511-558
Nyckelord [en]
adaptive mesh refinement algorithm, almost sure convergence, computational complexity, Monte Carlo method, stochastic differential equations, finite-element methods, error
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
URN: urn:nbn:se:kth:diva-14798DOI: 10.1081/sap-200056678ISI: 000229613900005Scopus ID: 2-s2.0-20444378209OAI: oai:DiVA.org:kth-14798DiVA, id: diva2:332839
Anmärkning
QC 20100525Tillgänglig från: 2010-08-05 Skapad: 2010-08-05 Senast uppdaterad: 2017-12-12Bibliografiskt granskad
Ingår i avhandling
1. Numerical Complexity Analysis of Weak Approximation of Stochastic Differential Equations
Öppna denna publikation i ny flik eller fönster >>Numerical Complexity Analysis of Weak Approximation of Stochastic Differential Equations
2002 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

The thesis consists of four papers on numerical complexityanalysis of weak approximation of ordinary and partialstochastic differential equations, including illustrativenumerical examples. Here by numerical complexity we mean thecomputational work needed by a numerical method to solve aproblem with a given accuracy. This notion offers a way tounderstand the efficiency of different numerical methods.

The first paper develops new expansions of the weakcomputational error for Itˆo stochastic differentialequations using Malliavin calculus. These expansions have acomputable leading order term in a posteriori form, and arebased on stochastic flows and discrete dual backward problems.Beside this, these expansions lead to efficient and accuratecomputation of error estimates and give the basis for adaptivealgorithms with either deterministic or stochastic time steps.The second paper proves convergence rates of adaptivealgorithms for Itˆo stochastic differential equations. Twoalgorithms based either on stochastic or deterministic timesteps are studied. The analysis of their numerical complexitycombines the error expansions from the first paper and anextension of the convergence results for adaptive algorithmsapproximating deterministic ordinary differential equations.Both adaptive algorithms are proven to stop with an optimalnumber of time steps up to a problem independent factor definedin the algorithm. The third paper extends the techniques to theframework of Itˆo stochastic differential equations ininfinite dimensional spaces, arising in the Heath Jarrow Mortonterm structure model for financial applications in bondmarkets. Error expansions are derived to identify differenterror contributions arising from time and maturitydiscretization, as well as the classical statistical error dueto finite sampling.

The last paper studies the approximation of linear ellipticstochastic partial differential equations, describing andanalyzing two numerical methods. The first method generates iidMonte Carlo approximations of the solution by sampling thecoefficients of the equation and using a standard Galerkinfinite elements variational formulation. The second method isbased on a finite dimensional Karhunen- Lo`eve approximation ofthe stochastic coefficients, turning the original stochasticproblem into a high dimensional deterministic parametricelliptic problem. Then, adeterministic Galerkin finite elementmethod, of either h or p version, approximates the stochasticpartial differential equation. The paper concludes by comparingthe numerical complexity of the Monte Carlo method with theparametric finite element method, suggesting intuitiveconditions for an optimal selection of these methods. 2000Mathematics Subject Classification. Primary 65C05, 60H10,60H35, 65C30, 65C20; Secondary 91B28, 91B70.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH, 2002. s. x, 28
Serie
Trita-NA, ISSN 0348-2952 ; 0220
Nyckelord
Adaptive methods, a posteriori error estimates, stochastic differential equations, weak approximation, Monte Carlo methods, Malliavin Calculus, HJM model, option price, bond market, stochastic elliptic equation, Karhunen-Loeve expansion, numerical co
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
urn:nbn:se:kth:diva-3413 (URN)KTH/NA/R--20/20--SE (ISRN)91-7283-350-5 (ISBN)
Disputation
2002-10-11, 00:00
Anmärkning
QC 20100825Tillgänglig från: 2002-09-26 Skapad: 2002-09-26 Senast uppdaterad: 2010-08-25Bibliografiskt granskad

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Moon, Kyoung-SookSzepessy, AndersTempone Olariaga, RaulZouraris, Georgios
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Numerisk Analys och Datalogi, NADA
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Stochastic Analysis and Applications
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