Browsing by Subject "problem osobliwy"

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Access status: Restricted ,
Optymalna aproksymacja całki Itô funkcji deterministycznych regularnych oraz osobliwych
(Data obrony: 2014-06-27) Dębowski, Jacek
Wydział Matematyki Stosowanej
Access status: Open Access ,
Złożoność obliczeniowa całkowania stochastycznego w sensie Itô
(Data obrony: 2011-11-30) Przybyłowicz, Paweł
Wydział Matematyki Stosowanej
In the thesis we study the computational complexity of the stochastic Itô integration. We first investigate the optimal approximation of Itô integrals when linear information about the Wiener process B, consisting of certain Riemann integrals of its trajectories, is available. We show upper and lower bounds on the complexity which, in some cases, turn out to be optimal. Obtained results indicates that algorithms which use integral information are more efficient than algorithms which use only discrete values of the Wiener process B. In the second part of the thesis we deal with the numerical approximation of stochastic Itô integrals of regular and singular deterministic functions $f:[0,T]->R$. In the regular case we show that the nonadaptive Ito-Taylor algorithm is optimal. In the singular case we show that any nonadaptive algorithm cannot efficiently handle such a problem, even in the case of a single singularity. Hence, in the case of a single singularity, we construct an adaptive Itô-Taylor algorithm which has the optimal error known from the regular case. Next, we consider the case of multiple singularities and we show that even adaptive algorithms cannot preserve the optimal rate of convergence known from the regular case. We show that also in the asymptotic setting nonadaptive algorithms cannot preserve the optimal error known from the regular case.