Optimal One-Point Approximation of Stochastic Heat Equations with Additive Noise.
Technische Universität, Darmstadt
[Ph.D. Thesis], (2008)
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|Item Type:||Ph.D. Thesis|
|Title:||Optimal One-Point Approximation of Stochastic Heat Equations with Additive Noise|
Let X be the mild solution of a stochastic heat equation taking values in a Hilbert space H=L^2((0,1)^d) driven by a (cylindrical) Brownian motion W with values in H. We study the strong approximation of X at a fixed time point t=T for equations with additive noise. The algorithms we consider, are based on evaluations of a finite number of one-dimensional components of W at a finite number of time nodes. For the first time, non-equidistant time discretizations are considered. We analyze the smallest possible error obtained by arbitrary algorithms that use at most a total of N evaluations. The main results of this thesis are the derivation of the weak asymptotic of these minimal errors, depending on the spatial dimension d and the smoothness of W, and further the construction of asymptotically optimal approximations. In particular, we show that asymptotic optimality, in general, is only achieved by approximation schemes based on non-equidistant time discretizations. We complete our analytical results with simulation studies.
|Place of Publication:||Darmstadt|
|Uncontrolled Keywords:||Stochastic heat equations, Strong approximation, Minimal errors, Lower bounds, Non-equidistant time discretization|
|Classification DDC:||500 Naturwissenschaften und Mathematik > 510 Mathematik|
|Divisions:||04 Department of Mathematics > Stochastik|
|Date Deposited:||21 Nov 2008 10:17|
|Last Modified:||07 Dec 2012 11:54|
|Referees:||Ritter, Prof. Dr. Klaus and Geiß, Prof. Dr. Stefan|