On approximation of a class of stochastic integrals and interpolation

Geiß, Christel; Geiß, Stefan

doi:10.1080/10451120410001728445

Cited by 36 publications

(53 citation statements)

References 8 publications

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“…This is due to the fact that a characterization of the L 2 -error proved in [11,Theorem 4.4] and [7,Lemma 3.2] is missing for higher dimensions. However, after Zhang [36] started with the regular case, Temam [34] extended results from [17] to higher dimensions, and Hujo [20] used nonuniform time nets to improve the approximation rates for certain irregular payoffs to the optimal rate 1/ √ n in this setting.…”

Section: Discussionmentioning

confidence: 89%

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Fractional Smoothness and Applications in Finance

Geiß

Gobet

2011

Advanced Mathematical Methods for Finance

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This overview article concerns the notion of fractional smoothness of random variables of the form g(X T ), where X = (X t ) t∈ [0,T ] is a certain diffusion process. We review the connection to the real interpolation theory, give examples and applications of this concept. The applications in stochastic finance mainly concern the analysis of discrete-time hedging errors. We close the review by indicating some further developments.

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Section: Discussionmentioning

confidence: 89%

“…In our context the fractional smoothness of jump functions (under different assumptions) was considered in [7,11,17] (1 − t)…”

Section: (X−y) 2 κ(T−s) = κγ κ(T−s) (X − Y)mentioning

confidence: 99%

Fractional Smoothness and Applications in Finance

Geiß

Gobet

2011

Advanced Mathematical Methods for Finance

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“…However, in [5,7] it is shown that for a large class of functions f one can verify this bound, i.e. there is a sequence of nets…”

Section: Approximation In B 2q ( )mentioning

confidence: 89%

“…Equidistant time nets give the optimal rate c/ √ n if and only if f (W 1 ) ∈ D 1,2 ( ) (see [5] and Theorem 3.2). So it seems that the necessary degree of concentration of the time-knots at time T = 1 of the nets f n indicates the distance of f to D 1,2 ( ).…”

Section: Approximation In B 2q ( )mentioning

confidence: 96%

“…After considering special cases the investigations indicated that there is a close relation between stochastic approximation and smoothness properties of these stochastic integrals. A first connection in this direction in terms of interpolation spaces was observed in [5]. The aim of this paper is to extend the results in the case that the underlying diffusion, which acts as integrator of the stochastic integrals, is the Brownian motion or the geometric Brownian motion.…”

Section: Introductionmentioning

confidence: 91%

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Interpolation and approximation in L2(γ)

Geiß

Hujo

2007

Journal of Approximation Theory

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Assume a standard Brownian motion W = (W t ) t∈ [0,1] , a Borel function f : R → R such that f (W 1 ) ∈ L 2 , and the standard Gaussian measure on the real line. We characterize that f belongs to the Besov space B 2,q ( ) := (L 2 ( ), D 1,2 ( )) ,q , obtained via the real interpolation method, by the behavior ofi=0 is a deterministic time net and P X : L 2 → L 2 the orthogonal projection onto a subspace of 'discrete'stochastic integrals x 0 + n i=1 v i−1 (X t i −X t i−1 ) with X being the Brownian motion or the geometric Brownian motion. By using Hermite polynomial expansions the problem is reduced to a deterministic one. The approximation numbers a X (f (X 1 ); ) can be used to describe the L 2 -error in discrete time simulations of the martingale generated by f (W 1 ) and (in stochastic finance) to describe the minimal quadratic hedging error of certain discretely adjusted portfolios.

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A correction note to “Discrete time hedging errors for options with irregular payoffs”

Gobet

2014

Finance Stoch

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On approximation of a class of stochastic integrals and interpolation

Cited by 36 publications

References 8 publications

Fractional Smoothness and Applications in Finance

Fractional Smoothness and Applications in Finance

Interpolation and approximation in L2(γ)

A correction note to “Discrete time hedging errors for options with irregular payoffs”

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