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Discrete approximation schemes for a class of Itô processes with random-jump component

Discrete approximation schemes for a class of Itô processes with random-jump component
Discrete approximation schemes for a class of Itô processes with random-jump component

In this thesis, discrete approximation schemes for a class of stochastic differential equations, defined in the sense of Itô, are suggested. The Itô process under consideration includes a Wiener process and a Poisson random measure with random-jump component. In the derivation of various schemes the Taylor expansion of the contraction semi-group of conditional expectations is applied.

Mean-square approximation schemes, based on fixed and jump-adapted time discretisations, are constructed. In particular, schemes depending on the values of the sampled driving processes at discrete time points only are considered and their maximum rate of convergence is established. Asymptotic properties as the discretisation parameter tends to zero are also investigated.

For similar time discretisations, the approximation of expectations of functionals of the solution, i.e. weak approximation, is examined. It is demonstrated that schemes constructed for this purpose are, in general, distinct from those approximating the solution in the mean-square sense. A particular weak scheme is derived and its order of mean-square convergence is also established.

The numerical performance of various schemes under both criteria is investigated, for fixed and jump-adapted time discretisations.

University of Southampton
Ieronimakis, Nikolaos George
Ieronimakis, Nikolaos George

Ieronimakis, Nikolaos George (1996) Discrete approximation schemes for a class of Itô processes with random-jump component. University of Southampton, Doctoral Thesis.

Record type: Thesis (Doctoral)

Abstract

In this thesis, discrete approximation schemes for a class of stochastic differential equations, defined in the sense of Itô, are suggested. The Itô process under consideration includes a Wiener process and a Poisson random measure with random-jump component. In the derivation of various schemes the Taylor expansion of the contraction semi-group of conditional expectations is applied.

Mean-square approximation schemes, based on fixed and jump-adapted time discretisations, are constructed. In particular, schemes depending on the values of the sampled driving processes at discrete time points only are considered and their maximum rate of convergence is established. Asymptotic properties as the discretisation parameter tends to zero are also investigated.

For similar time discretisations, the approximation of expectations of functionals of the solution, i.e. weak approximation, is examined. It is demonstrated that schemes constructed for this purpose are, in general, distinct from those approximating the solution in the mean-square sense. A particular weak scheme is derived and its order of mean-square convergence is also established.

The numerical performance of various schemes under both criteria is investigated, for fixed and jump-adapted time discretisations.

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Published date: 1996

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Local EPrints ID: 463008
URI: http://eprints.soton.ac.uk/id/eprint/463008
PURE UUID: c9e14263-bd04-48d2-89ef-5533f8ee651c

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Date deposited: 04 Jul 2022 20:37
Last modified: 04 Jul 2022 20:37

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Author: Nikolaos George Ieronimakis

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