The Ito calculus: a vector-integral approach
The Itô calculus is the theory of stochastic integrals ∫<sup>t</sup><sub>0</sub> X<sub>u</sub> dS<sub>u</sub>, where S is a semimartingale, and X is a suitable previsible process. The approach most commonly given in the literature is the ‘Strasbourg ap...
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| Format: | Thesis |
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1989
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| _version_ | 1826309977347719168 |
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| author | Ling, PD |
| author2 | Edwards, DA |
| author_facet | Edwards, DA Ling, PD |
| author_sort | Ling, PD |
| collection | OXFORD |
| description | The Itô calculus is the theory of stochastic integrals ∫<sup>t</sup><sub>0</sub> X<sub>u</sub> dS<sub>u</sub>, where S is a
semimartingale, and X is a suitable previsible process. The approach most commonly
given in the literature is the ‘Strasbourg approach’, in which S is taken first to be an
L<sup>2</sup>-martingale, and an L<sup>2</sup>-isometry is used to define the infegral. Then localization
and pathwise Stieltjes integration are used to extend to the case when § is a semimartingale.
In this thesis a different, and more direct approach is given. The space
L<sup>2</sup> is replaced by L<sup>p</sup> for an arbitrary p € [0,∞), with a particular interest shown in
the case when p = 0. The integral ∫ X dS is defined as an L<sup>p</sup>-valued Daniell integral
(the ‘Daniell’ analogue of L<sup>p</sup>-valued measures). The necessary theory of such Daniell
integrals is given in Chapters 2 and 3. In Chapter 4 the stochastic integral IPS
is defined and its properties given. It is necessary to place a dominated convergence
condition on 5. Processes satisfying this condition are called L<sup>p</sup>-integrators. A characterization
of L<sup>p</sup>-integrators in terms of uniform convergence is given at the end of
Chapter 4. A proof that all integrators have pathwise cadlag versions is given in Chapter
5. Localization and processes of finite-variation are also considered in Chapter 5.
Quadratic variation, [S], and mutual variation, [R,S], are defined in Chapter 6 by a
martingale-free method, and its properties proven. The Itô formula is then stated and
proved. Martingales are for the first time introduced in Chapters 7 and 8. Some results
from the discrete-time theory (which are presented in Chapter 7) are used in Chapter
8 to prove that all martingales are integrators. This Daniell method of stochastic
integration is seen to be consistent with and at least as powerful as the Strasbourg
method. Finally in Chapter 9 a proof is presented of the result of Bichteler, Dellacherie,
Mokobodzki and Meyer, that the class of all (L<sup>0</sup>-)integrators is precisely the
class of all semimartingales. As prerequisites for this proof, Doob-Meyer decompositions
for L<sup>1</sup>-integrators and local L<sup>1</sup>-integrators are obtained, and the class of all
semimartingales is proven to be invariant under change of the probability measure
(provided the two probabilities are equivalent). |
| first_indexed | 2024-03-07T07:43:48Z |
| format | Thesis |
| id | oxford-uuid:579ba15d-2b85-4134-9e00-17f2ab85f45b |
| institution | University of Oxford |
| last_indexed | 2024-03-07T07:43:48Z |
| publishDate | 1989 |
| record_format | dspace |
| spelling | oxford-uuid:579ba15d-2b85-4134-9e00-17f2ab85f45b2023-05-17T14:24:29ZThe Ito calculus: a vector-integral approachThesishttp://purl.org/coar/resource_type/c_db06uuid:579ba15d-2b85-4134-9e00-17f2ab85f45bORA41989Ling, PDEdwards, DAThe Itô calculus is the theory of stochastic integrals ∫<sup>t</sup><sub>0</sub> X<sub>u</sub> dS<sub>u</sub>, where S is a semimartingale, and X is a suitable previsible process. The approach most commonly given in the literature is the ‘Strasbourg approach’, in which S is taken first to be an L<sup>2</sup>-martingale, and an L<sup>2</sup>-isometry is used to define the infegral. Then localization and pathwise Stieltjes integration are used to extend to the case when § is a semimartingale. In this thesis a different, and more direct approach is given. The space L<sup>2</sup> is replaced by L<sup>p</sup> for an arbitrary p € [0,∞), with a particular interest shown in the case when p = 0. The integral ∫ X dS is defined as an L<sup>p</sup>-valued Daniell integral (the ‘Daniell’ analogue of L<sup>p</sup>-valued measures). The necessary theory of such Daniell integrals is given in Chapters 2 and 3. In Chapter 4 the stochastic integral IPS is defined and its properties given. It is necessary to place a dominated convergence condition on 5. Processes satisfying this condition are called L<sup>p</sup>-integrators. A characterization of L<sup>p</sup>-integrators in terms of uniform convergence is given at the end of Chapter 4. A proof that all integrators have pathwise cadlag versions is given in Chapter 5. Localization and processes of finite-variation are also considered in Chapter 5. Quadratic variation, [S], and mutual variation, [R,S], are defined in Chapter 6 by a martingale-free method, and its properties proven. The Itô formula is then stated and proved. Martingales are for the first time introduced in Chapters 7 and 8. Some results from the discrete-time theory (which are presented in Chapter 7) are used in Chapter 8 to prove that all martingales are integrators. This Daniell method of stochastic integration is seen to be consistent with and at least as powerful as the Strasbourg method. Finally in Chapter 9 a proof is presented of the result of Bichteler, Dellacherie, Mokobodzki and Meyer, that the class of all (L<sup>0</sup>-)integrators is precisely the class of all semimartingales. As prerequisites for this proof, Doob-Meyer decompositions for L<sup>1</sup>-integrators and local L<sup>1</sup>-integrators are obtained, and the class of all semimartingales is proven to be invariant under change of the probability measure (provided the two probabilities are equivalent). |
| spellingShingle | Ling, PD The Ito calculus: a vector-integral approach |
| title | The Ito calculus: a vector-integral approach |
| title_full | The Ito calculus: a vector-integral approach |
| title_fullStr | The Ito calculus: a vector-integral approach |
| title_full_unstemmed | The Ito calculus: a vector-integral approach |
| title_short | The Ito calculus: a vector-integral approach |
| title_sort | ito calculus a vector integral approach |
| work_keys_str_mv | AT lingpd theitocalculusavectorintegralapproach AT lingpd itocalculusavectorintegralapproach |