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The Ito calculus: a vector-integral approach

Abstract:
The Itô calculus is the theory of stochastic integrals ∫t0 Xu dSu, 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 L2-martingale, and an L2-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 L2 is replaced by Lp for an arbitrary p € [0,∞), with a particular interest shown in the case when p = 0. The integral ∫ X dS is defined as an Lp-valued Daniell integral (the ‘Daniell’ analogue of Lp-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 Lp-integrators. A characterization of Lp-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 (L0-)integrators is precisely the class of all semimartingales. As prerequisites for this proof, Doob-Meyer decompositions for L1-integrators and local L1-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).

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Division:
MPLS
Department:
Mathematical Institute
Oxford college:
St Catherine's College
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Funder identifier:
http://dx.doi.org/10.13039/501100005049

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Type of award:
DPhil
Level of award:
Doctoral
Awarding institution:
University of Oxford

Deposit date:
2023-05-17
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