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Chapter 7. The Kunita-Watanabe Extension

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Markov Processes from K. Itô's Perspective
This chapter is in the book Markov Processes from K. Itô's Perspective
CHAPTER 7 The Kunita-Watanabe Extension A careful examination of the results in §§5.1 and 5.3 reveals that they depend very little on detailed properties of Brownian motion and, in fact, that analogous results can be derived about any square-integrable martingale (M(t),Ft,JP>) with the properties that (1) the t""' M(t) is JP>-almost surely continuous; (2) there is an {Ft : t 2 0}-progressively measurable A : [0, oo) x 0 --t [0, oo) such that t ""' A(t) is JP>-almost surely continuous and nondecreasing, A(O) = 0, and (M(t)2-A(t),Ft,JP>) is a martingale. In the case of an IR-valued Brownian motion (,B(t), Ft, JP>), A(t) = t. In the case when t""' ,B(t) is 1Rn-valued and X(t) = (e, ,B(t))JRn for some e E IRn, A(t) = t!e\2. More generally, if(} E 82(JP>;IRn) and M = Io, then A(t) = f~ jB(r)\2 dr. Although J.L. Doob (cf. Chapter 6 of [6]) was the first to recognize that these are the only ingredients which are essential for Ito's theory, it was Kunita and Watanabe [21] who first accomplished the elegant extension of Ito's theory which we will present here. However, before we can do so, we need to have a special, and particularly simple, case of the renowned Doob-Meyer decomposition theorem for submartingales.1 Throughout, (f!,F,JP>) is a complete probability space and {Ft : t 2 0} is a nondecreasing family of JP>-complete sub a-algebras of F. Also, when I say that a stochastic process X on [0, oo) x 0 with values in a topological space is JP>-almost surely right continuous or continuous, I will mean that t""' X(t,w) is right continuous or continuous for JP>-almost every w. 7.1 Doob-Meyer for Continuous Martingales Recall (cf. Lemma 5.2.18 in [36]) Doob's decomposition lemma for dis-crete parameter, integrable submartingales (X(m),Fm,JP>): if Ao = 0 and 1 It should be recognized that A.V. Skorohod demonstrated in [30] and [31] that he already understood most of the ideas discussed below. What makes his treatment less palatable than Kunita and Watanabe's is his ignorance of the Doob--Meyer theorem.

CHAPTER 7 The Kunita-Watanabe Extension A careful examination of the results in §§5.1 and 5.3 reveals that they depend very little on detailed properties of Brownian motion and, in fact, that analogous results can be derived about any square-integrable martingale (M(t),Ft,JP>) with the properties that (1) the t""' M(t) is JP>-almost surely continuous; (2) there is an {Ft : t 2 0}-progressively measurable A : [0, oo) x 0 --t [0, oo) such that t ""' A(t) is JP>-almost surely continuous and nondecreasing, A(O) = 0, and (M(t)2-A(t),Ft,JP>) is a martingale. In the case of an IR-valued Brownian motion (,B(t), Ft, JP>), A(t) = t. In the case when t""' ,B(t) is 1Rn-valued and X(t) = (e, ,B(t))JRn for some e E IRn, A(t) = t!e\2. More generally, if(} E 82(JP>;IRn) and M = Io, then A(t) = f~ jB(r)\2 dr. Although J.L. Doob (cf. Chapter 6 of [6]) was the first to recognize that these are the only ingredients which are essential for Ito's theory, it was Kunita and Watanabe [21] who first accomplished the elegant extension of Ito's theory which we will present here. However, before we can do so, we need to have a special, and particularly simple, case of the renowned Doob-Meyer decomposition theorem for submartingales.1 Throughout, (f!,F,JP>) is a complete probability space and {Ft : t 2 0} is a nondecreasing family of JP>-complete sub a-algebras of F. Also, when I say that a stochastic process X on [0, oo) x 0 with values in a topological space is JP>-almost surely right continuous or continuous, I will mean that t""' X(t,w) is right continuous or continuous for JP>-almost every w. 7.1 Doob-Meyer for Continuous Martingales Recall (cf. Lemma 5.2.18 in [36]) Doob's decomposition lemma for dis-crete parameter, integrable submartingales (X(m),Fm,JP>): if Ao = 0 and 1 It should be recognized that A.V. Skorohod demonstrated in [30] and [31] that he already understood most of the ideas discussed below. What makes his treatment less palatable than Kunita and Watanabe's is his ignorance of the Doob--Meyer theorem.

Chapters in this book

  1. Frontmatter i
  2. Contents vii
  3. Preface xi
  4. Chapter 1. Finite State Space, a Trial Run 1
  5. Chapter 2. Moving to Euclidean Space, the Real Thing 35
  6. Chapter 3. Itô's Approach in the Euclidean Setting 73
  7. Chapter 4. Further Considerations 111
  8. Chapter 5. Itô's Theory of Stochastic Integration 125
  9. Chapter 6. Applications of Stochastic Integration to Brownian Motion 151
  10. Chapter 7. The Kunita-Watanabe Extension 189
  11. Chapter 8. Stratonovich's Theory 221
  12. Notation 260
  13. References 263
  14. Index 265
Readers are also interested in:
https://doi.org/10.1515/9781400835577-008

Chapters in this book

  1. Frontmatter i
  2. Contents vii
  3. Preface xi
  4. Chapter 1. Finite State Space, a Trial Run 1
  5. Chapter 2. Moving to Euclidean Space, the Real Thing 35
  6. Chapter 3. Itô's Approach in the Euclidean Setting 73
  7. Chapter 4. Further Considerations 111
  8. Chapter 5. Itô's Theory of Stochastic Integration 125
  9. Chapter 6. Applications of Stochastic Integration to Brownian Motion 151
  10. Chapter 7. The Kunita-Watanabe Extension 189
  11. Chapter 8. Stratonovich's Theory 221
  12. Notation 260
  13. References 263
  14. Index 265
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