# Markov Processes from K. Ito's Perspective (AM-155)

Daniel W. Stroock
Pages: 288
https://www.jstor.org/stable/j.ctt7zvdpb

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xviii)
Daniel W. Stroock
4. CHAPTER 1 Finite State Space, a Trial Run
(pp. 1-34)

In his famous article [19], Kolmogorov based his theory of Markov processes on what became known as Kolmogorov’sforward¹ andbackwardequations. In an attempt to explain Kolmogorov’s ideas, K. Itô [14] took a crucial step when he suggested that Kolmogorov’s forward equation can be thought of as describing the “flow of a vector field on the space of probability measures.” The purpose of this chapter is to develop Ito’s suggestion in the particularly easy case when the state spaceEis finite. The rest of the book is devoted to carrying out the analogous program when$E = {\mathbb{R}^n}$.

Suppose that...

5. CHAPTER 2 Moving to Euclidean Space, the Real Thing
(pp. 35-72)

The preceding chapter was an extended introduction which was offered in the hope that it would help to elucidate the ideas on which this book rests. Whether that hope was realistic will be apparent soon. In any case, in this and the next chapters we will be attempting to mimic for${\mathbb{R}^n}$what we did for${\mathbb{Z}_n}$in Chapter 1.

Throughout, the topology on${M_1}({\mathbb{R}^n})$will be the weak topology: the topology corresponding to convergence of measures when tested against bounded continuous functions. As is well known (cf. §3.1 [36]), the weak topology makes${M_1}({\mathbb{R}^n})$into a Polish space. Taht...

6. CHAPTER 3 Itô’s Approach in the Euclidean Setting
(pp. 73-110)

In §2.3 we used Kolmogorov’s approach to lift the integral curve of a vector field on${M_1}({\mathbb{R}^n})$to a measure on the pathspace$D([0,\infty );{\mathbb{R}^n})$(cf. Theorems 2.3.1 and 2.3.3), and we then showed in §2.4 that the measure associated with a Lévy process admits a much more transparent construction. Our goal in this chapter is, following Itô, to show how to build the paths corresponding to more general affine vector fields from Lévy processes. In other words, we want to implement here the analog for${\mathbb{R}^n}$of what we did for${\mathbb{Z}_n}$in §§1.5 and 1.6.

There are two major...

7. CHAPTER 4 Further Considerations
(pp. 111-124)

In Chapter 3, we explained how Itô’s method leads to stochastic processes which are the pathspace analog of the integral curves obtained in §2.2 when we integrated affine vector fields on${M_1}({\mathbb{R}^n})$. In this chapter we will point out a couple of the important properties with which Itô’s method endows these processes, and, because it is undoubtedly the most important, we begin with the Markov property.

Under the condition that we can solve the Kolmogorov’s backward equation (3.1.28) for a sufficiently rich class of initial dataφ, we showed in Corollary 3.1.27 that Itô’s method produces a process which is...

8. CHAPTER 5 Itô’s Theory of Stochastic Integration
(pp. 125-150)

Up to this point, I have been recognizing but not confronting the challenge posed by integrals of the sort in (3.1.22). There are several reasons for my decision to postpone doing so until now, perhaps the most important of which is my belief that, in spite of, or maybe because of, its elegance, Itô’s theory of stochastic integration tends to mask the essential simplicity and beauty of his ideas as we have been developing them heretofore. However, it is high time that I explain his theory of integration, and that is what I will be doing in this chapter. However,...

9. CHAPTER 6 Applications of Stochastic Integration to Brownian Motion
(pp. 151-188)

This chapter contains a highly incomplete selection of ways in which Itô’s theory of stochastic integration, especially his formula, has contributed to our understanding of Brownian motion. For a much more complete selection, see Revuz and Yor’s book [27].

Perhaps the single most beautiful application of Itô’s formula was made by H. Tanaka when (as reported in [22]) he applied it to prove the existence of local time for one-dimensional Brownian motion.

Before one can understand Tanaka’s idea, it is necessary to know what the preceding terminology means. Thus, let$(\beta (t),{F_t},P)$be a one-dimensional Brownian motion. What we are seeking...

10. CHAPTER 7 The Kunita–Watanabe Extension
(pp. 189-220)

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$(\beta (t),{F_t},P)$with the properties that

(1) the$t \to M(t)$is P-almost surely continuous;

(2) there is an$\{ {F_t}:t \geqslant 0\}$-progressively measurable$A:[0,\infty ) \times \Omega \to [0,\infty )$such that$t \to A(t)$is P-almost surely continuous and nondecreasing,$A(0) = 0$, and$(M{(t)^2} - A(t),{F_t},P)$is a martingale.

In the case of an R-valued Brownian motion$(\beta (t),{F_t},P),A(t) = t$. In the case when$t \to \beta (t)$is${\mathbb{R}^n}$-valued and$X(t) = {(\xi ,\beta (t))_{{\mathbb{R}^n}}}$for some$\xi \in {\mathbb{R}^n}$,$A(t) = t{\left| \xi \right|^2}$. More generally, if$\theta \in {\theta ^2}(P;{\mathbb{R}^n})$and$M = {I_\theta }$, then$A(t) = \int\limits_0^t {{{\left| {\theta (\tau )} \right|}^2}d\tau }$.

Although...

11. CHAPTER 8 Stratonovich’s Theory
(pp. 221-259)

From an abstract mathematical standpoint, Itô’s theory of stochastic integration has as a serious flaw: it behaves dreadfully under changes of coordinates (cf. Remark 3.3.6). In fact, Itô’s formula itself is the most dramatic manifestation of this problem.

The origin of the problems Itô’s theory has with coordinate changes can be traced back to its connection with independent increment processes. Indeed, the very notion of an independent increment process is inextricably tied to the linear structure of Euclidean space, and anything but a linear change of coordinates will wreak havoc to that structure. Generalizations of Itô’s theory like the one...

12. Notation
(pp. 260-262)
13. References
(pp. 263-264)
14. Index
(pp. 265-269)
15. Back Matter
(pp. 270-270)