Customer Reviews

Stochastic Flows and Stochastic Differential Equations (Cambridge Studies in Advanced Mathematics)

5 star:		(0)
4 star:		(2)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

This research monograph presents the notion of a stochastic flow and develops a theory of stochastic flows with the goal of applications to solving stochastic differential equations. There is very little material here on the interplay between the stochastic flows and differential geometry. Readers interested in geometry should consult Baudoin's An Introduction To The Geometry Of Stochastic Flows, Emery's Stochastic Calculus in Manifolds or Gliklikh's Global Analysis in Mathematical Physics: Geometric and Stochastic Models.

Kunita opens his book with very brief review of stochastic process and random fields in Chapter 1. Kunita offers an extremely brief survey of Ito & Stratonovich calculus in Chapter 2. In several instances, assumptions are introduced and proofs are offered without clearly stating where the assumptions are actually used. These two chapters are really the low point of the book. The reader is well advised to have prepared this material from a text dedicated to these introductory topics. I recommend Rogers & Williams two volume set Diffusions, Markov Processes, and Martingales: Volume 1, Foundations and Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus.

Ito calculus is generalized in Chapter 3 to cover the case of semimartingales with spatial parameters. This material is quite technical, particularly the treatment of the regularity conditions required on the quadratic variation of the martingale kernel. A number of the foundational results and key concepts, such as local characteristic representation, are delegated to the exercises. The development of the Ito and Stratonovich integrals to cover the spatial parameters scenario follows the well-know monotone class-style argument and begins with simple processes at the base. The author provides a two line extension from simple processes to predictable processes, which seems a bit terse to this reviewer. The careful reader will note that the author uses processes with continuous sample paths to define his notion of a predictable process, not merely left continuous as is often the case with other authors.

Chapter 4 is the real highlight of the text. Armed with the spatial parameter version of Ito calculus, Kunita introduces the stochastic flow in Chapter 4 and establishes the one-to-one correspondence between forward stochastic flows and continuous semimartingales. Asymptotic/Ergodic properties are investigated and the backward flow is introduced. There is a brief section covering stochastic flows on manifolds, however this is very cursory as a standalone section. It does however make good appendix material for readers of Emery's book.

Chapter 5 is devoted to establishing strong and weak approximation results for solutions of SDE's. The technique employed it to study the convergence properties of a family of semimartingales and their associated stochastic flows. This study is based on the theory of stochastic flows developed in Chapter 4. Convergence properties are then used to derive solution approximation results.

The final chapter considers existence and uniqueness of solutions of spatially parameterized SDE's. The approach here is to take a cue from the theory of 1st and 2nd order deterministic PDE's and consider the characteristic system of the SDE. Stochastic flows arise naturally in studying the characteristic system and establishing existence and uniqueness of solutions to the SDE. As an application of the existence results, solutions to certain non-linear filtering problems are studied.

There are a number of exercises at the end of most sections. Written in research monograph-style, the rest of the book requires real work on the part of the reader in order to get the most out of the material.

This is the stuff that makes you grit your teeth, but somehow keep coming back for more. For the reader, the best part was the definition of the proability spaces and events which broadened the readers knowledge of Queing theory considerably. Toward the end the writer's mathmatical exploration of the ricatti equation and the Kalman filter reflect, in a large degree, the heavy algebraic stuff put out in the 80's before the refinement of chaos theory, nnet, and fuzzy, which define these problems along more implemental routes.