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**Semimartingales and their stochastic calculus on manifolds. Ed. by I. Iscoe.**
*(English)*
Zbl 0539.60050

Collection de la Chaire Aisenstadt. Montréal (Québec), Canada: Les Presses de l’Université de Montréal. 187 p. $ 17.00 (1984).

This book divides into two parts. In the first part a general theory is expounded, concerning stochastic integration with respect to (possibly discontinuous) martingales. This first part is composed of chapters one to six. Points of note are: a discussion of the striking characterisation of (real) semimartingales as the most general ”reasonable” stochastic integrators, the use of vector-valued measures (taking values in the non- Banach, non-locally-convex space \(L^ 0(\Omega,\theta,P))\); the introduction of formal measures and formal semi-martingales as devices to defer discussions of convergence till the end of a calculation (playing in this a role analogous to that of distributions in differential equations); and finally a careful discussion of localisation.

The second part is composed of the remaining chapters seven to eleven. Interest is narrowed down to continuous semimartingales, but extended to consider such processes taking values in a manifold. Thus non-linearity is introduced, and this in turn provides strong motivation to try to formulate stochastic calculus on manifolds in an invariant manner. All who have tried their hands at problems in stochastic differential geometry will immediately appreciate the desirability of an invariant formulation, if only to rescue them from an exotic abundance of suffices and to reveal the underlying geometrical structure.

There are at least two approaches to an invariant formulation. On the one hand one can work with Stratonovich differentials, and this provides a direct route for suitably regular problems. The second approach is due to the author himself and is the one discussed here. It boldly tackles the problem head-on by providing an invariant interpretation of the (Itô) stochastic differential of a semimartingale. One is necessarily led to the theory of second-order differential geometry. In this interpretation stochastic differential equations (s.d.e.) are interpreted as tangential representations: a continuous semimartingale solves an s.d.e. if as an integrator it can be interpreted via a corresponding tangential representation. This is actually related to the famous Stroock-Varadhan martingale-problem formulation.

The final test of any such general theory as this must be: whether workers begin to use it in order to carry through effective solutions to problems. Techniques in stochastic differenial geometry are often tested out by applying them to the basic example of lifting a semimartingale through a connection. In conclusion to the book a solution to this exercise is described that employs the invariant formulation developed previously. The reviewer knows of further applications made by R. W. R. Darling.

The second part is composed of the remaining chapters seven to eleven. Interest is narrowed down to continuous semimartingales, but extended to consider such processes taking values in a manifold. Thus non-linearity is introduced, and this in turn provides strong motivation to try to formulate stochastic calculus on manifolds in an invariant manner. All who have tried their hands at problems in stochastic differential geometry will immediately appreciate the desirability of an invariant formulation, if only to rescue them from an exotic abundance of suffices and to reveal the underlying geometrical structure.

There are at least two approaches to an invariant formulation. On the one hand one can work with Stratonovich differentials, and this provides a direct route for suitably regular problems. The second approach is due to the author himself and is the one discussed here. It boldly tackles the problem head-on by providing an invariant interpretation of the (Itô) stochastic differential of a semimartingale. One is necessarily led to the theory of second-order differential geometry. In this interpretation stochastic differential equations (s.d.e.) are interpreted as tangential representations: a continuous semimartingale solves an s.d.e. if as an integrator it can be interpreted via a corresponding tangential representation. This is actually related to the famous Stroock-Varadhan martingale-problem formulation.

The final test of any such general theory as this must be: whether workers begin to use it in order to carry through effective solutions to problems. Techniques in stochastic differenial geometry are often tested out by applying them to the basic example of lifting a semimartingale through a connection. In conclusion to the book a solution to this exercise is described that employs the invariant formulation developed previously. The reviewer knows of further applications made by R. W. R. Darling.

Reviewer: W.S.Kendall

### MSC:

60Hxx | Stochastic analysis |

58J65 | Diffusion processes and stochastic analysis on manifolds |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

46G10 | Vector-valued measures and integration |