Stochastic integration in Banach spaces. Theory and applications.

*(English)*Zbl 1314.60007
Probability Theory and Stochastic Modelling 73. Cham: Springer (ISBN 978-3-319-12852-8/hbk; 978-3-319-12853-5/ebook). viii, 211 p. (2015).

Over the past decade, stochastic partial differential equations (SPDE) driven by continuous and discontinuous noises have attracted increasing attention. One approach is to understand an SPDE as an infinite-dimensional SDE, driven by an infinite-dimensional (often Hilbert or Banach space-valued) stochastic process. This requires the notion of a stochastic (Itō) integral driven by an infinite-dimensional noise and the corresponding stochastic calculus. Despite the wealth of research publications in the field, there are only few monographs on infinite dimensional stochastic integration driven by continuous processes and even less for integrals with non-continuous drivers – actually, the books by M. Métivier and J. Pellaumail [Stochastic integration. New York etc.: Academic Press (1980; Zbl 0463.60004)] and M. Métivier [Semimartingales. A course on stochastic processes. Berlin – New York: de Gruyter (1982; Zbl 0503.60054)] seem to be the only sources for the general case. Usually, authors rely on ad hoc constructions or refer to various journal publications.

The present monograph is an attempt to fill this gap, focusing on Banach-space valued stochastic integrals driven by jump-type infinite-dimensional stochastic processes, in particular, by infinite-dimensional Lévy processes. Inspired by J. Rosiński’s paper [Stud. Math. 78, 15–38 (1984; Zbl 0559.60050)], the authors use an approach based on (compensated) Poisson point processes, extending the presentations of A. V. Skorokhod [Studies in the theory of random processes. Reading: Addison-Wesley (1965; Zbl 0146.37701)] and N. Ikeda and S. Watanabe [Stochastic differential equations and diffusion processes. 2nd ed. Amsterdam etc.: North-Holland; Tokyo: Kodansha Ltd. (1989; Zbl 0684.60040)] to a Banach space setting. Most importantly, it is shown that this approach and Métivier’s notion of stochastic integration are essentially compatible.

The presentation is largely self-contained, but a certain degree of sophistication on the side of the readers is assumed, e.g., the notion of Bochner and Pettis integrals or measurability concepts from the general theory of stochastic processes are only very briefly discussed, and some familiarity with probability in Banach spaces is certainly helpful. Having said this, the book comes rather quickly to the heart of the matter: the definition of Wiener (with deterministic integrands) and Itō (with predictable random integrands) integrals in Banach space, based on compensated Poisson point processes. Using the Wiener integral, the authors are able to define Banach-space valued Lévy processes and more general martingales and their Lévy-Itō (or semimartingale) decompositions. This is then used to show that the martingale-based approach by Métivier is compatible with the present theory. A rather general Itō’s formula is developed which is useful, in particular, in connection with Lévy-driven SPDEs. Having in mind applications, the authors discuss existence, uniqueness, (non-)Markovianity and dependence on the initial data of Banach-space valued SDEs and – as an application – SPDEs in Hilbert space. The latter are treated in the functional analytic way as in [G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. 2nd ed. Cambridge: Cambridge University Press (2014; Zbl 1317.60077)] or [S. Peszat and J. Zabczyk, Stochastic partial differential equations with Lévy noise. An evolution equation approach. Cambridge: Cambridge University Press (2007; Zbl 1205.60122)] which requires a short excursion into semigroup theory. The last part of the monograph covers some applications of infinite-dimensional jump-type SDEs to mathematical finance, non-linear filtering and the stability of semilinear stochastic equations.

The present monograph is an attempt to fill this gap, focusing on Banach-space valued stochastic integrals driven by jump-type infinite-dimensional stochastic processes, in particular, by infinite-dimensional Lévy processes. Inspired by J. Rosiński’s paper [Stud. Math. 78, 15–38 (1984; Zbl 0559.60050)], the authors use an approach based on (compensated) Poisson point processes, extending the presentations of A. V. Skorokhod [Studies in the theory of random processes. Reading: Addison-Wesley (1965; Zbl 0146.37701)] and N. Ikeda and S. Watanabe [Stochastic differential equations and diffusion processes. 2nd ed. Amsterdam etc.: North-Holland; Tokyo: Kodansha Ltd. (1989; Zbl 0684.60040)] to a Banach space setting. Most importantly, it is shown that this approach and Métivier’s notion of stochastic integration are essentially compatible.

The presentation is largely self-contained, but a certain degree of sophistication on the side of the readers is assumed, e.g., the notion of Bochner and Pettis integrals or measurability concepts from the general theory of stochastic processes are only very briefly discussed, and some familiarity with probability in Banach spaces is certainly helpful. Having said this, the book comes rather quickly to the heart of the matter: the definition of Wiener (with deterministic integrands) and Itō (with predictable random integrands) integrals in Banach space, based on compensated Poisson point processes. Using the Wiener integral, the authors are able to define Banach-space valued Lévy processes and more general martingales and their Lévy-Itō (or semimartingale) decompositions. This is then used to show that the martingale-based approach by Métivier is compatible with the present theory. A rather general Itō’s formula is developed which is useful, in particular, in connection with Lévy-driven SPDEs. Having in mind applications, the authors discuss existence, uniqueness, (non-)Markovianity and dependence on the initial data of Banach-space valued SDEs and – as an application – SPDEs in Hilbert space. The latter are treated in the functional analytic way as in [G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. 2nd ed. Cambridge: Cambridge University Press (2014; Zbl 1317.60077)] or [S. Peszat and J. Zabczyk, Stochastic partial differential equations with Lévy noise. An evolution equation approach. Cambridge: Cambridge University Press (2007; Zbl 1205.60122)] which requires a short excursion into semigroup theory. The last part of the monograph covers some applications of infinite-dimensional jump-type SDEs to mathematical finance, non-linear filtering and the stability of semilinear stochastic equations.

Reviewer: René L. Schilling (Dresden)

##### MSC:

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

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60H05 | Stochastic integrals |

60G57 | Random measures |

60G51 | Processes with independent increments; Lévy processes |

91G30 | Interest rates, asset pricing, etc. (stochastic models) |

91G80 | Financial applications of other theories |

60G35 | Signal detection and filtering (aspects of stochastic processes) |

35B40 | Asymptotic behavior of solutions to PDEs |