Abstract
This chapter is about stochastic calculus for continuous martingales and local martingales in a Hilbert space. The topics include definitions and investigations of martingales, local martingales and a Wiener process in a Hilbert space, construction of stochastic integrals with respect to these processes, and a detailed proof of the Itô formula for the square of a norm of a continuous semimartingale.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Dellacherie, C.: Capacités et processus stochastiques. Springer, Berlin (1972)
Doob, J.L.: Stochastic Processes. Wiley, New York; Chapman and Hall, London (1953)
Dunford, N., Schwarz, J.T.: Linear Operators. Part I: General Theory. Interscience Publishers, New York (1958)
Gelafand, I.M., Vilenkin, N.Y.: Generalized Functions. Vol. 4: Applications of Harmonic Analysis. Academic Press, New York (1969)
Gihman, I.I., Skorokhod, A.V.: The Theory of Stochastic Processes. Springer, Berlin (1974)
Krein, S.G. (ed.): Functional Analysis. Wolters-Noordhoff, Groningen (1972)
Krein, S.G., Petunin, Y.U., Semenov, E.M.: Interpolation of Linear Operators. American Mathematical Society, Providence, RI (1982)
Lions, J.-L., Magenes, E.: Problemes aux limites non homogénes et applications, vol. 1. Dunod, Paris (1968)
Métivier, M.: Semimartingales, a Course on Stochastic Processes. Walter de Gruyter, Berlin (1982)
Métivier, M., Pellaumoil S.: Stochastic Integration. Academic, New York (1980)
Meyer, P.A.: Notes sur les integrales stochastiques I. Integrales Hilbertiennes. In: Séminaire de Prob. XI. Lecture Notes in Mathematics, vol. 581, pp. 446–463. Springer, Berlin (1977)
Pardoux, E.: Integrales stochastiques hilbertiennes, Univ. Paris-Dauphine, Cahiers de math. de la decis. no. 7617 (1976)
Riesz, F., Sz.-Nagy, B.: Lecons d’analyse fonctionelle. Akadémiai Kiado, Budapest (1972)
Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin (1987)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Rozovsky, B.L., Lototsky, S.V. (2018). Stochastic Integration in a Hilbert Space. In: Stochastic Evolution Systems. Probability Theory and Stochastic Modelling, vol 89. Springer, Cham. https://doi.org/10.1007/978-3-319-94893-5_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-94893-5_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-94892-8
Online ISBN: 978-3-319-94893-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)