Abstract
In this chapter, we introduce some basic techniques and notions which will be used throughout the sequel. Once and for all, we consider below, a filtered probability space (Ω, F, F t , P) and we suppose that each F t contains all the sets of P-measure zero in F. As a result, any limit (almost-sure, in the mean, etc.) of adapted processes is an adapted process; a process which is indistinguishable from an adapted process is adapted.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes and Comments
Itô, K., and Watanabe, S. Transformation of Markov processes by multiplicative functionals. Ann. Inst. Fourier 15(1965) 15–30.
Doléans-Dade, C., and Meyer, P.A. Intégrales stochastiques par rapport aux martingales locales. Sém. Prob. IV, Lect. Notes in Mathematics, vol. 124. Springer, Berlin Heidelberg New York 1970, pp. 77–107.
Dellacherie, C., and Meyer, P.A. Probabilités et potentiel. Hermann, Paris, vol. I 1976, vol. H 1980, vol. III 1983, vol. IV 1987.
Métivier, M. Semimartingales: a course on stochastic processes. de Gruyter, Berlin New York 1982.
Kunita, H. Stochastic differential equations and stochastic flows of diffeomorphisms. Ecole d’Eté de Probabilités de Saint-Flour XII, Lect. Notes in Mathematics, vol. 1097. Springer, Berlin Heidelberg New York 1984, pp. 143–303.
Getoor, R.K., and Sharpe, M.J. Conformal martingales. Invent. Math. 16(1972) 271–308.
Stricker, C. Quasi-martingales, martingales locales, semi-martingales et filtrations. Z.W. 39(1977) 55–63.
Yor, M. Sur quelques approximations d’intégrales stochastiques. Sém. Prob. XI, Lect. Notes in Mathematics, vol. 528. Springer, Berlin Heidelberg New York 1977, pp. 518–528.
Sharpe, M.J. Local times and singularities of continuous local martingales. Sém. Prob. XIV, Lect. Notes in Mathematics, vol. 784. Springer, Berlin Heidelberg New York 1980, pp. 76–101.
Maisonneuve, B. Une mise au point sur les martingales locales continues définies sur un intervalle stochastique. Sém. Prob. XI, Lect. Notes in Mathematics, vol. 528. Springer, Berlin heidelberg New York 1977, pp. 435–445.
Dellacherie, C., and Meyer, P.A. Probabilités et potentiel. Hermann, Paris, vol. I 1976, vol. H 1980, vol. III 1983, vol. IV 1987.
Paley, R., Wiener, N., and Zygmund, A. Note on random functions. Math. Z. 37(1933) 647–668.
Kunita, H., and Watanabe, S. On square-integrable martingales. Nagoya J. Math. 30(1967) 209–245.
Itô, K. Stochastic integral. Proc. Imp. Acad. Tokyo 20(1944) 519–524.
Métivier, M. Semimartingales: a course on stochastic processes. de Gruyter, Berlin New York 1982.
Isaacson, D. Stochastic integrals and derivatives. Ann. Math. Sci. 40(1969) 1610–1616.
Yoeurp, C. Sur la dérivation des intégrales stochastiques. Sém. Prob. XIV, Lect. Notes in Mathematics, vol. 784. Springer, Berlin Heidelberg New York 1980, pp. 249–253.
Donati-Martin, C., and Yor, M. Mouvement brownien et inégalité de Hardy dans L2. Sém. Prob. XXIII, Lect. Notes in Mathematics, vol. 1372. Springer, Berlin Heidelberg New York 1989, pp. 315–323.
Chan, J., Dean, D.S., Jansons, K.M., and Rogers, L.C.G. On polymer conformations in elongational flows. Comm. Math. Phys. 160(2) (1994) 239–257.
Dean, D.S., and Jansons, K.M. A note on the integral of the Brownian Bridge. Proc. R. Soc. London A 437(1992) 792–730.
Kunita, H., and Watanabe, S. On square-integrable martingales. Nagoya J. Math. 30(1967) 209–245.
Kunita, H. Some extensions of Itô’s formula. Sém. Prob. XV, Lect. Notes in Mathematics, vol. 850. Springer, Berlin Heidelberg New York 1981, pp. 118–141.
Kunita, H. Stochastic flows and stochastic differential equations. Cambridge University Press 1990.
Föllmer, H. Calcul d’Itô sans probabilités. Sém. Prob. XV, Lect. Notes in Mathematics, vol. 850. Springer, Berlin Heidelberg New York 1981, pp. 143–150.
Doléans-Dade, C. On the existence and unicity of solutions of stochastic integral equations. Z.W. 36(1976) 93–101.
Kazamaki, N. Continuous Exponential martingales and BMO. Lect. Notes in Mathematics, vol. 1579. Springer, Berlin Heidelberg New York 1994.
Kunita, H., and Watanabe, S. On square-integrable martingales. Nagoya J. Math. 30(1967) 209–245.
Ruiz de Chavez, J. Le théorème de Paul Lévy pour des mesures signées. Sém. Prob. XVIII, Lect. Notes in Mathematics, vol. 1059. Springer, Berlin Heidelberg New York 1984, pp. 245–255.
Robbins, H., and Siegmund, D. Boundary crossing probabilities for the Wiener process and sample sums. Ann. Math. Stat. 41(1970) 1410–1429.
Donati-Martin, C., and Yor, M. Mouvement brownien et inégalité de Hardy dans L2. Sém. Prob. XXIII, Lect. Notes in Mathematics, vol. 1372. Springer, Berlin Heidelberg New York 1989, pp. 315–323.
Calais, J.Y., and Génin, M. Sur les martingales locales continues indexées par] 0, oo[. Sém. Proba. XXII, Lect. Notes in Mathematics, vol. 986. Springer, Berlin Heidelberg New York 1988, pp. 454–466.
Carlen, E., and Krée, P. Sharp LP-estimates on multiple stochastic integrals. Ann. Prob. 19(1) (1991) 354–368.
Getoor, R.K., and Sharpe, M.J. Conformal martingales. Invent. Math. 16(1972) 271–308.
Lenglart, E. Relation de domination entre deux processus. Ann. I.H.P. 13(1977) 171–179.
Lenglart, E., Lépingle, D., and Pratelli, M. Une présentation unifiée des inégalités en théorie des martingales. Sém. Prob. XIV, Lect. Notes in Mathematics, vol. 784. Springer, Berlin Heidelberg New York 1980, pp. 26–48.
Burkholder, D.L. Distribution function inequalities for martingales. Ann. Prob. 1(1973) 19–42.
Bass, R.F. 4-inequalities for functionals of Brownian motion. Sém. Prob. XXI. Lect. Notes in Mathematics, vol. 1247. Springer, Berlin Heidelberg New York 1987, pp. 206–217.
Davis, B. On the Barlow-Yor inequalities for local time. Sém. Prob. XXI, Lect. Notes in Mathematics, vol. 1247. Springer, Berlin Heidelberg New York 1987, pp. 218–220.
Durrett, R. Brownian motion and martingales in analysis. Wadsworth, Belmont, Calif. 1984.
Yor, M. Inégalités de martingales continues arrètées à un temps quelconque, I, II. In: Grossissements de filtrations: exemples et applications. Lect. Notes in Mathematics, vol. 1118. Springer, Berlin Heidelberg New York 1985, pp. 110–171.
Chou, S. Sur certaines généralisations de l’inégalité de Fefferman Sém. Prob. XVIII, Lect. Notes in Mathematics, vol. 1059. Springer, Berlin Heidelberg New York 1984, pp. 219–222.
Chung, K.L., and Williams, R.T. Introduction to stochastic integration. Second edition. Birkhäuser, Boston 1989.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Revuz, D., Yor, M. (1999). Stochastic Integration. In: Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften, vol 293. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06400-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-662-06400-9_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08400-3
Online ISBN: 978-3-662-06400-9
eBook Packages: Springer Book Archive