Abstract
Suppose that pathwise uniqueness holds for the SDE Xt=x0+3 £ to σ(Xs)dBs where |σ is bounded and bounded away from 0, and B is a Brownian motion on a filtered probability space, (Ω,F,F t,P). We give conditions under which pathwise uniqueness continues to hold in the enlarged filtration (F Lt ), where L is the end of an (F t)-optional set.
Partially supported by an NSF grant through Cornell University.
Research partially supported by an NSERC of Canada operating grant.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M.T. Barlow: Study of a filtration expanded to include an honest time. Z.f.W. 44, 307–323 (1979).
M.T. Barlow: One dimensional stochastic differential equations with no strong solution. J. London Math. Soc. (2) 26, 335–347 (1982).
M.T. Barlow: Inequalities for upcrossings of semimartingales via Skorohod embedding. Z.f.W. 64, 457–474 (1983).
M.T. Barlow and E.A. Perkins: One dimensional stochastic differential equations involving a singular increasing process. Stochastics 12, 229–249 (1984).
M.T. Barlow and E.A. Perkins: Sample path properties of stochastic integrals and stochastic differentiation. Stochastics and Stochastic Reprots 27, 261–293 (1989).
M.T. Barlow and P. Protter: On convergence of semimartingales. To appear in Sém. Prob. XXIV (1990).
C. Dellacherie and P.A. Meyer: Probabilities and potential B. Theory of martingales. North Holland, Amsterdam (1982).
N. El-Karoui: Sur les montées des semi-martingales II. Le cas discontinu. In Temps Locaux, Astérisque 52–53, 73–88 (1978).
D.N. Hoover, Extending probability spaces and adapted distribution. Preprint (1989).
D.N. Hoover and H.J. Keisler: Adapted probability distributions. Trans. Amer. Math. Soc. 286, 159–201 (1984).
D.N. Hoover and E.A. Perkins: Nonstandard construction of the stochastic integral and applications to stochastic differential equations, I, II. Trans. Amer. Math. Soc. 275, 1–58 (1983).
J.Jacod and J. Memin: Weak and strong solutions of stochastic differential equations: Existence and uniqueness. In Stochastic Integrals, Lect. Notes Math. 851 Springer (1981).
T. Jeulin: Semimartingales et grossissement d'une filtration. Lect. Notes. Math 833 Springer (1980).
H.J. Keisler: An infinitesmal approach to stochastic analysis. Mem. A.M.S. 297 (1984).
J.-F. Le Gall: Applications du temps local aux equations différentielles stochastiques unidimensionelles. Sem. Prob. XVII, 15–31 Lect. Notes. Math. 986 Springer (1983).
C. Stricker: Quasimartingales, martingales locales, semimartingales et filtration naturelle. Z.f.W. 39, 55–63 (1977).
M. Yor: Sur le balayage des semi-martingales continues. Sém Prob XIII 453–471. Lect. Notes Math. 721 (1979).
M. Yor: Rappels et préliminaires généraux. In Temps Locaux, Astérisque 52–53, 17–22 (1978).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1990 Springer-Verlag
About this paper
Cite this paper
Barlow, M.T., Perkins, E.A. (1990). On pathwise uniqueness and expansion of filtrations. In: Azéma, J., Yor, M., Meyer, P.A. (eds) Séminaire de Probabilités XXIV 1988/89. Lecture Notes in Mathematics, vol 1426. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083766
Download citation
DOI: https://doi.org/10.1007/BFb0083766
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-52694-0
Online ISBN: 978-3-540-47098-4
eBook Packages: Springer Book Archive