RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

 Mat. Sb. (N.S.), 1978, Volume 107(149), Number 3(11), Pages 364–415 (Mi msb2679)

Absolute continuity and singularity of locally absolutely continuous probability distributions. I

Yu. M. Kabanov, R. Sh. Liptser, A. N. Shiryaev

Abstract: Let $(\Omega,\mathscr F)$ be a measurable space provided with a nondecreasing family of $\sigma$-algebras ($\mathscr F_t)_{t\geqslant0}$ with $\mathscr F=\bigvee_{t\geqslant0}\mathscr F_t$ and $\widetilde{\mathsf P}$ and $\mathsf P$ two locally absolutely continuous probability measures on $(\Omega,\mathscr F)$, i.e., such that $\widetilde{\mathsf P}_t\ll\mathsf P_t$ for $t\geqslant0$ ($\widetilde{\mathsf P}_t$ and $\mathsf P_t$ are the restrictions of $\widetilde{\mathsf P}$ and $\mathsf P$ to $\mathscr F_t$). One asks when $\widetilde{\mathsf P}\ll \mathsf P$ or $\widetilde{\mathsf P}\perp\mathsf P$. An answer to this question is given in terms of the convergence set of a certain increasing predictable process constructed for the martingale $\mathfrak Z=(\mathfrak Z_t,\mathscr F_t,\mathsf P)$ with $\mathfrak Z_t=d\widetilde{\mathsf P}_t/d\mathsf P_t$. Actually, the somewhat more general situation of $\theta$-local absolute continuity of measures is studied. The proof of the fundamental theorem is based on a series of results that are of independent interest.
In § 2 the theory of integration with respect to random measures is developed. § 4 deals with the convergence sets of semimartingales, and § 5 with the transformation of the predictable characteristics of a semimartingale under a locally absolutely continuous change of measure. Sufficient conditions are given in § 7 for the uniform integrability of nonnegative local martingales.
Bibliography: 24 titles.

Full text: PDF file (4001 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1979, 35:5, 631–680

Bibliographic databases:

UDC: 519.2
MSC: Primary 60G30, 60G45, 60H05; Secondary 28A40, 60G25, 60G40

Citation: Yu. M. Kabanov, R. Sh. Liptser, A. N. Shiryaev, “Absolute continuity and singularity of locally absolutely continuous probability distributions. I”, Mat. Sb. (N.S.), 107(149):3(11) (1978), 364–415; Math. USSR-Sb., 35:5 (1979), 631–680

Citation in format AMSBIB
\Bibitem{KabLipShi78} \by Yu.~M.~Kabanov, R.~Sh.~Liptser, A.~N.~Shiryaev \paper Absolute continuity and singularity of locally absolutely continuous probability distributions.~I \jour Mat. Sb. (N.S.) \yr 1978 \vol 107(149) \issue 3(11) \pages 364--415 \mathnet{http://mi.mathnet.ru/msb2679} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=515738} \zmath{https://zbmath.org/?q=an:0426.60039|0402.60039} \transl \jour Math. USSR-Sb. \yr 1979 \vol 35 \issue 5 \pages 631--680 \crossref{https://doi.org/10.1070/SM1979v035n05ABEH001615} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1979JG48000003} 

• http://mi.mathnet.ru/eng/msb2679
• http://mi.mathnet.ru/eng/msb/v149/i3/p364

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles
Cycle of papers

This publication is cited in the following articles:
1. A. A. Novikov, “On conditions for absolute continuity of probability measures”, Math. USSR-Sb., 35:5 (1979), 697–707
2. A. V. Melnikov, “On the theory of stochastic equations in components of semimartingales”, Math. USSR-Sb., 38:3 (1981), 381–394
3. L. G. Vetrov, “On a filtration of semimartingales”, Russian Math. Surveys, 34:4 (1979), 189–190
4. Yu. M. Kabanov, R. Sh. Liptser, A. N. Shiryaev, “On the representation of integral-valued random measures and local martingales by means of random measures with deterministic compensators”, Math. USSR-Sb., 39:2 (1981), 267–280
5. L. I. Gal'chuk, “Optional martingales”, Math. USSR-Sb., 40:4 (1981), 435–468
6. R. Sh. Liptser, A. N. Shiryayev, “A Functional Central Limit Theorem for Semimartingales”, Theory Probab Appl, 25:4 (1981), 667
7. R. Sh. Liptser, A. N. Shiryaev, “On weak convergence of semimartingales to stochastically continuous processes with independent and conditionally independent increments”, Math. USSR-Sb., 44:3 (1983), 299–323
8. Shiryayev A., “Martingales - Recent Developments, Results and Applications”, Int. Stat. Rev., 49:3 (1981), 199–233
9. A. A. Butov, “The equivalence of measures corresponding to canonical Gaussian processes”, Russian Math. Surveys, 37:5 (1982), 162–163
10. Yu. M. Kabanov, “On the existence of a solution in a problem of controlling a counting process”, Math. USSR-Sb., 47:2 (1984), 425–438
11. R. Sh. Liptser, A. N. Shiryaev, “Weak convergence of a sequence of semimartingales to a process of diffusion type”, Math. USSR-Sb., 49:1 (1984), 171–195
12. Lavrentiev V., “A Functional Central Limit-Theorem for Hilbert Space-Valued Semimartingales”, 270, no. 1, 1983, 41–44
13. Chitashvili R., “Martingale Ideology in the Theory of Controlled Stochastic-Processes”, 1021, 1983, 73–92
14. Glonti O., “Transmission of Television Type Signals Through a Feedback Channel”, 1021, 1983, 157–166
15. A. I. Ekushov, “A strong invariance principle and some of its applications”, Russian Math. Surveys, 39:4 (1984), 119–120
16. È. I. Kolomiets, “Relations between triplets of local characteristics of semimartingales”, Russian Math. Surveys, 39:4 (1984), 123–124
17. N. E. Kordzahiya, “Weak likelihood convergence of a process”, Russian Math. Surveys, 39:4 (1984), 127–128
18. Keigo Yamada, “A stability theorem for stochastic differential equations and application to stochastic control problems”, Stochastics, 13:4 (1984), 257
19. Kabanov I., Liptser R., Shiriaev A., “Estimates of Proximity in the Variation of Probability-Measures”, 278, no. 2, 1984, 265–268
20. A. F. Taraskin, “On the limiting behaviour of the likelihood ratio for semimartingales”, Russian Math. Surveys, 40:2 (1985), 237–238
21. Memin J., Shiryayev A., “Hellinger-Kakutani Distance in Laws Corresponding to 2 Processes with Independent Increments”, 70, no. 1, 1985, 67–89
22. Musiela M., “Divergence, Convergence and Moments of Some Integral Functionals of Diffusions”, 70, no. 1, 1985, 49–65
23. C. R. Baker, A. F. Gualtierotti, “Discrimination with respect to a Gaussian process”, Probab Theory Relat Fields, 71:2 (1986), 159
24. Yu. M. Kabanov, “An Estimate of Closeness in Variation of Probability Measures”, Theory Probab Appl, 30:2 (1986), 413
25. V. V. Lavrent'ev, “The existence of a Hilbert space valued process with given jumps”, Russian Math. Surveys, 41:5 (1986), 151–152
26. Musiela M., “On Kac Functionals of One-Dimensional Diffusions”, Stoch. Process. Their Appl., 22:1 (1986), 79–88
27. Kabanov Y., Liptser R., Shiryaev A., “On the Variation Distance for Probability-Measures Defined on a Filtered Space”, Probab. Theory Relat. Field, 71:1 (1986), 19–35
28. A. Yu. Veretennikov, “On Strong Solutions of Ito^ Stochastic Equations with Jumps”, Theory Probab Appl, 32:1 (1987), 148
29. Knut K. Aase, Peter Guttorp, “Estimation in models for security prices”, Scandinavian Actuarial Journal, 1987:3-4 (1987), 211
30. L Vostrikva, “On the weak convergence of likelihood ratio processes of general statistical parametric models”, Stochastics, 23:3 (1988), 277
31. Knut K. Aase, “Contingent claims valuation when the security price is a combination of an Ito process and a random point process”, Stochastic Processes and their Applications, 28:2 (1988), 185
32. Víctor Pérez-Abreu, “Decompositions of semimartingales on”, Journal of Functional Analysis, 80:2 (1988), 358
33. Lebreton A., Musiela M., “Laws of Large Numbers for Semimartingales with Applications to Stochastic Regression”, Probab. Theory Relat. Field, 81:2 (1989), 275–290
34. V. I. Bogachev, O. G. Smolyanov, “Analytic properties of infinite-dimensional distributions”, Russian Math. Surveys, 45:3 (1990), 1–104
35. Yashin A., “An Extension of the Cameron-Martin Result”, J. Appl. Probab., 30:1 (1993), 247–251
36. Yashin A., Manton K., “Modifications of the Em Algorithm for Survival Influenced by an Unobserved Stochastic-Process”, Stoch. Process. Their Appl., 54:2 (1994), 257–274
37. N. V. Kvashko, “A Forward Interpolation Equation of a Semimartingale by Observations over a Point Process”, Theory Probab Appl, 40:1 (1995), 162
38. A. Barchielli, A.S. Holevo, “Constructing quantum measurement processes via classical stochastic calculus”, Stochastic Processes and their Applications, 58:2 (1995), 293
39. V. Kanišauskas, “Asymptotic parameter estimation for multivariate point processes”, Lith Math J, 37:4 (1997), 352
40. V. Kanišauskas, “Asymptotically minimax separation of two simple hypotheses”, Lith Math J, 38:2 (1998), 131
41. W. Schachermayer, W. Schachinger, “Is There a Predictable Criterion for Mutual Singularity of Two Probability Measures on a Filtered Space?”, Theory Probab Appl, 44:1 (2000), 51
42. Galtchouk L., “Optimality of the Wald Sprt for Processes with Continuous Time Parameter”, Moda6 Advances in Model-Oriented Design and Analysis, Contributions to Statistics, ed. Atkinson A. Hackl P. Muller W., Physica-Verlag Gmbh & Co, 2001, 97–110
43. A. A. Gushchin, É. Mordecki, “Bounds on Option Prices for Semimartingale Market Models”, Proc. Steklov Inst. Math., 237 (2002), 73–113
44. Lars Peter Hansen, Thomas J. Sargent, Gauhar Turmuhambetova, Noah Williams, “Robust control and model misspecification”, Journal of Economic Theory, 128:1 (2006), 45
45. Kardaras C., “Market Viability via Absence of Arbitrage of the First Kind”, Financ. Stoch., 16:4 (2012), 651–667
46. F. Klebaner, R. Liptser, “When a stochastic exponential is a true martingale. Extension of the Beneš method”, Theory Probab. Appl., 58:1 (2014), 38–62
47. Irina Penner, Anthony Réveillac, “Risk measures for processes and BSDEs”, Finance Stoch, 2014
•  Number of views: This page: 825 Full text: 177 References: 34 First page: 2