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

This article is cited in 47 scientific papers (total in 48 papers)

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.

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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
Received: 11.01.1978

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
\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
\jour Math. USSR-Sb.
\yr 1979
\vol 35
\issue 5
\pages 631--680

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    1. A. A. Novikov, “On conditions for absolute continuity of probability measures”, Math. USSR-Sb., 35:5 (1979), 697–707  mathnet  crossref  mathscinet  zmath  isi
    2. A. V. Melnikov, “On the theory of stochastic equations in components of semimartingales”, Math. USSR-Sb., 38:3 (1981), 381–394  mathnet  crossref  mathscinet  zmath  isi
    3. L. G. Vetrov, “On a filtration of semimartingales”, Russian Math. Surveys, 34:4 (1979), 189–190  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath  isi
    5. L. I. Gal'chuk, “Optional martingales”, Math. USSR-Sb., 40:4 (1981), 435–468  mathnet  crossref  mathscinet  zmath  isi
    6. R. Sh. Liptser, A. N. Shiryayev, “A Functional Central Limit Theorem for Semimartingales”, Theory Probab Appl, 25:4 (1981), 667  mathnet  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  mathscinet  zmath
    8. Shiryayev A., “Martingales - Recent Developments, Results and Applications”, Int. Stat. Rev., 49:3 (1981), 199–233  crossref  mathscinet  isi
    9. A. A. Butov, “The equivalence of measures corresponding to canonical Gaussian processes”, Russian Math. Surveys, 37:5 (1982), 162–163  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
    12. Lavrentiev V., “A Functional Central Limit-Theorem for Hilbert Space-Valued Semimartingales”, 270, no. 1, 1983, 41–44  mathscinet  isi
    13. Chitashvili R., “Martingale Ideology in the Theory of Controlled Stochastic-Processes”, 1021, 1983, 73–92  mathscinet  zmath  isi
    14. Glonti O., “Transmission of Television Type Signals Through a Feedback Channel”, 1021, 1983, 157–166  mathscinet  zmath  isi
    15. A. I. Ekushov, “A strong invariance principle and some of its applications”, Russian Math. Surveys, 39:4 (1984), 119–120  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    16. È. I. Kolomiets, “Relations between triplets of local characteristics of semimartingales”, Russian Math. Surveys, 39:4 (1984), 123–124  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    17. N. E. Kordzahiya, “Weak likelihood convergence of a process”, Russian Math. Surveys, 39:4 (1984), 127–128  mathnet  crossref  mathscinet  zmath  adsnasa  isi
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    19. Kabanov I., Liptser R., Shiriaev A., “Estimates of Proximity in the Variation of Probability-Measures”, 278, no. 2, 1984, 265–268  mathscinet  zmath  isi
    20. A. F. Taraskin, “On the limiting behaviour of the likelihood ratio for semimartingales”, Russian Math. Surveys, 40:2 (1985), 237–238  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    21. Memin J., Shiryayev A., “Hellinger-Kakutani Distance in Laws Corresponding to 2 Processes with Independent Increments”, 70, no. 1, 1985, 67–89  crossref  mathscinet  zmath  isi
    22. Musiela M., “Divergence, Convergence and Moments of Some Integral Functionals of Diffusions”, 70, no. 1, 1985, 49–65  crossref  mathscinet  zmath  isi
    23. C. R. Baker, A. F. Gualtierotti, “Discrimination with respect to a Gaussian process”, Probab Theory Relat Fields, 71:2 (1986), 159  crossref  mathscinet  zmath
    24. Yu. M. Kabanov, “An Estimate of Closeness in Variation of Probability Measures”, Theory Probab Appl, 30:2 (1986), 413  mathnet  crossref  mathscinet  zmath  isi
    25. V. V. Lavrent'ev, “The existence of a Hilbert space valued process with given jumps”, Russian Math. Surveys, 41:5 (1986), 151–152  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    26. Musiela M., “On Kac Functionals of One-Dimensional Diffusions”, Stoch. Process. Their Appl., 22:1 (1986), 79–88  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    28. A. Yu. Veretennikov, “On Strong Solutions of Ito^ Stochastic Equations with Jumps”, Theory Probab Appl, 32:1 (1987), 148  mathnet  crossref  mathscinet  zmath  isi
    29. Knut K. Aase, Peter Guttorp, “Estimation in models for security prices”, Scandinavian Actuarial Journal, 1987:3-4 (1987), 211  crossref  mathscinet
    30. L Vostrikva, “On the weak convergence of likelihood ratio processes of general statistical parametric models”, Stochastics, 23:3 (1988), 277  crossref  mathscinet
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    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  crossref  mathscinet  isi
    34. V. I. Bogachev, O. G. Smolyanov, “Analytic properties of infinite-dimensional distributions”, Russian Math. Surveys, 45:3 (1990), 1–104  mathnet  crossref  mathscinet  zmath  isi
    35. Yashin A., “An Extension of the Cameron-Martin Result”, J. Appl. Probab., 30:1 (1993), 247–251  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    37. N. V. Kvashko, “A Forward Interpolation Equation of a Semimartingale by Observations over a Point Process”, Theory Probab Appl, 40:1 (1995), 162  mathnet  crossref  mathscinet  isi
    38. A. Barchielli, A.S. Holevo, “Constructing quantum measurement processes via classical stochastic calculus”, Stochastic Processes and their Applications, 58:2 (1995), 293  crossref  mathscinet  zmath
    39. V. Kanišauskas, “Asymptotic parameter estimation for multivariate point processes”, Lith Math J, 37:4 (1997), 352  crossref  mathscinet  zmath
    40. V. Kanišauskas, “Asymptotically minimax separation of two simple hypotheses”, Lith Math J, 38:2 (1998), 131  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  isi
    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  crossref  mathscinet  isi
    43. A. A. Gushchin, É. Mordecki, “Bounds on Option Prices for Semimartingale Market Models”, Proc. Steklov Inst. Math., 237 (2002), 73–113  mathnet  mathscinet  zmath
    44. Lars Peter Hansen, Thomas J. Sargent, Gauhar Turmuhambetova, Noah Williams, “Robust control and model misspecification”, Journal of Economic Theory, 128:1 (2006), 45  crossref  mathscinet  zmath
    45. Kardaras C., “Market Viability via Absence of Arbitrage of the First Kind”, Financ. Stoch., 16:4 (2012), 651–667  crossref  mathscinet  zmath  isi
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    47. Irina Penner, Anthony Réveillac, “Risk measures for processes and BSDEs”, Finance Stoch, 2014  crossref  mathscinet
    48. V. M. Abramov, B. M. Miller, E. Ya. Rubinovich, P. Yu. Chiganskii, “Razvitie teorii stokhasticheskogo upravleniya i filtratsii v rabotakh R. Sh. Liptsera”, Avtomat. i telemekh., 2020, no. 3, 3–13  mathnet  crossref
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