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 Electronic Journal of Probability > Vol. 2 (1997) > Paper 4 open journal systems 


A Non-Skorohod Topology on the Skorohod Space

Adam Jakubowski, Nicholas Copernicus University


Abstract
A new topology (called $S$) is defined on the space $D$ of functions $x: [0,1] to R^1$ which are right-continuous and admit limits from the left at each $t > 0$. Although $S$ cannot be metricized, it is quite natural and shares many useful properties with the traditional Skorohod's topologies $J_1$ and $M_1$. In particular, on the space $P(D)$ of laws of stochastic processes with trajectories in $D$ the topology $S$ induces a sequential topology for which both the direct and the converse Prohorov's theorems are valid, the a.s. Skorohod representation for subsequences exists and finite dimensional convergence outside a countable set holds.


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Pages: 1-21

Published on: July 4, 1997
Bibliography
  1. Dellacherie, C., Meyer, P.A. (1980) Probabilit'es et potentiel, Chapters V-VIII, Hermann, Paris 1980. Math. Review link.
  2. Engelking, R., General Topology, Helderman, Berlin 1989. Math. Review link.
  3. Fernique, X., Processus lin'eaires, processus g'en'eralis'es, Ann. Inst. Fourier (Grenoble), 17 (1967), 1-92. Math. Review link.
  4. Fernique, X., Convergence en loi de variables al'etoires et de fonctions al'etoires, propri'etes de compacit'e des lois, II, in: J. Az'ema, P.A. Meyer, M. Yor, (Eds.) S'eminaire de Probabilit'es XXVII, Lecture Notes in Math., 1557 , 216-232, Springer, Berlin 1993. Math. Review link.
  5. Jakubowski, A., The a.s. Skorohod representation for subsequences in nonmetric spaces, Theory Probab. Appl., 42 (1997), 209-216, (preprint). Math. Review number not available.
  6. Jakubowski, A., Convergence in various topologies for stochastic integrals driven by semimartingales, Ann. Probab., 24 (1996), 2141-2153. Math. Review number not available.
  7. Jakubowski, A., From convergence of functions to convergence of stochastic processes. On Skorokhod's sequential approach to convergence in distribution., to appear in A Volume in Honour of A.V. Skorokhod, VSP 1997, (preprint). Math. Review number not available.
  8. Jakubowski, A., M'emin, J., Pages, G., Convergence en loi des suites d'int'egrales stochastiques sur l'espace $GD^1$ de Skorokhod, Probab. Th. Rel. Fields, 81 (1989), 111-137. Math. Review link.
  9. Kantorowich, L.V., Vulih, B.Z. & Pinsker, A.G., Functional Analysis in Partially Ordered Spaces (in Russian), Gostekhizdat, Moscow 1950. Math. Review number not available.
  10. Kisy'nski, J., Convergence du type L, Colloq. Math., 7 (1960), 205-211. Math. Review link.
  11. Kurtz, T., Random time changes and convergence in distribution under the Meyer-Zheng conditions, Ann. Probab., 19 (1991), 1010-1034. Math. Review link.
  12. Kurtz, T., Protter, P., Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab., 19 (1991), 1035-1070. Math. Review link.
  13. M'emin, J., Sl omi'nski, L. Condition UT et stabilit'e en loi des solutions d'equations diff'erentieles stochastiques, S'em. de Probab. XXV, Lecture Notes in Math., 1485, 162-177, Springer, Berlin 1991. Math. Review link.
  14. Meyer, P.A., Zheng, W.A., Tightness criteria for laws of semimartingales, Ann. Inst. Henri Poincar'e B, 20 (1984), 353-372. Math. Review link.
  15. Protter, Ph., Stochastic Integration and Differential Equations. A New Approach., 2nd Ed., Springer 1992. Math. Review link.
  16. Skorohod,A.V., Limit theorems for stochastic processes, Theor. Probability Appl., 1 (1956), 261-290. Math. Review link.
  17. Sl omi'nski, L., Stability of strong solutions of stochastic differential equations, Stoch. Proc. Appl., 31 (1989), 173-202. Math. Review link.
  18. Sl omi'nski, L., Stability of stochastic differential equations driven by general semimartingales, Dissertationes Math., CCCXLIX (1996), 113 p. Math. Review number not available.
  19. Stricker, C., Lois de semimartingales et crit`eres de compacit'e, S'eminares de probabilit'es XIX. Lect. Notes in Math., 1123, Springer, Berlin 1985. Math. Review link.
  20. Topso e, F., A criterion for weak convergence of measures with an application to convergence of measures on $GD,[0,1]$, Math. Scand. 25 (1969), 97-104. Math. Review link.
















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Electronic Journal of Probability. ISSN: 1083-6489