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Weak convergence of stochastic integrals and differential equations II: Infinite dimensional case

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1627)

Keywords

  • Stochastic Differential Equation
  • Weak Convergence
  • Random Measure
  • Stochastic Integral
  • Stochastic Partial Differential Equation

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References

  1. Araujo, A., Giné, E., The Central Limit Theorem for Real and Banach Valued Random Variables, Wiley, New York (1980).

    MATH  Google Scholar 

  2. Bhatt, A.G., Mandrekar, V., “On weak solution of stochastic PDE's”, preprint (1995).

    Google Scholar 

  3. Blount, D., “Comparison of stochastic and deterministic models of a linear chemical reaction with diffusion”, Ann. Probab. 19, 1440–1462 (1991).

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Blount, D., “A simple inequality with applications to SPDE's”, preprint (1995).

    Google Scholar 

  5. Brzeźniak, Z., Capiński, M., Flandoli, F., “A convergence result for stochastic partial differential equations”, Stochastics 24, 423–445 (1988).

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Brown, T.C., A martingale approach to the Poisson convergence of simple point processes. Ann. Probab. 6, 615–628 (1978).

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Burkholder, D.L., “Distribution function inequalities for martingales”, Ann. Probab. 1, 19–42, (1973).

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Cho, N., Weak convergence of stochastic integrals and stochastic differential equations driven by martingale measure and its applications, PhD Dissertation, University of Wisconsin—Madison (1994).

    Google Scholar 

  9. Cho, N., “Weak convergence of stochastic integrals driven by martingale measure”, Stochastic Process. Appl. (to appear) (1995).

    Google Scholar 

  10. Çinlar, E., Jacod, J., “Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures”, Seminar on Stochastic Processes 1981, (E. Çinlar, K.L. Chung, R.K. Getoor, eds.), Birkhäuser, Boston, 159–242 (1981).

    CrossRef  Google Scholar 

  11. Da Prato, G., Zabczyk, J., Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992).

    CrossRef  MATH  Google Scholar 

  12. Da Prato, G., Zabczyk, J., “A note on stochastic convolution”, Stoch. Anal. Appl. 10, 143–153. (1992).

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Dellacherie, C., Maisonneuve, B., Meyer, P.A., Probabilites et Potentiel, Chapitres XVII a XXIV: Processus de Markov (fin); Compléments de Calcul Stochastique, Hermann, Paris (1992).

    Google Scholar 

  14. Donnelly, P.E., Kurtz, T.G., “A countable representation of the Fleming-Viot measure-valued diffusion”, Ann Probab., to appear (1996).

    Google Scholar 

  15. Donnelly, P.E., Kurtz, T.G., “Particle representations for measure-valued population models”, (in preparation (1996).

    Google Scholar 

  16. Engelbert, H.J., “On the theorem of T. Yamada and S. Watanabe”, Stochastics 35, 205–216 (1991).

    MathSciNet  MATH  Google Scholar 

  17. Ethier, S.N., Kurtz, T.G., Markov Processes: Characterization and Convergence. Wiley, New York (1986).

    CrossRef  MATH  Google Scholar 

  18. Feller, W., An Introduction to Probability Theory and Its Applications II, 2nd ed., Wiley, New York (1971).

    MATH  Google Scholar 

  19. Fichtner, K.H., Manthey, R., “Weak approximation of stochastic equations”, Stochastics Stochastics Rep. 43, 139–160. (1993).

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. Graham, C., “McKean-Vlasov Ito-Skorohod equations and nonlinear diffusions with discrete jump sets”, Stochastic Process. Appl. 40, 69–82 (1992).

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. Gyöngy, I., “On the approximation of stochastic partial differential equations I, II”, Stochastics 25, 59–85, 26, 129–164 (1988,1989).

    CrossRef  MATH  Google Scholar 

  22. Ichikawa, A., “Some inequalities for martingales and stochastic convolutions”, Stoch. Anal. Appl. 4, 329–339 (1986).

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. Ikeda, N., Watanabe, S., Stochastic Differential Equations and Diffusion Processes, North Holland, Amsterdam (1981).

    MATH  Google Scholar 

  24. Itô, K., “On stochastic differential equations”, Mem. Amer. Math. Soc. 4 (1951).

    Google Scholar 

  25. Jakubowski, A., “Continuity of the Ito stochastic integral in Hilbert spaces”, preprint (1995).

    Google Scholar 

  26. Jakubowski, A., “A non-Skorohod topology on the Skorohod space” (1995).

    Google Scholar 

  27. Jakubowski, A. Mémin, J., Pagès, G., “Convergence in loi des suites d'intégrales stochastique sur l'espace D 1 de Skorohod”, Probab. Theory Related Fields 81, 111–137 (1989).

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. Jetschke, G., “Lattice approximation of a nonlinear stochastic partial differential equation with white noise”, Random Partial Differential Equations, Oberwolfach (1989), Birkhäuser Verlag, Basel, 107–126 (1991).

    CrossRef  Google Scholar 

  29. Kallianpur, G., Pérez-Abreu, V., “Weak convergence of solutions of stochastic evolution equations on nuclear spaces”, Stochastic Partial Differential Equations and Applications II LNM 1390, 119–131 (1989).

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. Kallianpur, G., Xiong, J., “Atochastic models of environmental pollution”, J. Appl. Probab. (1994).

    Google Scholar 

  31. Kallianpur, G., Xiong, J., “Asymptotic behavior of a system of interacting nuclear space valued stochastic differential equations driven by Poisson random measures”, Appl. Math. Optim. 30, 175–201 (1994).

    CrossRef  MathSciNet  MATH  Google Scholar 

  32. Kasahara, Y., Yamada, K., “Stability theorem for stochastic differential equations with jumps”, Stochastic Process. Appl. 38, 13–32 (1991).

    CrossRef  MathSciNet  MATH  Google Scholar 

  33. Khas'minskii, R.Z., “On stochastic processes defined by differential equations with a small parameter”, Theory Probab. Appl. 11, 211–228 (1966).

    CrossRef  MathSciNet  Google Scholar 

  34. Khas'minskii, R.Z., “A limit theorem for the solutions of differential equations with random right-hand sides”, Theory Probab. Appl. 11, 390–406 (1966).

    CrossRef  MATH  Google Scholar 

  35. Kotelenez, P., “A submartingale type inequality with applications to stochastic evolution equations”, Stochastics. 8, 139–151 (1982).

    CrossRef  MathSciNet  MATH  Google Scholar 

  36. Kotelenez, P., “A stopped Doob inequality for stochastic convolution integrals and stochastic evolution equations”, Stoch. Anal. Appl. 2, 245–265 (1984).

    CrossRef  MathSciNet  MATH  Google Scholar 

  37. Kotelenez, P., “A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation”, Probab. Theory Relat. Fields 102, 159–188 (1995).

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. Kurtz, T.G., “Representations of Markov processes as multiparameter time changes”, Ann. Probab. 8, 682–715 (1980).

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. Kurtz, T.G., “Random time changes and convergence in distribution under the Meyer-Zheng conditions”, Ann. Probab. 19, 1010–1034 (1991).

    CrossRef  MathSciNet  MATH  Google Scholar 

  40. Kurtz, T.G., “Averaging for martingale problems and stochastic approximation”, Applied Stochastic Analysis. Proceedings of the US-French Workshop. Lect. Notes. Control. Inf. Sci. 177, 186–209 (1992).

    MathSciNet  Google Scholar 

  41. Kurtz, T.G., Marchetti, F., “Averaging stochastically perturbed Hamiltonian systems”, Proceedings of Symposia in Pure Mathematics, 57, 93–114 (1995).

    CrossRef  MathSciNet  MATH  Google Scholar 

  42. Kurtz, T.G., Protter, P., “Weak limit theorems for stochastic integrals and stochastic differential quations”, Ann. Probab. 19, 1035–1070 (1991).

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. Kurtz, T.G., Protter, P., “Wong-Zakai corrections, random evolutions, and simulation schemes for sde's”, Stochastic Analysis: Liber Amicorum for Moshe Zakai, Academic Press, San Diego. 331–346 (1991).

    MATH  Google Scholar 

  44. Lenglart, E., Lepingle, D., Pratelli, M., “Presentation unifiée de certaines inegalités des martingales”, Séminaires de Probabilités, XIV, LNM, Springer, Berlin, 26–61 (1980).

    MATH  Google Scholar 

  45. Liggett, T.M., “Existence theorems for infinite particle systems”, Trans. Amer. Math. Soc. 165, 471–481 (1972).

    CrossRef  MathSciNet  MATH  Google Scholar 

  46. Liggett, T.M., Interacting Particle Systems, Springer-Verlag, New York (1985).

    CrossRef  MATH  Google Scholar 

  47. Méléard, S., “Representation and approximation of martingale measures”, Stochastic PDE's and their Applications. Lect. Notes in Control and Information Sci. 176, Springer, Berlin-New York, 188–199 (1992).

    MATH  Google Scholar 

  48. Méléard, S., “Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models”, this volume (1996).

    Google Scholar 

  49. Merzbach, E., Zakai, M., “Worthy martingales and integrators”, Stat. Prob. Letters. 16, 391–395 (1993).

    CrossRef  MathSciNet  MATH  Google Scholar 

  50. Métivier, M., Pellaumail, J., Stochastic Integration, Academic Press, New York (1980).

    MATH  Google Scholar 

  51. Meyer, P.A., Zheng, W.A., “Tightness criteria for laws of semimartingales”, Ann. Inst. H. Poincaré, Probab. Statist. 20, 353–372 (1984).

    MathSciNet  MATH  Google Scholar 

  52. Mikulevicius, R., Rozovskii, B.L., “On stochastic integrals in topological vector spaces”, Stochastic Analysis. Proceedings of Symposia in Pure Mathematics. 57, 593–602 (1994).

    CrossRef  MathSciNet  MATH  Google Scholar 

  53. Protter, P., Stochastic Integration and Differential Equations, Spriger-Verlag, New York (1990).

    CrossRef  MATH  Google Scholar 

  54. Perkins, E., “On the martingale problem for interactive measure-valued branching diffusions”, Mem. Amer. Math. Soc. 549 (1995).

    Google Scholar 

  55. Shiga, T., Shimizu, A., “Infinite-dimensional stochastic differential equations and their applications”, J. Mat. Kyoto Univ. 20, 395–416 (1980).

    MathSciNet  MATH  Google Scholar 

  56. Stricker, C., “Lois de semimartingales et critères de compacité”, Séminares de Probabilités XIX, LNM 1123 (1985).

    Google Scholar 

  57. Tubaro, L., “An estimate of Burkholder type for stochastic processes defined by the stochastic integral”, Stochastic Analysis and Appl., 187–192 (1984).

    Google Scholar 

  58. Twardowska, K., “Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions”, Dissertationes Mathematicae CCXXV (1993).

    Google Scholar 

  59. Ustunel, S., “Stochastic integration on nuclear spaces and its applications”, Ann. Inst. Henri Poincaré 18, 165–200 (1982).

    MathSciNet  MATH  Google Scholar 

  60. Walsh, J., “An introduction to stochastic partial differential equations”, Lect. Notes in Math. 1180, 265–439 (1986).

    CrossRef  MathSciNet  Google Scholar 

  61. Yamada, T., Watanabe, S., “On the uniqueness of solutions of stochastics differential equations” J. Math. Kyota Univ. 11, 155–167 (1971).

    MathSciNet  MATH  Google Scholar 

  62. Zabczyk, J., “The fractional calculus and stochastic evolution equations”, Barcelona Seminar on Stochastic Analysis (St. Feliu de Goixols, 1991). Progr. Probab. 32, 222–234. Birkhäuser, Basel (1993).

    MATH  Google Scholar 

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Kurtz, T.G., Protter, P.E. (1996). Weak convergence of stochastic integrals and differential equations II: Infinite dimensional case. In: Talay, D., Tubaro, L. (eds) Probabilistic Models for Nonlinear Partial Differential Equations. Lecture Notes in Mathematics, vol 1627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0093181

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  • DOI: https://doi.org/10.1007/BFb0093181

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