Wong-Zakai type convergence in infinite dimensions

Arnab Ganguly (Brown University)

Abstract


The paper deals with convergence of solutions of a class of stochastic differential equations driven by infinite-dimensional semimartingales. The infinite dimensional semimartingales considered in the paper are Hilbert-space valued. The theorems presented generalize the convergence result obtained by Wong and Zakai for stochastic differential equations driven by linear interpolations of a finite-dimensional Brownian motion. In particular, a general form of the correction factor is derived. Examples are given illustrating the use of the theorems to obtain other kinds of approximation results.

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Pages: 1-34

Publication Date: March 5, 2013

DOI: 10.1214/EJP.v18-2650

References

  • Acquistapace, P.; Terreni, B. An approach to Ito linear equations in Hilbert spaces by approximation of white noise with coloured noise. Stochastic Anal. Appl. 2 (1984), no. 2, 131--186. MR0746434
  • Brzeźniak, Z.; Capiński, M.; Flandoli, F. A convergence result for stochastic partial differential equations. Stochastics 24 (1988), no. 4, 423--445. MR0972973
  • Cho, Nhansook. Weak convergence of stochastic integrals driven by martingale measure. Stochastic Process. Appl. 59 (1995), no. 1, 55--79. MR1350256
  • Cho, Nhansook. Weak limit theorems for stochastic differential equations driven by martingale measures. Stochastics Stochastics Rep. 59 (1996), no. 1-2, 1--20. MR1427257
  • Da Prato, Giuseppe; Zabczyk, Jerzy. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. xviii+454 pp. ISBN: 0-521-38529-6 MR1207136
  • Gihman, I. I.; Skorohod, A. V. The theory of stochastic processes. III. Translated from the Russian by Samuel Kotz. With an appendix containing corrections to Volumes I and II. Grundlehren der Mathematischen Wissenschaften, 232. Springer-Verlag, Berlin-New York, 1979. iii+387 pp. ISBN: 3-540-90375-5 MR0651015
  • Itō, Kiyosi. Foundations of stochastic differential equations in infinite-dimensional spaces. CBMS-NSF Regional Conference Series in Applied Mathematics, 47. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1984. ix+70 pp. ISBN: 0-89871-193-2 MR0771478
  • A. Jakubowski, J. Mémin, and G. Pagès. Convergence en loi des suites d'intégrales stochastiques sur l'espace D^1 de Skorokhod. phProbab. Theory Related Fields, 81penalty0 (1):penalty0 111--137, 1989.
  • Jakubowski, Adam. Continuity of the Ito stochastic integral in Hilbert spaces. Stochastics Stochastics Rep. 59 (1996), no. 3-4, 169--182. MR1427737
  • Konecny, Franz. On Wong-Zakai approximation of stochastic differential equations. J. Multivariate Anal. 13 (1983), no. 4, 605--611. MR0727043
  • Kurtz, Thomas G.; Protter, Philip. Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 (1991), no. 3, 1035--1070. MR1112406
  • Kurtz, Thomas G.; Protter, Philip E. Weak convergence of stochastic integrals and differential equations. II. Infinite-dimensional case. Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), 197--285, Lecture Notes in Math., 1627, Springer, Berlin, 1996. MR1431303
  • Kurtz, Thomas G.; Pardoux, Étienne; Protter, Philip. Stratonovich stochastic differential equations driven by general semimartingales. Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), no. 2, 351--377. MR1324812
  • Marcus, Steven I. Modeling and approximation of stochastic differential equations driven by semimartingales. Stochastics 4 (1980/81), no. 3, 223--245. MR0605630
  • Métivier, Michel. Semimartingales. A course on stochastic processes. de Gruyter Studies in Mathematics, 2. Walter de Gruyter & Co., Berlin-New York, 1982. xi+287 pp. ISBN: 3-11-008674-3 MR0688144
  • Métivier, Michel; Pellaumail, Jean. Stochastic integration. Probability and Mathematical Statistics. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. xii+196 pp. ISBN: 0-12-491450-0 MR0578177
  • Nakao, Shintaro; Yamato, Yuiti. Approximation theorem on stochastic differential equations. Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976), pp. 283--296, Wiley, New York-Chichester-Brisbane, 1978. MR0536015
  • Protter, Philip. Approximations of solutions of stochastic differential equations driven by semimartingales. Ann. Probab. 13 (1985), no. 3, 716--743. MR0799419
  • Ryan, Raymond A. Introduction to tensor products of Banach spaces. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2002. xiv+225 pp. ISBN: 1-85233-437-1 MR1888309
  • Słomiński, Leszek. Stability of stochastic differential equations driven by general semimartingales. Dissertationes Math. (Rozprawy Mat.) 349 (1996), 113 pp. MR1377600
  • Stratonovich, R. L. A new representation for stochastic integrals and equations. SIAM J. Control 4 1966 362--371. MR0196814
  • Stroock, Daniel W.; Varadhan, S. R. S. On the support of diffusion processes with applications to the strong maximum principle. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pp. 333--359. Univ. California Press, Berkeley, Calif., 1972. MR0400425
  • Tessitore, Gianmario; Zabczyk, Jerzy. Wong-Zakai approximations of stochastic evolution equations. J. Evol. Equ. 6 (2006), no. 4, 621--655. MR2267702
  • Twardowska, Krystyna. Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions. Dissertationes Math. (Rozprawy Mat.) 325 (1993), 54 pp. MR1215779
  • Ustunel, S. Stochastic integration on nuclear spaces and its applications. Ann. Inst. H. Poincaré Sect. B (N.S.) 18 (1982), no. 2, 165--200. MR0662449
  • Walsh, John B. An introduction to stochastic partial differential equations. École d'été de probabilités de Saint-Flour, XIV—1984, 265--439, Lecture Notes in Math., 1180, Springer, Berlin, 1986. MR0876085
  • Wong, Eugene; Zakai, Moshe. On the relation between ordinary and stochastic differential equations. Internat. J. Engrg. Sci. 3 1965 213--229. MR0183023
  • Wong, Eugene; Zakai, Moshe. On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Statist. 36 1965 1560--1564. MR0195142
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