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Lithuanian Mathematical Journal

, Volume 39, Issue 2, pp 196–202 | Cite as

Some approximations of stochastic θ-integrals

  • N. V. Lazakovich
  • S. P. Stashulenok
  • O. L. Yablonskii
Article

Abstract

In this paper, we consider problems of approximation of stochastic θ-integrals (θ) 0 t f(B(s))dB(s) with respect to a Brownian motion by sums of the form ∑ k=1 p fn(B n θ (tk-1))[B n θ (tk)-B n θ (tk-1], where the sequences {fn,n∈∕#x007D; and {[B n θ ,n∈∕} are convolution-type approximations of the functionf and Brownian motionB.

Key words

approximation of stochastic θ-integrals Brownian motion 

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References

  1. 1.
    V. S. Pugachev and I. N. Sinicyn,Stochastic Differential Systems. Analysis and Filtration [in Russian], Moscow (1990).Google Scholar
  2. 2.
    B. S. Ogawa, On a Riemann definition of the stochastic integral. I,Proc. Japan Acad.,21(46), 153–157 (1970).CrossRefGoogle Scholar
  3. 3.
    B. S. Ogawa, On a Riemann definition of the stochastic integral. II,Proc. Japan Acad.,21(46), 158–161 (1970).MathSciNetGoogle Scholar
  4. 4.
    E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals,Ann. Math. Statist.,36(5), 1560–1564 (1965).MATHMathSciNetGoogle Scholar
  5. 5.
    A. P. Yurachkovskii, A limit theorem for stochastic difference schemes without retardation,Dokl. Akad. Nauk Ukr. SSR,A(7), 24–26 (1986).MATHMathSciNetGoogle Scholar
  6. 6.
    V. Mackevičius, Stability of solutions of stochastic differential equations under uniform perturbations of driving semimartingales, in:Statistics and Control of Stochastic Processes [in Russian], Nauka, Moscow (1989), pp. 143–147.Google Scholar
  7. 7.
    N. V. Lazakovich, Stochastic differentials in the algebra of generalized stochastic processes,Dokl. Akad. Nauk Belarusi,38(5), 23–27 (1994).MATHMathSciNetGoogle Scholar
  8. 8.
    N. V. Lazakovich, S. P. Stashulenok and I. V. Yufereva, Stochastic differential equations in the algebra of generalized random processes,Differ. Equations,31(12), 2056–2058 (1995).MATHMathSciNetGoogle Scholar
  9. 9.
    N. V. Lazakovich and S. P. Stashulenok, An approximation of the Ito and Stratonovich stochastic integrals by elements of a direct product of algebras of generalized random processes,Theory Probab. Appl.,41(4), 695–715 (1996).MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    N. Ikeda and S. Watanabe,Stochastic Differential Equations and Diffusion Processes, Kodansha, Tokyo (1981).MATHGoogle Scholar
  11. 11.
    T. Hida,Brownian Motion, Springer-Verlag, New York (1980).MATHGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • N. V. Lazakovich
  • S. P. Stashulenok
  • O. L. Yablonskii

There are no affiliations available

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