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Convergence of Riemann sums for stochastic integrals

  • Peter Sjögren
Article
  • 52 Downloads

Summary

Let b be a Brownian motion and f a function in L2[0,1]. If Δ is a partition of [0,1], denote by f Δ the step function obtained by replacing f by its mean values in each subinterval. As Δ becomes fine, the martingale ∫f Δ db converges to ∫fdb in L2 but not necessarily almost surely. We determine precisely which Lipschitz conditions on f imply a.s. convergence. A similar thing is done for non-anticipating random functions.

Keywords

Stochastic Process Brownian Motion Probability Theory Mathematical Biology Step Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Hayes, C.A., Pauc, C.Y.: Derivation of martingales. Berlin-Heidelberg-New York: Springer 1970Google Scholar
  2. 2.
    McKean, H.P.: Stochastic integrals. New York-London: Academic Press 1969Google Scholar
  3. 3.
    Wong, E., Zakai, M.: On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Statist. 36, 1560–1564 (1965)Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Peter Sjögren
    • 1
  1. 1.Department of MathematicsUniversity of UmeåUmeåSweden

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