Convergence of Riemann sums for stochastic integrals

  • Peter Sjögren


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.


Stochastic Process Brownian Motion Probability Theory Mathematical Biology Step Function 
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  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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