Convergence of Riemann sums for stochastic integrals
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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.
KeywordsStochastic Process Brownian Motion Probability Theory Mathematical Biology Step Function
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