CONTENTS 11

32.3 The Quadratic Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 86432.4 The Covariation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87132.5 The Burkholder Davis Gundy Inequality . . . . . . . . . . . . . . . . . . 87432.6 Approximation With Step Functions . . . . . . . . . . . . . . . . . . . . 880

33 Quadratic Variation and Stochastic Integration 88333.1 The Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89133.2 The Stochastic Integral When f (s) ∈L2 (U,H) . . . . . . . . . . . . . . 89333.3 More on Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . . 89833.4 Local Martingales as Integrators . . . . . . . . . . . . . . . . . . . . . . . 90133.5 The Stochastic Integral and the Quadratic Variation . . . . . . . . . . . . 90333.6 The Case of f ∈ L2 (Ω× [0,T ] ;L2 (U,H)) . . . . . . . . . . . . . . . . . 904Copyright © 2018, You are welcome to use this, including copying it for use in classes

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