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Brownian Motion, 2nd Edition by Björn Böttcher, Lothar Partzsch, René L. Schilling

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5 Brownian motion as a martingale

Let I be some subset of [0, ∞] and e9783110307290_i0595.jpg a filtration, i. e. an increasing family of sub-σ-algebras of e9783110307290_i0596.jpg. Recall that a martingale e9783110307290_i0597.jpg is a real or complex stochastic process e9783110307290_i0598.jpg or e9783110307290_i0599.jpg satisfying

  • a) for all tI;
  • b) Xt is ℱt

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