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Basic Concepts and Definitions
Martingales and the Wiener Process
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adapted assume assumption Borel bounded variation Chapter coefficients condition continuous L2-martingale continuous local martingale continuous version converges covariance d-dimensional defined definition denote derivatives exists filtering problem finite Gaussian process Girsanov's theorem given Hence Hilbert space implies increasing process inequality innovation process Ito's formula jointly measurable kernel Lemma linear Markov process nonanticipative notation observation process obtain oo a.s. P-null sets probability measure probability space process Xt proof of Theorem proved r-field random variables Remark representation right-continuous right-hand side satisfies Section semimartingale sequence set is empty square-integrable stochastic differential equation stochastic equation stochastic integral stochastic process stopping strong solution subspace supermartingale Suppose t e 0,T t e R+ theory uniformly integrable unique solution Volterra operator weak solution white noise Wiener integral Wiener martingale Wiener measure Wiener process write Wt(co Xt(co