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Lévy's Characterisation

Senior Quant · Stochastic Calculus

Question

State Lévy's characterisation of Brownian motion. Use it to conclude that W~t=Wt+∫0tθs ds\widetilde{W}_t = W_t + \int_0^t \theta_s\,ds is a Q\mathbb{Q}-Brownian motion after a Girsanov change of measure. Why is the quadratic variation argument pathwise?
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