Brownian Motion, Martingales, and Stochastic Calculus by Jean-François Le Gall

Author:Jean-François Le Gall
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

and Theorem 5.22 shows that is a continuous local martingale under Q. Furthermore the bracket of this continuous local martingale with any continuous local martingale N under Q is equal to , and it follows from Theorem 5.6 that hence also (H ⋅ M) P  = (H ⋅ M) Q .

With the notation of Theorem 5.22, set . Then maps the set of all P-continuous local martingales onto the set of all Q-continuous local martingales. One easily verifies, using the remarks in (a) above, that . Furthermore, the mapping commutes with the stochastic integral: if H is a locally bounded progressive process, .

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