... s is property For A F s , we have F s -measurable (1. 1) We check the partial averaging IE XZ tjF s dIP = IE 1 IE XZ tjF s f f A Z s A Z s = IE 1A IE XZ tjF s (Lemma 1. 53) ... because Z t; t T , is a martingale under IP 19 2 Lemma 1. 54 (Baye s Rule) If X is F t-measurable and s t T , then f IE X jF s = Z s IE XZ tjF s : Proof: It is clear that Z 1s ... 2T ,1 Z T + b exp , b + T 2 db =p 2T 2T ,1 2 Z1 y exp , y2 dy (Substitute y = T + b) y = T + b = p 2T ,1 = 0: CHAPTER 17 Girsanov stheorem and the risk-neutral measure 19 1 fe...