# Download PDF by K.L. Chung, R.J. Williams: Introduction to Stochastic Integration

1R+ is called a predictable time (or simply predictable) if there is a sequence of optional times {7n } which increases to 7 such that each 7n is strictly less than 7 on {7 'i= O}. Such a sequence {7n } is called an announcing sequence for 7.

10. 7) that {MtMk' J t , t E 1R+} is an V-martingale. 1. 3 that {Mt, J t , t E 1R+} is an V-martingale. 7(i) that {IMlfMk' k E IN} is uniformly integrable for each fixed t. I If M is a continuous local martingale, then there is a natural choice of localizing sequence for M which shows that M is a local V -martingale for any p E [1,00). This sequence is exhibited below. 9. Suppose M is a continuous local martingale and let Tk = inf{t > 0: IMt - Mol> k} for each k E IN. Then, for each p E [1,00), M is a local V-martingale and {rd is a localizing sequence for it.

Ii) All stochastic intervals with both end-points predictable are predictable. (iii) The predictable a-field is generated by the class of stochastic intervals of the form [7,00) where 7 is a predictable time. (iv) The optional a-field is generated by the class of stochastic intervals of the form [7,00) where 7 is an optional time. Proof. To prove (i), suppose 7 is a predictable time and {7n } is an announcing sequence for 7. Since 7 n i 7 and 7 n < 7 on {7 'i= O}, we 32 2. 1. Hence [r, 00) is predictable, proving (i).