Assume that the hypotheses of Proposition 5.9.3.1 hold and that $1/p_{\infty }(\cdot ,L)$ is integrable with expectation $1/c$. Prove that under the probability $R$ defined as

$${dR|}_{F_{\infty }}=c/{p_{\infty }(\cdot ,L)dP|}_{F_{\infty }}$$

the r.v. $L$ is independent of $F_{\infty }$.

Example 5.9.4.8:

Let $B$ be a Brownian motion and define the stopping time $T_1=inf\{t:B_t=1\}$ and the random time $\theta =sup\{t<T_1:B_t=0\}$. Set

$$\tau =sup\{s<\theta :B_s=M_s^B\}$$

where $M_s^B$ is the running maximum of the Brownian motion. Then, $\tau$ is a pseudo-stopping time. Note that $E\left(B_{\tau }\right)$ is not equal to 0 ; this illustrates the fact we cannot take any martingale in Definition 5.9.4.6. The martingale $(B_{t\wedge T_1},t\ge 0)$ is neither bounded, nor uniformly integrable. In fact, since the maximum $M_{\theta }^B(=B_{\tau })$ is uniformly distributed on $[0,1]$, one has $E\left(B_{\tau }\right)=1/2$.

Transcribed Image Text:

Example 5.9.4.11 Let W be a Brownian motion, and r = 91, the last time when the BM reaches 0 before time 1, i.e., = sup{t <1 W = 0}. Using the computation of Z91 in Subsection 5.6.4 and Proposition 5.9.4.10, we obtain the decomposition of the Brownian motion in the enlarged filtration $' 8 sgn(W) ds W = - [*101) (+) SATA) wr So 110,7] (8) +1{rt} sgn(W) 1 $ t [ $' |W| -8 ds where (x)= exp(-u/2)du.