Justify the first equality in (29.4). Show that if Mt is a martingale with respect to is a martingale with respect to Suppose W is a Brownian motion and {Ft} is its minimal augmented filtration....

Justify the first equality in (29.4).

Show that if Mt is a martingale with respect to

is a martingale with respect to

Suppose W is a Brownian motion and {Ft} is its minimal augmented filtration. Let T be a bounded stopping time with respect to {Ft}. Suppose Y is a FT measurable random variable with EY²
<>_s
with

such that

(1) Show that the solution to (29.9) is a Gaussian process. (2) Show that the solutions (Xt, Zt) to

(29.9)–(29.10) form a Gaussian process.

Chapter 30