Show that if f (t) is a purely discontinuous function, then is also. Suppose M is a non-negative right-continuous martingale and Show that Suppose P and Q are two equivalent probability...

Show that if f (t) is a purely discontinuous function, then

is also.

Suppose M is a non-negative right-continuous martingale and

Show that

Suppose P and Q are two equivalent probability measures,

is the Radon–Nikodym derivative of Q with respect to P, and

Show that Y_t
is a local martingale with respect to Q if and only if Y_tM_t
is a local martingale with respect to P.