Answer all questions. Kindly explain the answers fully.

Let {N, : 0 > 0 defined on the probability space (0, F, P) with respect to the filtration F,, 0 X, OSIST i=1 we let > > 0 and consider the Radon-Nikodym derivative process N. nQ(X;) AP (X;) if X; is discrete, P(X;) > 0 Z = N nfo (X;) AfP ( X, ) if X, is continuous where Q(X;) (f(X;)) is the probability mass (density) function of X;, i = 1,2, ... under the Q measure. For the case of continuous random variables X;, i = 1, 2, ... we also let Q be absolutely continuous with respect to P on F. Show that E" (Z, ) = 1 and Z, is a positive P martingale for O SI ST.In a Markov jump process model of sickness and death there are three states: healthy, sick and dead, denoted by { H.S.D} . The transition rates are given by the matrix: -0 (1)-1(1) (1) H (1) A (1) = p(1) - p (1) - v (1 ) v (! ) 0 0 0 State whether each of the following is a correct equation for this process. PHS (5,1) = PHH (S,") PHS (u,1) + PHS (5,u) Pss (u,1)