2. Definitions, Preliminaries and Generalities.
2.1 The General Setting.
If y D(S) let yt(s) = y(sAt).
Definition 2.1.0 Let (S,^) be a metrizable Lusin space (i.e. S
is homeomorphic to a Borel subset of a compact metric space) with
its Borel cr-field. Let Ee%( [0,oo) ) xff and set S = {x:(t,x)=E} .
We say Z =
, Z. ,P ) is an inhomogeneous Borel
strong Markov process (IBSMP) with cadlag paths in S c S iff:
(i) (Q,^ ) is a measurable space and {^ [s,t]:s^t} is a
non-decreasing collection of sub-a-fields of indexed by compact
intervals. ^°[s,t+] = p J°[s,t+l/n], ^°[s,») =vy°[s,n].
n=l n
* . o
(n) V (s,z)€E, P is a probability on (Q,^ [s,oo)) such that
VA€^°[u,oo) , (s,z) —-» P (A) is Borel measurable on En([0,u]xS).
(iii) V t^O, Z.:(Q,£°[t,t]) —* (S,?f) is measurable and satisfies
P (Z(s)=z, Z+-€S Vt^s and Z is cadlag on [s,oo)) = l
(iv) If (s,z)€E, ^€bS( [s,oo)xD(S)) and T^s is a stopping time
with respect to {^°[s,t+]:t^s}, then
Ps^z(0(T/Z(T+.)) |y°[sfT+])(w) = PT(CJ)#Z(T)(CJ) (*T(u),Z(T(u)+.)),
P -a.s. on {Too}.
s, z
We say that Z is an inhomogeneous Hunt process if, in addition,
(v) V(s,z)«=E, if T ^s are {^°[s,t+]:t^s}-stopping times which
increase to T and P (Tco) = l, then Z(T ) Z(T) P^ -a.s.
s, z
' s, z
It is easy to check that (ii) and (iii) imply that
Previous Page Next Page