4.3 Path Decomposition of L´evy Processes
4.3.1 Poisson Random Measures and Point Processes
Definition 4. LetS be a Borel subset ofRd, letS be the set of Borel subsets of S, and letmbe afinite measure onS. A collectionfM.B/ W B 2 SgofZNC- valued random variables defined on a probability space.;F;P/is called a Poisson random measure (PRM) (or process) onSwith mean measuremif:
(1) For everyB2S,M.B/is a Poisson random variable with meanm.B/. (2) IfB1; : : : ; Bn2Sare disjoint, thenM.B1/; : : : ; M.B1/are independent.
(3) For every sample outcome!2,M.I!/is a measure onS.
Above, we used some basic terminology of real analysis. For all practical purposes, Borel sets ofRd are those subsets that can be constructed from basic operations (complements, countable unions, and intersections) of elementary sets of the form .a1; b1: : : .ad; bd. A measuremis a mapping fromStoŒ0;1such that
m.;/D0; and m [1 iD1
Bi
! DX1
iD1
m.Bi/;
for any mutually disjoints Borel setsfBigi1. A measure is said to be -finite if there exists mutually disjointsfBigi1 such thatRd D S1
iD1Bi andm.Bi/ <1, for anyi.
It can be proved that (a.s.), a Poisson random measureM.I!/ is an atomic measure; that is, there exist countably many (random) pointsfxigi S (called atoms) such that
M.B/D#fiWxi 2Bg D X1 iD1
ıxi.B/: (4.21)
Similarly, if a sequence of finitely many or countably many random pointsfxigi
is such that the measure (4.21) satisfies (1)–(3) above, then we say thatfxigi is a Poisson point process onS with mean measurem. The following is a common procedure to construct a realization of a Poisson random measure or point process:
1. Suppose thatB1; B2; : : :is a partition ofSsuch thatm.Bj/ <1 2. Generatenj Poiss.m.Bj//
3. Independently, generate nj-points, say fxjigniD1j , according to the distribution m./=m.Bj/
4. DefineM.B/D#f.i; j /Wxji 2Bg
4.3.1.1 Transformation of Poisson Random Measures
Among the most useful properties of PRM is that certain transformations of a Poisson point process are still a Poisson point process. The following is the simplest version:
Proposition 1. Suppose thatT W S ! S0 Rd0 is a one-to-one measurable function. Then, the random measure associated with the transformed points x0i WD T .xi/, namelyM0./DP1
iD1ıx0i./;is also a Poisson random measure with mean measurem0.B/WDm.fxWT .x/2Bg/:
The following result shows that a marked Poisson point process is still a Poisson point process. Suppose that we associate aRd0-valued score x0i to each point xi ofM. The scores are assigned independently from one another. The distribution of the scores can actually depend on the point xi. Concretely, let.x; dx0/be a probability measure onS0Rd0, for each x2S(hence,.x; S0/D1). For eachi, generate a r.v. x0i according .xi; dx0/ (independently from any other variable).
Consider the so-called marked Poisson process M0./D
X1 iD1
ı.xi;x0i/./:
Proposition 2. M0is a Poisson random measure onSS0with mean measure m0.dx;dx0/D.x;dx0/m.dx/:
As an example, consider the following experiment. We classify the points of the Poisson process M into k different types. The probability that the point xi is of typej ispj.xi/(necessarilypj./ 2 Œ0; 1), independently from any other classification. Letfyjigbe the points offxigof typej and letMj be the counting measure associated withfyjig:
Mj WDX ıfyj
ig
We say that the processM1is constructed fromMby thinning.
Proposition 3. M1; : : : ; Mk are independent Poisson random measures with respective mean measuresm1.dx/WDp1.x/m.dx/; : : : ; mk.dx/WDpk.x/m.dx/. Example 1. Suppose that we want to simulate a Poisson point process on the unit circleSWD f.x; y/Wx2Cy21gwith mean measure:
m0.B/D
“
B\S
px2Cy2dxdy:
A method to do this is based on the previous thinning method. Suppose that we generate a “homogeneous” Poisson point process M on the square R WD f.x; y/Wjxj 1;jyj 1gwith an intensity of D 8 points per unit area. That is, the mean measure ofM is
m.B/D 1 4
“
B
dxdy:
Letf.xi; yi/gi denote the atoms of the Poisson random measureM. Now, consider the following thinning process. We classify the point .xi; yi/ of type 1 with probabilityp.xi; yi/WD12q
xi2Cyi2 and of type 2 with probability1p.xi; yi/. Suppose thatf.xi1; yi1/gi are the point of type 1. Then, this process is a Poisson point process with mean measurem0.
4.3.1.2 Integration with Respect to a Poisson Random Measure
LetM be a Poisson random measure as Definition4. SinceM.I!/is an atomic random measure for each !, say M.I!/DP1
iD1ıxi.!/./, one can define the integral
M.f /WD Z
S
f .x/M.dx/D X1 iD1
f .xi/;
for any measurable nonnegative deterministic functionf. This is aRNCDR[f1g- valued r.v. such that
Eh e
Rf .x/M.dx/i Dexp
Z
1ef .x/ m.dx/
; E
Z
f .x/M.dx/
D Z
f .x/m.dx/I seeKallenberg 1997, Lemma 10.2. Also, ifB2Sis such thatm.B/ <1, then
Z
Bf .x/M.dx/WD X
iWxi2B
f .xi/;
is a well-defined Rdvalued r.v. for any measurable functionf W S ! Rd. Its characteristic function is given by
Eh
eihRBf .x/M.dx/;uii Dexp
Z
B
eihf .x/;ui1
m.dx/
: Furthermore, ifB1; : : : ; Bmare disjoints sets inSwith finite measure, then
Z
B1
f .x/M.dx/; : : : ; Z
Bm
f .x/M.dx/:
are independent (see (Sato,1999, Proposition 19.5)).
In the general case, determining conditions for the integralR
Sf .x/M.dx/to be well-defined requires some care. Let us assume thatmis a radon measure (that is, m.K/ <1, for any compactKS). Then,R
Sf .x/M.dx/DP1
iD1f .xi/is well- defined for any bounded functionf WS !Rof compact support. We say that the integralR
Sf .x/M.dx/exists if Z
Sfn.x/M.dx/!P X; as n! 1;
for a random variableX and any sequencefnof bounded functions with compact support such thatjfnj jfjandfn!f. In that case, the so-called Poisson integral R
Sf .x/M.dx/is defined to be that common limitX. We define in a similar way the so-called compensated Poisson integral off, denoted byR
Sf .x/.Mm/.dx/. The following theorem gives conditions for the existence of the Poisson integrals (see (Kallenberg,1997, Theorem 10.15)):
Proposition 4. LetM be a Poisson random measure as in Definition4. Then, (a) M.f /DR
Sf .x/M.dx/exists iffR
S.jf .x/j ^1/m.dx/ <1. (b) .Mm/.f /WDR
Sf .x/.Mm/.dx/exists iffR
S.jf .x/j2^jf .x/j/m.dx/ <1.