Stochastic Analysis for Poisson Processes

Abstract

This chapter develops some basic theory for the stochastic analysis of Poisson process on a general σ-finite measure space. After giving some fundamental definitions and properties (as the multivariate Mecke equation) the chapter presents the Fock space representation of square-integrable functions of a Poisson process in terms of iterated difference operators. This is followed by the introduction of multivariate stochastic Wiener–Itô integrals and the discussion of their basic properties. The chapter then proceeds with proving the chaos expansion of square-integrable Poisson functionals, and defining and discussing Malliavin operators. Further topics are products of Wiener–Itô integrals and Mehler’s formula for the inverse of the Ornstein–Uhlenbeck generator based on a dynamic thinning procedure. The chapter concludes with covariance identities, the Poincaré inequality, and the FKG-inequality.

Appendix

In this appendix we prove Proposition 1. If χ ∈ N is given by

$$ \displaystyle\begin{array}{rcl} \chi =\sum _{ j=1}^{k}\delta _{ x_{j}}& &{}\end{array}$$

(90)

for some $k \in \mathbb{N}_{0} \cup \{\infty \}$ and some points $x_{1},x_{2},\ldots \in \mathbb{X}$ (which are not assumed to be distinct) we define, for $m \in \mathbb{N}$, the factorial measure $\chi ^{(m)} \in \mathbf{N}(\mathbb{X}^{m})$ by

$$\displaystyle\begin{array}{rcl} \chi ^{(m)}(C) ={{}{\sum } ^{\neq }}_{ i_{1},\ldots,i_{m}\leq k}\mathbb{1}\{(x_{i_{1}},\ldots,x_{i_{m}}) \in C\},\quad C \in \mathcal{X}^{m}.& &{}\end{array}$$

(91)

These measures satisfy the recursion

$$\displaystyle\begin{array}{rcl} \chi ^{(m+1)}& =& \int \bigg[\int\mathbb{1}\{(x_{ 1},\ldots,x_{m+1}) \in \cdot \}\,\chi (\mathrm{d}x_{m+1}) \\ & & -\sum _{j=1}^{m}\mathbb{1}\{(x_{ 1},\ldots,x_{m},x_{j}) \in \cdot \}\bigg]\,\chi ^{(m)}(\mathrm{d}(x_{ 1},\ldots,x_{m})).{}\end{array}$$

(92)

Let N _< ∞ denote the set of all χ ∈ N with $\chi (\mathbb{X}) <\infty$. For χ ∈ N _< ∞ the recursion (92) is solved by

$$\displaystyle\begin{array}{rcl} \chi ^{(m)} =\idotsint\mathbb{1}\{(x_{ 1},\ldots,x_{m}) \in \cdot \}\,\bigg(\chi -\sum _{j=1}^{m-1}\delta _{ x_{j}}\bigg)(\mathrm{d}x_{m})\cdots \chi (\mathrm{d}x_{1}),& &{}\end{array}$$

(93)

where the integrations are with respect to finite signed measures. Note that χ ^(m) is a signed measure such that $\chi ^{(m)}(C) \in \mathbb{Z}$ for all $C \in \mathcal{X}^{m}$. At this stage it might not be obvious that χ ^(m)(C) ≥ 0. If, however, χ is given by (90) with $k \in \mathbb{N}$, then (93) coincides with (91). Hence χ ^(m) is a measure in this case. For any χ ∈ N _< ∞ we denote by χ ^(m) the signed measure (93). This is in accordance with the recursion (92). The next lemma shows that χ ^(m) is a measure.

Lemma 7

Let χ ∈ N _<∞ and $m \in \mathbb{N}$ . Then χ ^(m) (C) ≥ 0 for all $C \in \mathcal{X}^{m}$ .

Proof

Let $B_{1},\ldots,B_{m} \in \mathcal{X}$ and let Π _m denote the set of partitions of [m]. The definition (93) implies that

$$\displaystyle\begin{array}{rcl} \chi ^{(m)}(B_{ 1} \times \cdots \times B_{m}) =\sum _{\pi \in \varPi _{m}}c_{\pi }\prod _{J\in \pi }\chi (\cap _{i\in J}B_{i}),& &{}\end{array}$$

(94)

where the coefficients $c_{\pi } \in \mathbb{R}$ do not depend on B ₁, …, B _m and χ. For instance

$$\displaystyle\begin{array}{rcl} & & \chi ^{(3)}(B_{ 1} \times B_{2} \times B_{3}) =\chi (B_{1})\chi (B_{2})\chi (B_{3}) -\chi (B_{1})\chi (B_{2} \cap B_{3}) {}\\ & & \qquad -\chi (B_{2})\chi (B_{1} \cap B_{3}) -\chi (B_{3})\chi (B_{1} \cap B_{2}) + 2\chi (B_{1} \cap B_{2} \cap B_{3}). {}\\ \end{array}$$

It follows that the left-hand side of (94) is determined by the values of χ on the algebra generated by B ₁, …, B _m . The atoms of this algebra are all nonempty sets of the form

$$\displaystyle{B = B_{1}^{i_{1} } \cap \cdots \cap B_{m}^{i_{m} },}$$

where i ₁, …, i _m  ∈ { 0, 1} and, for $B \subset \mathbb{X}$, B ¹: = B and $B^{0}:= \mathbb{X}\setminus B$. Let $\mathcal{A}$ denote the set of all these atoms. For $B \in \mathcal{A}$ we take x ∈ B and let χ _B : = χ(B)δ _x . Then the measure

$$\displaystyle{\chi ':=\sum _{B\in \mathcal{A}}\chi _{B}}$$

is a finite sum of Dirac measures and (94) implies that

$$\displaystyle{(\chi ')^{(m)}(B_{ 1} \times \cdots \times B_{m}) =\chi ^{(m)}(B_{ 1} \times \cdots \times B_{m}).}$$

Therefore it follows from (91) (applied to χ′) that χ ^(m)(B ₁ ×⋯ × B _m ) ≥ 0.

Let $\mathcal{A}_{m}$ be the system of all finite and disjoint unions of sets B ₁ ×⋯ × B _m . This is an algebra; see Proposition 3.2.3 in [2]. From the first step of the proof and additivity of χ ^(m) we obtain that χ ^(m)(A) ≥ 0 holds for all $A \in \mathcal{A}_{m}$. The system $\mathcal{M}$ of all sets $A \in \mathcal{X}^{m}$ with the property χ ^(m)(A) ≥ 0 is monotone. Hence a monotone class theorem (see e.g. Theorem 4.4.2 in [2]) implies that $\mathcal{M} = \mathcal{X}^{m}$. Therefore χ ^(m) is nonnegative. □ 

Lemma 8

Let χ,ν ∈ N _<∞ and assume that χ ≤ν. Let $m \in \mathbb{N}$ . Then χ ^(m) ≤ν ^(m) .

Proof

By a monotone class argument it suffices to show that

$$\displaystyle\begin{array}{rcl} \chi ^{(m)}(B_{ 1} \times \cdots \times B_{m}) \leq \nu ^{(m)}(B_{ 1} \times \cdots \times B_{m})& &{}\end{array}$$

(95)

for all $B_{1},\ldots,B_{m} \in \mathcal{X}$. Fixing the latter sets we define the system $\mathcal{A}$ of atoms of the generated algebra as in the proof of Lemma 7. For $B \in \mathcal{A}$ we choose x ∈ B and define χ _B : = χ(B)δ _x and ν _B : = ν(B)δ _x . Then

$$\displaystyle\begin{array}{rcl} \chi ':=\sum _{B\in \mathcal{A}}\chi _{B},\quad \nu ':=\sum _{B\in \mathcal{A}}\nu _{B}& & {}\\ \end{array}$$

are finite sums of Dirac measures satisfying χ′ ≤ ν′. By (94) we have

$$\displaystyle\begin{array}{rcl} \chi ^{(m)}(B_{ 1} \times \cdots \times B_{m}) = (\chi ')^{(m)}(B_{ 1} \times \cdots \times B_{m}).& & {}\\ \end{array}$$

A similar identity holds for ν ^(m) and (ν′)^(m). Therefore (91) (applied to χ′ and ν′) implies the asserted inequality (95). □ 

We can now prove a slightly more detailed version of Proposition 1.

Proposition 6

For any χ ∈ N _σ there is a unique sequence χ ^(m) , $m \in \mathbb{N}$ , of symmetric σ-finite measures on $(\mathbb{X}^{m},\mathcal{X}^{m})$ satisfying χ ⁽¹⁾ := χ and the recursion (92) . Moreover, the mapping χ ↦ χ ^(m) is measurable. Finally, χ ^(m) (B ^m ) ≤χ(B) ^m for all $m \in \mathbb{N}$ and $B \in \mathcal{X}$ .

Proof

For χ ∈ N _< ∞ the functionals defined by (93) satisfy the recursion (92) and are measures by Lemma 7.

For a general χ ∈ N _σ we proceed by induction. For m = 1 we have χ ⁽¹⁾ = χ and there is nothing to prove. Assume now that m ≥ 1 and that the measures χ ⁽¹⁾, …, χ ^(m) satisfy the first m − 1 recursions and have the properties stated in the proposition. Then (92) enforces the definition

$$\displaystyle\begin{array}{rcl} \chi ^{(m+1)}(C):=\int K(x_{ 1},\ldots,x_{m},\chi,C)\,\chi ^{(m)}(\mathrm{d}(x_{ 1},\ldots,x_{m}))& &{}\end{array}$$

(96)

for $C \in \mathcal{X}^{m+1}$, where

$$\displaystyle\begin{array}{rcl} & & K(x_{1},\ldots,x_{m},\chi,C) {}\\ & & \quad \:=\int\mathbb{1}\{(x_{1},\ldots,x_{m+1}) \in C\}\,\chi (\mathrm{d}x_{m+1}) -\sum _{j=1}^{m}\mathbb{1}\{(x_{ 1},\ldots,x_{m},x_{j}) \in C\}.{}\\ \end{array}$$

The function $K: \mathbb{X}^{m} \times \mathbf{N}_{\sigma } \times \mathcal{X}^{m} \rightarrow (-\infty,\infty ]$ is a signed kernel in the following sense. The mapping (x ₁, …, x _m , χ) ↦ K(x ₁, …, x _m , χ, C) is measurable for all $C \in \mathcal{X}^{m+1}$, while K(x ₁, …, x _m , χ, ⋅ ) is σ-additive for all $(x_{1},\ldots,x_{m},\chi ) \in \mathbb{X}^{m} \times \mathbf{N}_{\sigma }$. Hence it follows from (96) and the measurability properties of χ ^(m) (which are part of the induction hypothesis) that χ ^(m+1)(C) is a measurable function of χ.

Next we show that

$$\displaystyle\begin{array}{rcl} K(x_{1},\ldots,x_{m},\chi,C) \geq 0\quad \chi ^{(m)}\mbox{ -a.e. $(x_{ 1},\ldots,x_{m}) \in \mathbb{X}^{m}$}& &{}\end{array}$$

(97)

holds for all χ ∈ N _σ and all $C \in \mathcal{X}^{m+1}$. Since χ ^(m) is a measure (by induction hypothesis) (96), (97) and monotone convergence then imply that χ ^(m+1) is a measure. Fix χ ∈ N _σ and choose a sequence (χ _n ) of finite measures in N _σ such that χ _n ↑ χ. Lemma 7 (applied to χ _n and m + 1) implies that

$$\displaystyle\begin{array}{rcl} K(x_{1},\ldots,x_{m},\chi _{n},C) \geq 0\quad (\chi _{n})^{(m)}\mbox{ -a.e. $(x_{ 1},\ldots,x_{m}) \in \mathbb{X}^{m}$},\,n \in \mathbb{N}.& & {}\\ \end{array}$$

Indeed, we have for all $B \in \mathcal{X}^{m}$ that

$$\displaystyle\begin{array}{rcl} \int \limits _{B}K(x_{1},\ldots,x_{m},\chi _{n},C)\,(\chi _{n})^{(m)}(\mathrm{d}(x_{ 1},\ldots,x_{m})) = (\chi _{n})^{(m+1)}((B \times \mathbb{X}) \cap C) \geq 0.& & {}\\ \end{array}$$

Since K(x ₁, …, x _m , ⋅ , C) is increasing, this implies

$$\displaystyle\begin{array}{rcl} K(x_{1},\ldots,x_{m},\chi,C) \geq 0\quad (\chi _{n})^{(m)}\mbox{ -a.e. $(x_{ 1},\ldots,x_{m}) \in \mathbb{X}^{m}$},\,n \in \mathbb{N}.& & {}\\ \end{array}$$

By induction hypothesis we have that (χ _n )^(m) ↑ χ ^(m) so that (97) follows.

Finally we note that χ ^(m)(B ^m) ≤ χ(B)^m follows by induction. In particular, χ ^(m) is σ-finite. To prove the symmetry of χ ^(m) it is then sufficient to show that the restriction of χ ^(m) to B ^m is symmetric, for any $B \in \mathcal{X}$ with χ(B) < ∞. This fact follows from (94). □ 

For any χ ∈ N, $B \in \mathcal{X}$ with χ(B) < ∞, and $m \in \mathbb{N}$ it follows by induction that

$$\displaystyle\begin{array}{rcl} \chi ^{(m)}(B^{m}) =\chi (B)(\chi (B) - 1)\cdots (\chi (B) - m + 1).& & {}\\ \end{array}$$

Since χ and χ ^(m) are σ-finite, this extends to any $B \in \mathcal{X}$. In particular χ ^(m) is the zero measure whenever $\chi (\mathbb{X}) <m$.