[Paper Review] Moments of Poisson stochastic integrals with random integrands

Nicolas Privault|arXiv (Cornell University)|Apr 22, 2012

Probabilistic and Robust Engineering Design5 references3 citations

TL;DR

This paper derives a general moment formula for Poisson stochastic integrals with random integrands by extending classical results from deterministic to random settings. It introduces a partition-based formula involving the addition operator $\varepsilon^{+}_{\mathfrak{s}_k}$, which adds points to Poisson configurations, and recovers known identities for deterministic integrands as special cases.

ABSTRACT

We compute the moment of order n of the Poisson stochastic integral of a random process u over a metric space X as a sum that runs over all partitions of {1,...,n} and involves the addition of points to Poisson configurations. This formula recovers known results in case u is a deterministic function on X.

Motivation & Objective

• To generalize the known moment formula for Poisson stochastic integrals with deterministic integrands to the case of random integrands.
• To establish a systematic method for computing the nth-order moments of stochastic integrals driven by Poisson random measures with random integrands.
• To unify and extend existing moment identities through a combinatorial framework based on set partitions and point addition operators.
• To recover classical results such as those involving Bell polynomials and Stirling numbers as special cases of the proposed formula.

Proposed method

• The paper uses a partition-based decomposition of the moment $\mathbb{E}\left[\left(\int_X u_x(\omega)\,\omega(dx)\right)^n\right]$ over all disjoint partitions $P_1 \cup \cdots \cup P_k$ of the index set $\{1,\ldots,n\}$.
• It introduces the operator $\varepsilon^{+}_{\mathfrak{s}_k}$, which adds points $s_1, \ldots, s_k$ to a Poisson configuration $\omega$, and applies it to products of powers of the random integrand $u_x(\omega)$.
• The key formula expresses the nth moment as a sum over all such partitions, with each term involving the expectation of the $\varepsilon^{+}_{\mathfrak{s}_k}$-transformed product of integrand powers.
• The method leverages the combinatorial structure of set partitions and the independence properties of Poisson random measures to derive the moment identity.
• The approach is validated by recovering known results: for deterministic $u_x$, the formula reduces to the classical moment identity involving Bell polynomials.
• The framework is further applied to indicator functions and polynomials, leading to identities involving Stirling numbers and the $\mathcal{U}$-transform on Poisson chaos.

Experimental results

Research questions

• RQ1 How can the moment formula for Poisson stochastic integrals be extended from deterministic to random integrands?
• RQ2 What is the role of the point-addition operator $\varepsilon^{+}_{\mathfrak{s}_k}$ in expressing moments of random integrals?
• RQ3 How do the new moment identities relate to classical results such as those involving Bell polynomials and Stirling numbers?
• RQ4 Can the proposed formula recover known moment identities for Poisson random variables and polynomial functionals?
• RQ5 What is the connection between the moment formula and the $\mathcal{U}$-transform or Chen-Stein identities on the Poisson chaos?

Key findings

• The paper establishes a general moment identity: $\mathbb{E}\left[\left(\int_X u_x(\omega)\,\omega(dx)\right)^n\right] = \sum_{P_1,\ldots,P_k} \mathbb{E}\left[\int_{X^k} \varepsilon^{+}_{\mathfrak{s}_k}(u^{\vert P_1\vert}_{s_1} \cdots u^{\vert P_k\vert}_{s_k}) \sigma(ds_1)\cdots\sigma(ds_k)\right]$, where the sum is over all disjoint partitions of $\{1,\ldots,n\}$.
• When $u_x(\omega)$ is deterministic, the formula reduces to the known identity involving Bell polynomials and multinomial coefficients.
• For $u_x(\omega) = \omega(X)$, the third moment yields $\mathbb{E}[(\omega(X))^6] = \lambda + 7\lambda^2 + 6\lambda^3 + \lambda^4 = B_4(\lambda)$, recovering the Bell polynomial expression.
• The formula recovers the classical moment identity $\mathbb{E}[Z^n] = B_n(\lambda)$ for a Poisson random variable $Z \sim \text{Pois}(\lambda)$, when $u_x(\omega) = \mathbf{1}_A(x)$.
• The identity extends to covariance and exponential moment formulas, such as $\mathbb{E}[F e^{tZ}] = \sum_{k=0}^\infty \frac{\lambda^k}{k!}(e^t - 1)^k \mathbb{E}[f(Z + k)]$, linking to the Chen-Stein method.
• The framework yields a new representation of the $\mathcal{U}$-transform on the Poisson space via the $\varepsilon^{+}$ operator and Stirling numbers of the second kind.