Ambit Processes and Stochastic Partial Differential Equations

Abstract

Ambit processes are general stochastic processes based on stochastic integrals with respect to Lévy bases. Due to their flexible structure, they have great potential for providing realistic models for various applications such as in turbulence and finance. This papers studies the connection between ambit processes and solutions to stochastic partial differential equations. We investigate this relationship from two angles: from the Walsh theory of martingale measures and from the viewpoint of the Lévy noise analysis.

Appendix: Lévy Bases and Integration

This section reviews the integration theory of [38] (for a survey, see also [40]), since this concept of integration is used for defining stochastic integrals in the context of ambit fields.

2.1.1 A.1 Introduction

Throughout the text, let S denote a nonempty set, and let $\mathcal{A}$ denote a σ-finite δ-ring on S, i.e. $\mathcal{A}$ is a family of subsets of S such that for every pair of sets in $\mathcal{A}$, the union, the intersection, and the set difference is in $\mathcal{A}$ (hence $\mathcal{A}$ is a ring), and if $(A_{n})_{n \geq1} \subseteq\mathcal{A}$, then $\bigcap A_{n} \in \mathcal {A}$; also, there exists a sequence $(A_{n}^{*})_{n \geq1} \subseteq\mathcal{A}$ such that $\bigcup A_{n}^{*} = S$.

Note that we call a real stochastic process $\varLambda =\{\varLambda (A):A\in \mathcal{A}\}$ on some probability space $(\varOmega , \mathcal {F},\mathbb {P})$ an independently scattered random measure if for every sequence of disjoint sets (A _n ) _n≥1, the random variables Λ(A _n ), n=1,2,…, are independent, and if $\bigcup_{n}A_{n} \in \mathcal{S}$, then $\varLambda (\bigcup_{n} A_{n}) = \sum_{n} \varLambda (A_{n})$ almost surely.

2.1.2 A.2 Representation of the Characteristic Function of a Lévy Basis

If Λ(A) is infinitely divisible for every $A \in\mathcal{A}$, we call it a Lévy basis. Its characteristic function for $A\in\mathcal{A}$ is then given by

(2.47)

(2.48)

where $\nu_{0}:\mathcal{S}\to\mathbb{R}$ is a signed measure, $\nu _{1}:\mathcal {A}\to[0,\infty)$ is a measure, and F _A is a Lévy measure on ℝ for every $A \in \mathcal{A}$, while A↦F _A (B)∈[0,∞) is a measure for every $B\in\mathcal{B}(\mathbb{R})$ whenever $0 \notin\overline{B}$. Also, the centering function τ is defined by τ(x)=x if ‖x‖≤1 and by τ(x)=x/‖x‖ if ‖x‖>1.

Further, let

It can be shown that $\lambda:\mathcal{A}\to[0,\infty)$ is a measure on $\mathcal{A}$ such that if, for every $(A_{n})_{n\geq1} \subset \mathcal{A}$, λ(A _n )→0, then Λ(A _n )→0 in probability. Also, if, for every sequence $(A_{n}')_{n\geq1} \subset \mathcal{A}$ with $A_{n}' \subset A_{n} \in\mathcal{A}$, we have $\varLambda (A_{n}')\to0$ in probability, then λ(A _n )→0.

Note that the measure λ satisfies $\lambda(A_{n}^{*})< \infty$ for n=1,2,…. Hence, it can be extended to a σ-finite measure on $(S,\sigma(\mathcal{A}))$. This measure is then called the control measure of Λ.

It turns out that the characteristic function of an infinitely divisible random measure has also an alternative representation than the one given above.

In order to state it, we first need a preliminary result (see [38, Lemma 2.3]). Let F _⋅ be as above. Then there exists a unique σ-finite measure F on $\sigma(\mathcal{A})\times\mathcal{B}(\mathbb{R})$ such that F(A×B)=F _A (B) for all $A \in\mathcal{A}$, $B \in \mathcal{B}(\mathbb{R})$. Furthermore, there exists a function $\rho:S\times\mathcal{B}(\mathbb{R})\to[0,\infty]$ such that

• 1. ρ(s,⋅) is a Lévy measure on $\mathcal{B}(\mathbb {R})$ for every s∈S,

• 2. ρ(⋅,B) is a Borel measurable function for every $B\in \mathcal{B}(\mathbb{R})$,

• 3. ∫ _S×ℝ h(s,x)F(ds,dx)=∫ _S (∫_ℝ h(s,x)ρ(s,dx))λ(ds) for every $\sigma(\mathcal {A})\times \mathcal{B}(\mathbb{R})$-measurable function h:S×ℝ→[0,∞]. Under some restrictions regarding the behaviour at ±∞, this equality can be extended to real and complex-valued functions h.

Using the above notation, we can now rewrite the characteristic function of Λ(A) (see [38, Proposition 2.4]):

(2.49)

where

where $a(s)=\frac{d\nu_{0}}{d\lambda}(s)$, $\sigma^{2}(s)= \frac{d\nu _{1}}{d\lambda}(s)$, and ρ is defined as above. Furthermore,

2.1.3 A.3 Integration with Respect to a Lévy Basis

Next, we review the definition of a stochastic integral with respect to an infinitely divisible random measure Λ as defined in [38].

First, we define integration of a real simple function on S, which is given by $f=\sum_{j=1}^{n}x_{j} \mathrm{1}_{A_{j}}$ for disjoint $A_{j} \in \mathcal{A}$. Then, for every $A\in\sigma(\mathcal{A})$, the stochastic integral with respect to Λ is defined by

The generalisation to general functions works as follows. We call a measurable function $f:(S,\sigma(\mathcal{A})) \to(\mathbb {R},\mathcal {B}(\mathbb{R}))$ Λ-integrable if there exists a sequence of simple functions (f _n ) _n≥1 such that f _n →f λ-a.e. and for every $A \in\sigma(\mathcal{A})$, the sequence (∫ _A f _n  dΛ) _n≥1 converges in probability as n→∞. In that case, we define

The above integral is well defined in the sense that it does not depend on the approximating sequence (f _n ) _n≥1.

2.1.4 A.4 Criteria for Integrability

Now we provide a characterisation of Λ-integrable functions. Necessary and sufficient conditions will depend on the characteristics given in the Lévy form of the characteristic function of Λ.

According to [38, Theorem 2.7], the integrability conditions are as follows.

Let f:S→ℝ be a $\sigma(\mathcal{A})$-measurable function. Then f is integrable w.r.t. Λ if and only if the following three conditions are satisfied:

• 1. ∫ _S |U(f(s),s)|λ(ds)<∞,

• 2. ∫ _S |f(s)|² σ ²(s)λ(ds)<∞, and

• 3. ∫ _S V ₀(f(s),s)λ(ds)<∞, where

  

Further, if f is integrable w.r.t. Λ, then the characteristic function of ∫ _S fdΛ can be expressed as

where

and