Metatimes constitute an extension of time-change to general measurable spaces, defined as mappings between two

In this paper we study the connection between metatimes and linear operators on Hilbert spaces on one hand, and orthogonal random measures and cylindrical random variables on the other hand. Given an orthogonal random measure (or Lévy basis)

In stochastic modelling, time-changing a Brownian motion or Lévy process provides an alternative to amplitude scaling by a volatility, the latter being a stochastic integral with respect to the driving noise. Metatimes extend this flexible modeling device to Lévy bases (see [

In this paper, we first analyse some of the basic properties of metatimes. Next, we establish a link between metatimes and linear operators on some canonically defined Hilbert spaces. Indeed, equipping the image

As a second step, we study orthogonal random measures. These measures are closely related to cylindrical random variables, and we show that we can lift the orthogonal random measures to cylindrical variables where orthogonality is preserved. This coincides with more classical studies by [

As an application of our results, we extend the class of trawl processes by constructing real-valued trawl processes from cylindrical random variables and curves in a Hilbert space. Next, we define cylindrical trawl processes by looking at operator-changes being time dependent. Some basic properties are derived, in particular for the case of semigroups and Hilbert–Schmidt-valued operators.

Let

A mapping

For

For disjoint sets

The next lemma contains some basic properties of metatimes.

This follows from (i) in Definition

This follows from (ii) in Definition

The image

A mapping

For all

For disjoint sets

The next lemma states some properties of

From (i) in Definition

From (i) Definition

Let

We show that we have equivalence between

If

Let

In view of this result one may redefine any metatime

The next lemma gives an equivalent characterization of injective metatimes and

Assume that

To show implication in the other direction, assume that

Let

The following result shows that metatimes and

Define

Next we show that for both metatimes and

For

Let us express

By Proposition

We are now ready to show that

For any

Let us consider a simple, canonical example of a

Consider a measurable function

In many applications one typically chooses

The next example of translation metatimes will be a guiding case in the sequel of this paper.

Let

Let us show that

Trawl processes, first introduced in [

The following proposition shows a natural algebraic property of metatimes, namely that they are closed under concatenation.

Obviously

Let

If

To end this section, we show that given a measure on

As

Since

If

Let

We first define the operator

We say that

Let

Following similar arguments as in the proof of linearity in Lemma

By Prop. 6.7 in [

Since for any elementary

Suppose that

Let

Let

Let us now ask for a (canonical) definition of a metatime

Let

The following concluding lines of arguments are elementary set theory (see top of page 4 in [

Suppose

It is well known that if a linear bounded operator is surjective, it is invertible (see [

We end this section by going back to translation metatimes in Example

Let

The aim of this section is to construct a cylindrical random variable from an orthogonal random measure, and to show that it satisfies an orthogonality preserving property. We denote by

A

We notice that if

An

We restrict our attention to random measures which have finite variance. Orthogonality means that

There are some simple properties of orthogonal random measures. First, metatimes act invariantly on orthogonal random measures (see [

Since

For property (ii) in Definition

We next collect in Lemmas

From this result, we see that for a sequence

An orthogonal random measure

We have the following important lemma for countably additive orthogonal random measures, showing that

An orthogonal random measure

For the remainder of this section, we assume that

Consider the space

An elementary function in

From [

Under the mild condition that the orthogonal random measure

Since the cylindrical random variable constructed from an orthogonal random measure as above is isometric, it preserves the inner product and therefore also the orthogonality. In the case of a Gaussian cylindrical random variable,

We say that a cylindrical random variable

Let

As another example, consider a measure space

To this end, we have shown how to lift a metatime

Let

On the other hand, we can lift the metatime

In this section we look at some applications of changing the argument in a cylindrical random variable to define stochastic processes. To this end, we consider

Let

As the cylindrical random variable

As the next example shows, our Definition

Let us consider an example of a classical trawl process. Let

Hence, Definition

An example of a trawl process beyond the classical one could be the following. Let

Suppose now that

It is clear that any set in the generator of

We can create a trawl process with independent increments using the filtration

Let

As

Note that if

From the above, we see that we need

We next consider an extension of trawl processes. First recall the definition of a cylindrical process.

A

Let

We see that the cylindrical trawl process is a cylindrical process, since for a fixed

Consider now a separable Hilbert space

Fix

Let

If

A natural class of time-parametric operators is a

Let

Let us return to the examples with translation metatimes, Examples

We are grateful to Almut Veraart for discussions. Three anonymous referees and the editor Professor Yuliya Mishura are thanked for their careful reading and critics of an earlier version of the paper, leading to significant improvements of its presentation.