Semimartingales on duals of nuclear spaces

Abstract

This work is devoted to the study of semimartingales on the dual of a general nuclear space. We start by establishing conditions for a cylindrical semimartingale in the strong dual Φ ′ of a nuclear space Φ to have a Φ ′ -valued semimartingale version whose paths are right-continuous with left limits. Results of similar nature but for more specific classes of cylindrical semimartingales and examples are also provided. Later, we will show that under some general conditions every semimartingale taking values in the dual of a nuclear space has a canonical representation. The concept of predictable characteristics is introduced and is used to establish necessary and sufficient conditions for a Φ ′ -valued semimartingale to be a Φ ′ -valued Lévy process.

Stochastic integration with respect to cylindrical semimartingales

“…We have (1) ⇒ (2) since the canonical inclusion from Φ into Φ θ is continuous. For (2) ⇔ (3) ⇔ (5) and (2) ⇒ (1) see Propositions 3.14 and 3.15 in [14]. For (3) ⇔ (4) see Proposition IV.3.4 in [52].…”

“…Semimartingales and cylindrical semimartingales in the dual of nuclear space Φ where studied in [14]. In the next paragraphs we recall some of the key properties of such semimartingales that we will need for our study of the stochastic integral.…”

“…Let Z = (Z t : t ≥ 0) be a R d -valued semimartingale and let δ x denotes the Dirac measure at x ∈ R d . As an application of Theorem 4.2 it is shown in Example 3.10 in [14] (see also Example III.1 in [49]) that the cylindrical processes in D (R d ) defined for each t ≥ 0 by…”

“…Then, using the properties of the Itô stochastic integral one can show that X = (X t : t ≥ 0) is a S (R)-valued process which defines a M 2 ∞ -cylindrical martingale satisfying Assumption 3.1 (for details see Example 3.20 in [14]). Therefore X is a good integrator by Proposition 4.12.…”

“…Motivated by the above, in [14] the present author initiated a systematic study of semimartingales in duals of general nuclear spaces by considering them as Φ -valued processes whose induced cylindrical process is a cylindrical semimartingale in Φ . In [14] we obtained several criteria to determine when a given cylindrical semimartingale in Φ defines a Φ -valued càdlàg semimartingale (this is known as the technique of regularization), and we proved in [14] that semimartingales in Φ do enjoy a canonical semimartingale representation as the sum of a predictable process which is weakly of bounded variation, a continuous process which is weakly a continuous local martingale, and a càdlàg process which is weakly a purely discontinuous local martingale. However, such a decomposition cannot be used to define a stochastic integral with respect to semimartingales in the dual of a nuclear space.…”