Let X and Y be an m-dimensional F-semi-martingale and an n-dimensional H-semi-martingale, respectively, on the same probability space (Omega, F, P), both enjoying the predictable representation property. We propose two representation results for the square-integrable G-martingales, where G = F boolean OR H. As a first application, we identify the biggest possible value of the multiplicity in the sense of Davis and Varaiya of Vi-1(d) F-i where, fixed i is an element of (1, ..., d), F-i is the reference filtration of a martingale M-i, which enjoys the (P, F-i)-predictable representation property. This result helps us to identify a basis of martingales for the Poisson filtration enlarged by a general random time. A second application falls into the framework of credit risk modelling and in particular into the study of progressive enlargement of the market filtration by a default time. We present a new proof of the analogous of classical Kusuoka's theorem, when the risky asset price is a multidimensional semi-martingale enjoying the predictable representation property and the default time satisfies the density hypothesis.

Calzolari, A., Torti, B. (2019). Martingale representations in progressive enlargement by the reference filtration of a semi-martingale: a note on the multidimensional case. STOCHASTICS, 91(2), 265-287 [10.1080/17442508.2018.1533015].

Martingale representations in progressive enlargement by the reference filtration of a semi-martingale: a note on the multidimensional case

Calzolari A.;Torti B.
2019-01-01

Abstract

Let X and Y be an m-dimensional F-semi-martingale and an n-dimensional H-semi-martingale, respectively, on the same probability space (Omega, F, P), both enjoying the predictable representation property. We propose two representation results for the square-integrable G-martingales, where G = F boolean OR H. As a first application, we identify the biggest possible value of the multiplicity in the sense of Davis and Varaiya of Vi-1(d) F-i where, fixed i is an element of (1, ..., d), F-i is the reference filtration of a martingale M-i, which enjoys the (P, F-i)-predictable representation property. This result helps us to identify a basis of martingales for the Poisson filtration enlarged by a general random time. A second application falls into the framework of credit risk modelling and in particular into the study of progressive enlargement of the market filtration by a default time. We present a new proof of the analogous of classical Kusuoka's theorem, when the risky asset price is a multidimensional semi-martingale enjoying the predictable representation property and the default time satisfies the density hypothesis.
2019
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/06 - PROBABILITA' E STATISTICA MATEMATICA
English
Semi-martingales; predictable representations property; enlargement of filtration; completeness of a financial market
Calzolari, A., Torti, B. (2019). Martingale representations in progressive enlargement by the reference filtration of a semi-martingale: a note on the multidimensional case. STOCHASTICS, 91(2), 265-287 [10.1080/17442508.2018.1533015].
Calzolari, A; Torti, B
Articolo su rivista
File in questo prodotto:
File Dimensione Formato  
multidimensional case.pdf

solo utenti autorizzati

Tipologia: Versione Editoriale (PDF)
Licenza: Copyright dell'editore
Dimensione 1.84 MB
Formato Adobe PDF
1.84 MB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/215614
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 5
  • ???jsp.display-item.citation.isi??? 4
social impact