Abstract
Motivated by applications to SPDEs we extend the Itô formula for the square of the norm of a semimartingale y(t) from Gyöngy and Krylov (Stochastics 6(3):153–173, 1982) to the case
∑i=1m∫(0,t]v∗i(s)dA(s)+h(t)=:y(t)∈VdA×P-a.e.,
where A is an increasing right-continuous adapted process, v∗i is a progressively measurable process with values in V∗i, the dual of a Banach space Vi, h is a cadlag martingale with values in a Hilbert space H, identified with its dual H∗, and V:=V1∩V2∩⋯∩Vm is continuously and densely embedded in H. The formula is proved under the condition that ∥y∥piVi and ∥v∗i∥qiV∗i are almost surely locally integrable with respect to dA for some conjugate exponents pi,qi. This condition is essentially weaker than the one which would arise in application of the results in Gyöngy and Krylov (Stochastics 6(3):153–173, 1982) to the semimartingale above.
∑i=1m∫(0,t]v∗i(s)dA(s)+h(t)=:y(t)∈VdA×P-a.e.,
where A is an increasing right-continuous adapted process, v∗i is a progressively measurable process with values in V∗i, the dual of a Banach space Vi, h is a cadlag martingale with values in a Hilbert space H, identified with its dual H∗, and V:=V1∩V2∩⋯∩Vm is continuously and densely embedded in H. The formula is proved under the condition that ∥y∥piVi and ∥v∗i∥qiV∗i are almost surely locally integrable with respect to dA for some conjugate exponents pi,qi. This condition is essentially weaker than the one which would arise in application of the results in Gyöngy and Krylov (Stochastics 6(3):153–173, 1982) to the semimartingale above.
| Original language | English |
|---|---|
| Pages (from-to) | 428-455 |
| Number of pages | 28 |
| Journal | Stochastics and Partial Differential Equations: Analysis and Computations |
| Volume | 5 |
| Issue number | 3 |
| Early online date | 17 Mar 2017 |
| DOIs | |
| Publication status | Published - Sep 2017 |
