1

The definition for Locally bounded is utilized in the following link. Which states that for the process $\Phi:[0,T]\times\Omega\to H$ ($H$ is a Hilbert space), $\Phi$ is locally bounded provided $$ \sup_{\Omega} \big\Vert \Phi_t(\omega)\big\Vert_{H}<\infty, \ \text{ for all }\ t$$

But I could'nt find any reference/books stating such definition. I'm new in this stuffs, any help is highly appreciated.

How can we prove that $\langle\int_0^t\Phi_s{\rm d}W_s,x\rangle_H=\sum_{n\in\mathbb N}\int_0^t\langle\sqrt{λ_n}\Phi_se_n,x\rangle_H{\rm d}B_s^{(n)}$?

TheBridge
  • 5,721
raijin
  • 155
  • Usually a property "P" is said to hold locally for a process $X_t$ when there exists an increasing sequence of stopping times $\tau_n$ tending a.s. to $\infty$ (or here $T$) such that the stopped processes $X_t^{\tau_n}=X_{t \wedge \tau_n}$ verify the propperty "P" for every $n$. You can look here for example Defintion 1 : https://almostsure.wordpress.com/2009/12/23/localization/ – TheBridge Jul 02 '20 at 12:03
  • The local boundedness given in your reference seems to imply local boundedness by taking the following localizing sequence $\tau_n(\omega))=inf{t, s.t. \big\Vert \Phi_t(\omega)\big\Vert_{H}<n}$ unless mistaken. – TheBridge Jul 02 '20 at 12:07

0 Answers0