- #1
GSpeight
- 31
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Hi there,
I'm starting revision for Stochastic Analysis and have a few questions relating to the notes I'm reading. I'd much appreciate any clarification as I'm not as up to speed as I'd like.
1) In the definition of classical Wiener space I have [itex]H=L_{0}^{2,1}([0,T]; \mathbb{R}^{n})[/itex] the space of continuous paths starting at 0 with first derivative in [itex]L^{2}[/itex] but I'm a little confused as to what this means. Are we assuming the paths are everywhere differentiable in the classical sense with derivative in [itex]L^{2}[/itex] or only that for all [itex]\sigma \in L_{0}^{2,1}([0,T]; \mathbb{R}^{n})[/itex] there exists [itex]\phi \in L^{2}([0,T];\mathbb{R}^{n})[/itex] such that [itex]\sigma (t)=\int_{0}^{t}\phi(s)ds[/itex]? I imagine in the second case one has [itex]\sigma '(t)=\phi(t)[/itex] for almost every [itex]t[/itex]. In case it's important [itex]E=C_{0}([0,T];\mathbb{R}^{n})[/itex] and [itex]i[/itex] is the inclusion from [itex]H[/itex] to [itex]E[/itex].
2) Shortly after the definition of AWS I have that the inclusion from [itex]L^{2}[/itex] to [itex]L^{1}[/itex] is an AWS. It's clear that the inclusion is continuous linear and injective with dense range but I can't see easily that it radonifies the canonical Gaussian CSM. Is this easy to prove?
3) Throughout the notes I have (Gaussian measures, CSM's, Paley-Wiener map, Ito's integral etc.) it's assumed all Banach or Hilbert spaces are separable but I can't see where we actually use that. Where is it important?
Thanks for any help.
I'm starting revision for Stochastic Analysis and have a few questions relating to the notes I'm reading. I'd much appreciate any clarification as I'm not as up to speed as I'd like.
1) In the definition of classical Wiener space I have [itex]H=L_{0}^{2,1}([0,T]; \mathbb{R}^{n})[/itex] the space of continuous paths starting at 0 with first derivative in [itex]L^{2}[/itex] but I'm a little confused as to what this means. Are we assuming the paths are everywhere differentiable in the classical sense with derivative in [itex]L^{2}[/itex] or only that for all [itex]\sigma \in L_{0}^{2,1}([0,T]; \mathbb{R}^{n})[/itex] there exists [itex]\phi \in L^{2}([0,T];\mathbb{R}^{n})[/itex] such that [itex]\sigma (t)=\int_{0}^{t}\phi(s)ds[/itex]? I imagine in the second case one has [itex]\sigma '(t)=\phi(t)[/itex] for almost every [itex]t[/itex]. In case it's important [itex]E=C_{0}([0,T];\mathbb{R}^{n})[/itex] and [itex]i[/itex] is the inclusion from [itex]H[/itex] to [itex]E[/itex].
2) Shortly after the definition of AWS I have that the inclusion from [itex]L^{2}[/itex] to [itex]L^{1}[/itex] is an AWS. It's clear that the inclusion is continuous linear and injective with dense range but I can't see easily that it radonifies the canonical Gaussian CSM. Is this easy to prove?
3) Throughout the notes I have (Gaussian measures, CSM's, Paley-Wiener map, Ito's integral etc.) it's assumed all Banach or Hilbert spaces are separable but I can't see where we actually use that. Where is it important?
Thanks for any help.