Stochastic Fubini (Da Prato & Zabcyzk)
I've been studying Da Prato & Zabcyzk's text on SPDE, and there's a part of their proof of the stochastic Fubini theorem that I don't quite get. The setting is that $Phi: (Omega_Ttimes E, mathcal{P}_T times mathcal{B}(E))to (L_2^0, mathcal{B}(L_2^0))$, such that
$$
int_E |Phi(cdot, cdot,x)|_Tmu(dx)< infty
$$
where $mu$ is a finite positive measure and the $T$ norm corresponds to
$$
|Phi|_T^2 = mathbb{E}int_0^T|Phi(s)|_{L_2^0}^2 ds
$$
The authors have an intermediate result, Prop 4.34 in the 2014 edition, that $Phi$ can be approximated by a sequence $Phi_n$ of analogously measurable processes that can be written as simple processes and converge in the sense of
$$
int_E |Phi(cdot, cdot,x)-Phi_n(cdot, cdot,x)|_Tmu(dx) to 0.
$$
I find myself unsure of the proof of this intermediate result in that the authors first assert the existence of a sequence $Psi_n$ (not neccessarily simple) which are bounded and converge in the above sense. It is clear from the above assumption that $mu$-a.e., there are processes $Psi_n(cdot,cdot,x)$ which converge in the $T$ norm. This follows from a Prop 4.22 of the text, and it is clear that they can be taken such that
$$
|Psi_n(cdot,cdot,x)|_T leq 2|Phi(cdot,cdot,x)|_T.
$$
How can I infer that this sequence ${Psi_n}$ is then bounded as a mapping over all three arguments $(x,t,omega)$?
functional-analysis measure-theory stochastic-processes stochastic-pde
asked Dec 2 at 2:39
user2379888
23719
add a comment |
I've been studying Da Prato & Zabcyzk's text on SPDE, and there's a part of their proof of the stochastic Fubini theorem that I don't quite get. The setting is that $Phi: (Omega_Ttimes E, mathcal{P}_T times mathcal{B}(E))to (L_2^0, mathcal{B}(L_2^0))$, such that
$$
int_E |Phi(cdot, cdot,x)|_Tmu(dx)< infty
$$
where $mu$ is a finite positive measure and the $T$ norm corresponds to
$$
|Phi|_T^2 = mathbb{E}int_0^T|Phi(s)|_{L_2^0}^2 ds
$$
The authors have an intermediate result, Prop 4.34 in the 2014 edition, that $Phi$ can be approximated by a sequence $Phi_n$ of analogously measurable processes that can be written as simple processes and converge in the sense of
$$
int_E |Phi(cdot, cdot,x)-Phi_n(cdot, cdot,x)|_Tmu(dx) to 0.
$$
I find myself unsure of the proof of this intermediate result in that the authors first assert the existence of a sequence $Psi_n$ (not neccessarily simple) which are bounded and converge in the above sense. It is clear from the above assumption that $mu$-a.e., there are processes $Psi_n(cdot,cdot,x)$ which converge in the $T$ norm. This follows from a Prop 4.22 of the text, and it is clear that they can be taken such that
$$
|Psi_n(cdot,cdot,x)|_T leq 2|Phi(cdot,cdot,x)|_T.
$$
How can I infer that this sequence ${Psi_n}$ is then bounded as a mapping over all three arguments $(x,t,omega)$?
functional-analysis measure-theory stochastic-processes stochastic-pde
asked Dec 2 at 2:39
user2379888
23719
add a comment |
I've been studying Da Prato & Zabcyzk's text on SPDE, and there's a part of their proof of the stochastic Fubini theorem that I don't quite get. The setting is that $Phi: (Omega_Ttimes E, mathcal{P}_T times mathcal{B}(E))to (L_2^0, mathcal{B}(L_2^0))$, such that
$$
int_E |Phi(cdot, cdot,x)|_Tmu(dx)< infty
$$
where $mu$ is a finite positive measure and the $T$ norm corresponds to
$$
|Phi|_T^2 = mathbb{E}int_0^T|Phi(s)|_{L_2^0}^2 ds
$$
The authors have an intermediate result, Prop 4.34 in the 2014 edition, that $Phi$ can be approximated by a sequence $Phi_n$ of analogously measurable processes that can be written as simple processes and converge in the sense of
$$
int_E |Phi(cdot, cdot,x)-Phi_n(cdot, cdot,x)|_Tmu(dx) to 0.
$$
I find myself unsure of the proof of this intermediate result in that the authors first assert the existence of a sequence $Psi_n$ (not neccessarily simple) which are bounded and converge in the above sense. It is clear from the above assumption that $mu$-a.e., there are processes $Psi_n(cdot,cdot,x)$ which converge in the $T$ norm. This follows from a Prop 4.22 of the text, and it is clear that they can be taken such that
$$
|Psi_n(cdot,cdot,x)|_T leq 2|Phi(cdot,cdot,x)|_T.
$$
How can I infer that this sequence ${Psi_n}$ is then bounded as a mapping over all three arguments $(x,t,omega)$?
functional-analysis measure-theory stochastic-processes stochastic-pde
asked Dec 2 at 2:39
user2379888
23719
I've been studying Da Prato & Zabcyzk's text on SPDE, and there's a part of their proof of the stochastic Fubini theorem that I don't quite get. The setting is that $Phi: (Omega_Ttimes E, mathcal{P}_T times mathcal{B}(E))to (L_2^0, mathcal{B}(L_2^0))$, such that
$$
int_E |Phi(cdot, cdot,x)|_Tmu(dx)< infty
$$
where $mu$ is a finite positive measure and the $T$ norm corresponds to
$$
|Phi|_T^2 = mathbb{E}int_0^T|Phi(s)|_{L_2^0}^2 ds
$$
The authors have an intermediate result, Prop 4.34 in the 2014 edition, that $Phi$ can be approximated by a sequence $Phi_n$ of analogously measurable processes that can be written as simple processes and converge in the sense of
$$
int_E |Phi(cdot, cdot,x)-Phi_n(cdot, cdot,x)|_Tmu(dx) to 0.
$$
I find myself unsure of the proof of this intermediate result in that the authors first assert the existence of a sequence $Psi_n$ (not neccessarily simple) which are bounded and converge in the above sense. It is clear from the above assumption that $mu$-a.e., there are processes $Psi_n(cdot,cdot,x)$ which converge in the $T$ norm. This follows from a Prop 4.22 of the text, and it is clear that they can be taken such that
$$
|Psi_n(cdot,cdot,x)|_T leq 2|Phi(cdot,cdot,x)|_T.
$$
How can I infer that this sequence ${Psi_n}$ is then bounded as a mapping over all three arguments $(x,t,omega)$?
functional-analysis measure-theory stochastic-processes stochastic-pde
functional-analysis measure-theory stochastic-processes stochastic-pde
asked Dec 2 at 2:39
user2379888
23719
asked Dec 2 at 2:39
user2379888
23719
asked Dec 2 at 2:39
user2379888
23719
asked Dec 2 at 2:39
user2379888
23719
asked Dec 2 at 2:39
user2379888
23719
23719
add a comment |
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