Timeline for If $(Φ^x)_{x∈ℝ}$ is a family of real-valued stochastic processes and $B$ is a Brownian motion, then $\int_0^tΦ^x_s\:dB_s=(\int_0^t\Phi_s\:dB_s)(x)$
Current License: CC BY-SA 3.0
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| when toggle format | what | by | license | comment | |
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| Dec 6, 2016 at 11:03 | comment | added | 0xbadf00d | @NateEldredge I didn't want to bloat the question with too much details. You're free to assume any technical assumption you need (just write these assumption down). | |
| Dec 5, 2016 at 21:58 | comment | added | Nate Eldredge | Do you have any assumptions on the joint measurability of $\Phi^x_t(\omega)$ with respect to $(x,t,\omega)$? As it stands I only see separate measurability in $x$ and in $(t,\omega)$, and I suspect that's not enough to make sense of the integral of $\Psi$. | |
| Dec 5, 2016 at 21:05 | comment | added | 0xbadf00d | Now, I want to eliminate the dependence on $x_0$ in $(4)$ and transform this SPDE into a SDE on $L^2(Λ,ℝ^d)$. I've asked a special case of this question in a separate thread. The problem in this question seems to be a starting point for a solution to the other question. | |
| Dec 5, 2016 at 21:05 | comment | added | 0xbadf00d | $(2)$ is motivated by the following problem: Let $U$ be a separable $ℝ$-Hilbert space, $Q∈\mathfrak L(U)$ be nonnegative and self-adjoint with $\text{tr}Q<∞$, $W$ be a $Q$-Wiener process on $U$, $x_0∈Λ$, $F:[0,T]×ℝ^d→ℝ^d$, $G:[0,T]×ℝ^d→\mathfrak L(U,ℝ^d)$ and $X^{x_0}$ be a strong solution of $$X_t=x_0+\int_0^tF(s,X_s)\:{\rm d}s+\int_0^tG(s,X_s)∘{\rm d}W_s\;\;\;\text{for all }t∈[0,T]\;.\tag 3$$ Then we obtain $${\rm d}F(t,X_t^{x_0})=\left[\frac{∂F}{∂t}(t,X_t^{x_0})+(F(t,X_t^{x_0})⋅∇)F(t,X_t^{x_0})\right]{\rm d}t+(G(t,X_t^{x_0})⋅∇)F(t,X_t^{x_0})∘{\rm d}W_t\tag 4$$ for all $t∈[0,T]$. | |
| Dec 5, 2016 at 21:04 | history | asked | 0xbadf00d | CC BY-SA 3.0 |