Timeline for Correction term in the relation between the Itō and Stratonovich integrals in Hilbert spaces
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Aug 28, 2016 at 12:58 | comment | added | 0xbadf00d | I think that the discussion here is complete. However, I still got a question related to your last comment and asked a new question. | |
| Aug 27, 2016 at 19:34 | comment | added | 0xbadf00d | However, cause I want to use well-known existence theory for solutions as, for example, presented in Da Prato's book (Google books link), I would need to convert my equation into the corresponding Itō equation. The result of the paper discussed here is exactly what I want, but now I'm unsure whether or not the drift of my equation will satisfy (A3) or not. | |
| Aug 27, 2016 at 19:34 | comment | added | 0xbadf00d | (a) But then the series representation above (2.1) is wrong, isn't it? (b) The remaining question is: How restrictive is the assumption (A3) for the drift of (2.3)? With part (b) of my last two comments in the other question in mind, the reason why I'm asking this question is cause I'm thinking about interpreting equation $(1)$ in the other question in the Stratonovich sense. If I'm doing that, I don't need to deal with the (bad?) trace term in the SDE for $u$ which comes from the application of the Itō formula. | |
| Aug 27, 2016 at 13:39 | comment | added | Nawaf Bou-Rabee | They really mean that $w$ is a $Q$-Wiener process: please note that the left-hand-side of (2.1) involves the eigenvalues of $Q$. I think (2.4) is right, since the role of this term is to eliminate the Ito correction to the standard chain rule -- that is the essence of the Stratonovich description. | |
| Aug 27, 2016 at 13:31 | comment | added | 0xbadf00d | Of course. I didn't accept it so far, cause I wanted to make sure that I've understood what you've written. Let me ask you a last question on the paper: They write that $w$ is a $Q$-Wiener process, but from the series representation on page 2 it seems like $w$ is a cylindrical Wiener process (cause otherwise the square-roots $\sqrt{\lambda_i}$ of the eigenvalues of $Q$ should occur in that series). Is there anything I'm missing? | |
| Aug 27, 2016 at 13:05 | vote | accept | 0xbadf00d | ||
| Aug 26, 2016 at 19:26 | comment | added | 0xbadf00d | I'm not sure what you mean with "corrected nonlinear part of the drift", but I guess $$C_1(x)=C(x)+\frac12\tilde{\operatorname{tr}}\:Q{\rm D}B(x)B(x)\;,$$ right? | |
| Aug 25, 2016 at 13:18 | comment | added | Nawaf Bou-Rabee | Indeed, since the convolution defining the correction term is bounded and $C^1$, it is straightforward to prove that it is globally Lipschitz. Moreover, math.stackexchange.com/questions/1227314/… | |
| Aug 25, 2016 at 13:09 | comment | added | Nawaf Bou-Rabee | No. They do not mean (5'') and (6''). They mean that the corrected nonlinear part of the drift still satisfies (A3). To be sure, if we label the correction term as $C_1(x)$, they mean: $\| C_1(x) \|_{H_1}^2 \le K (1+ \|x\|_{H_1}^2)$ (linear growth condition) and $\| C_1(x) - C_1(y) \|_{H_1}^2 \le K \| x - y \|_{H_1}^2$ (globally Lipschitz). With the caveat given in my answer, the proof of this result seems math.stackexchange level. | |
| Aug 24, 2016 at 19:40 | comment | added | 0xbadf00d | You didn't respond the crucial part of my question. Do you think that they mean that $(5'')$ and $(6'')$ are satisfied, instead of $(5)$ and $(6)$? And do you think we can achieve a similar result for the trace term in the Itō formula? I've asked another question for that and hope you can help there too. | |
| Aug 24, 2016 at 19:34 | comment | added | 0xbadf00d | I've made a typo. I've intended to write down $$\left\|C(x)\right\|_{H_1}^2+\left\|\tilde{\operatorname{tr}}\:Q{\rm D}B(x)B(x)\right\|_{H_1}\le K(1+\left\|x\right\|_{H_1}^2)\;\;\;\text{for all }x\in H_1\tag{5''}$$ and $$\left\|C(x)-C(y)\right\|_{H_1}^2+\left\|\tilde{\operatorname{tr}}\:Q({\rm D}B(x)-{\rm D}B(y))(B(x)-B(y))\right\|_{H_1}\le K\left\|x-y\right\|_{H_1}^2\;\;\;\text{for all }x,y\in H_1\tag{6''}\;.$$ | |
| Aug 24, 2016 at 14:17 | comment | added | 0xbadf00d | If I got you right, you think that they mean that, instead of $(5)$ and $(6)$, $$\left\|C(x)\right\|_{H_1}^2+\tilde{\operatorname{tr}}\:Q{\rm D}B(x)B(x)\le K(1+\left\|x\right\|_{H_1}^2)\;\;\;\text{for all }x\in H_1\tag{5'}$$ and $$\left\|C(x)-C(y)\right\|_{H_1}^2+\tilde{\operatorname{tr}}\:Q({\rm D}B(x)-{\rm D}B(y))(B(x)-B(y))\le K\left\|x-y\right\|_{H_1}^2\tag{6'}$$ are satisfied, right? If that's the case, could you write down how exactly $(5')$ and $(6')$ follow from (A4) and the convergence of the series in $(4)$? | |
| Aug 24, 2016 at 14:16 | comment | added | 0xbadf00d | Did you mean ${\rm D}B(x)B(x)$ instead of ${\rm D}^2B(x)B(x)$ in your last sentence? They don't talk about the second Fréchet derivative of $B$ in the paper. | |
| Aug 24, 2016 at 13:48 | history | answered | Nawaf Bou-Rabee | CC BY-SA 3.0 |