Intereting Posts

Is a dense subset of the plane always dense in some line segment?
Fundamental Optimization question consisting of two parts.
Map bounded if composition is bounded
Series for envelope of triangle area bisectors
Weak categoricity in first order logic
The inverse of a matrix in which the sum of each row is $1$
Can Three Equilateral Triangles with Sidelength $s$ Cover A Unit Square?
Prove that $\|UVU^{-1}V^{-1}-I\|\leq 2\|U-I\|\|V-I\|$
Use a lemma to prove that $A_4$ has no subgroup of order $6$.
Derivative of a vector with respect to a matrix
Every metrizable space with a countable dense subset has a countable basis
invariance of integrals for homotopy equivalent spaces
Simple function approximation of a function in $L^p$
Is every set countable according to some outer model?
Is $\lor$ definable in intuitionistic logic?

Let

- $U$ and $H$ be separable Hilbert spaces
- $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace
- $U_0:=Q^{1/2}U$
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(W_t)_{t\ge 0}$ be a $Q$-Wiener process on $(\Omega,\mathcal A,\operatorname P)$
- $X_0$ be a $H$-valued random variable on $(\Omega,\mathcal A,\operatorname P)$
- $v:[0,\infty)\times H\to H$ be continuously Fréchet differentiable with respect to the first argument and twice continuously Fréchet differentiable with respect to the second argument
- $\xi:\Omega\times[0,\infty)\times H\to\operatorname{HS}(U_0,H)$ suitable for the following

Assuming that $$X_t=X_0+\int_0^tv_s(X_s)\;{\rm d}s+\int_0^t\xi_s(X_s)\;{\rm d}W_s\;\;\;\text{for }t>0\;,\tag 1$$ I would like to use an Itō-like formula to find an expression for $$Y_t:=v_t(X_t)\;\;\;\text{for }t\ge 0\;.\tag 2$$ However, the Itō formula (see Da Prato, Theorem 4.32) for $(1)$ can only be used to find an expression for $f_t(X_t)$, if $f$ is a continuously partially Fréchet differentiable mapping $[0,\infty)\times H\to\color{red}{\mathbb R}$.

Since I want to derive a SPDE for $Y$, I’m unsure what I need to do. Maybe we can do the following: Let $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ and $$v^{(n)}_t(x):=\langle v_t(x),e_n\rangle_H\;\;\;\text{for }n\in\mathbb N\;.$$ Then, each $v^{(n)}$ is continuously partially Fréchet differentiable and we can apply the Itō formula to find an expression for $$Y^{(n)}_t:=v_t^{(n)}(X_t)\;\;\;\text{for }t\ge 0\;.$$ Is this a good idea? We should be able to obtain $(2)$ by $$Y=\sum_{n\in\mathbb N}Y^{(n)}e_n\;.$$ Later, I’m interested in numerically obtaining $Y$.

- Name for kind of big O notation with leading coefficient
- Iterative refinement algorithm for computing exp(x) with arbitrary precision
- Fastest curve from $p_0$ to $p_1$
- Bifurcation Example Using Newton's Method
- Show that the LU decomposition of matrices of the form $\begin{bmatrix}0& x\\0 & y\end{bmatrix}$ is not unique
- What is the difference between Finite Difference Methods, Finite Element Methods and Finite Volume Methods for solving PDEs?

**EDIT**: By the referenced version of the Itō formula, we obtain

\begin{equation}

\begin{split}

\langle v(t,X_t),e_n\rangle_H&=\langle v(0,X_0),e_n\rangle_H+\int_0^t\langle{\rm D}u(s,X_s)(\xi_s(X_s){\rm d}W_s),e_n\rangle_H\\

&+\int_0^t\langle\frac{\partial u}{\partial t}(s,X_s),e_n\rangle_H+\langle{\rm D}u(s,X_s)(u_s(X_s)),e_n\rangle_H\\

&+\frac 12\operatorname{tr}\langle{\rm D}^2u(s,X_s)\left(\tilde\xi_s(X_s)\tilde\xi_s^\ast(X_s)\right),e_n\rangle_H{\rm d}s

\end{split}\tag 3

\end{equation}

where $\tilde\xi:=\xi Q^{1/2}$. I’ve got two questions:

- Can we write the infinite system of equations $(3)$ as one equation, as it is possible in the case $H=\mathbb R^d$ and $e_n=n\text{-th standard basis vector}$? It seems like that’s the case, cause almost all terms are simply projections to the $n$-th basis vector, but I don’t know how I need to deal with the trace term.
- In my real application, I have another known expression for $v(t,X_t)$. Thus, my SPDE would be obtained by equating the SPDE of the question with that other expression. Now the question is: I want to solve that SPDE numerically. Is there any recommended method for such a type of equation? [I should note that I will consider something like $H=[L^2(\mathcal V)]^3$ or $H=[H_0^1(\mathcal V)]^3$ for some bounded domain $\mathcal V\subseteq\mathbb R^3$].

- Connections between SDE and PDE
- Method of characteristics with constant PDE
- Derive $u(x,t)$ as a solution to the initial/boundary-value problem.
- Detecting perfect squares faster than by extracting square root
- Good reference texts for introduction to partial differential equation?
- Reformulation of Goldbach's Conjecture as optimization problem correct?
- Claims in Pinchover's textbook's proof of existence and uniqueness theorem for first order PDEs
- Why do odd dimensions and even dimensions behave differently?
- A numerical evaluation of $\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1x^{(n)} dx$
- Derivation of weak form for variational problem

This longer version of my comment addresses only the first question.

Yes, you can do that. Let, for simplicity, $v$ be independent of $t$, i.e. $v\colon H\to H$, and let it be twice continuously differentiable with bounded derivatives. Then $D^2 v(x)$ can be thought as a bilinear map $H\times H\mapsto H$. On the other hand, $\xi\xi^*(x)$ can be regarded as a trace class operator on $H$. In turn, the space of trace class operators is the closure of $H\otimes H$ with respect to the trace norm. Hence, the map $H\otimes H\mapsto H$ which sends $h\otimes g$ into $D^2v(x) (h,g)$ can be extended to the space of trace class operators by continuity; denote this by $\operatorname{tr}_H(D^2v(x)\cdot)$. This makes the writing $\operatorname{tr}_H\big(D^2v(X_s) \xi Q\xi^*(X_s)\big)$ meaningful as an element of $H$.

So you can write

$$

v(X_t) = v(X_0) + \int_0^t Dv(X_s)\Big(v(X_s)\mathrm{d}s + \xi_s(X_s) \mathrm{d}W_s\Big) \\

+ \frac12 \int_0^t \operatorname{tr}_H\big(D^2v(X_s) \xi Q\xi^*(X_s)\big)\mathrm{d}s.

$$

Long story short, it is just a matter of making some notation to write this in a coordinate-free form on $H$.

- How many different subsets of a $10$-element set are there where the subsets have at most $9$ elements?
- Sum of digits and product of digits is equal (3 digit number)
- Proof of transitivity in Hilbert Style
- Girth and monochromatic copy of trees
- Can you prove the following formula for hypergeometric functions?
- What is combinatorics?
- Why is the 2 norm “special”?
- Show that $ \mathbb{E} < \infty \Longleftrightarrow \sum_{n=1}^\infty \mathbb{P} < \infty $ for random variable $X$
- $\sigma$- compact clopen subgroup.
- Infinite number of rationals between any two reals.
- What is the name of this theorem, and are there any caveats?
- How to generalize the Thue-Morse sequence to more than two symbols?
- Accumulation Points for $S = \{(-1)^n + \frac1n \mid n \in \mathbb{N}\}$
- 1-1 correspondence between and
- How to compute the residue of a complex function with essential singularity