All Questions
37 questions
2
votes
1
answer
236
views
Self-adjointness of generator and semigroup of an SDE
$
\newcommand{\bR}{\mathbb{R}}
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\newcommand{\bP}{\mathbb{P}}
\newcommand{\bF}{\mathbb{F}}
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\newcommand{\eps}{\...
- 835
2
votes
1
answer
86
views
Smoothness of resolvent of the infinitesimal generator of an Ito diffusion acting on bounded continuous function
Let $dX_t=\sigma(X_t)\,dW_t+\mu(X_t)\,dt$ be an Ito diffusion with Lipschitz coefficients and $\sigma(x)>0$. Let $f(x)$ be a continuous and bounded and non decreasing function. Can we prove that ...
- 41
7
votes
0
answers
151
views
Stochastic analysis on nuclear Fréchet spaces
This is a reference request question, so to make it clear what I am after, I will give a quick outline of the area I am thinking in and some questions that arise.
A lot of the time in infinite-...
- 439
1
vote
0
answers
70
views
On calculating the second quantization operator $\Gamma(A)$ of the Ornstein-Uhlenbeck operator $A$
Let $A$ be a self-adjoint operator on a Hilbert space , and let $d\Gamma(A)$ be the generator of the second quantization of $A$. Consider the following theorem from Segal's "Non-Linear Quantum ...
- 90
2
votes
0
answers
103
views
Find a function $f\geq 0$ such that $e^{-t[(x-\partial_x)\partial_x]^2} f$ is not non-negative for some $t\geq 0$
Consider the square of the Ornstein-Uhlenbeck operator $$A=[(x-\partial_x)\partial_x]^2=(x-\partial_x)\partial_x (x-\partial_x)\partial_x.$$ We know that $[(x-\partial_x)\partial_x]^2$ cannot be a ...
- 90
2
votes
0
answers
62
views
Continuous-time Wold decomposition
I'm looking for a reference for the Wold–Zasukhin decomposition in continuous time for stationary random processes on the real line.
I am aware of the classic result in the book from Rozanov, which ...
- 21
0
votes
0
answers
176
views
A convergence question in $L^2$ construction of Brownian motion
I feel confused with a particular step in the $L^2$ consturction of Brownian motion.
Let $\{\xi_n \sim N(0,1)\}_{n\geq 1}$ be a sequence of i.i.d Gaussian random variables on some probability space $(\...
- 227
1
vote
0
answers
177
views
A question on Gaussian small ball probability
Consider the random variable $$ G = \sum_{j=1}^{\infty} \lambda_j Z_j^2 $$
where $Z_j \sim_{\substack{i.i.d}} N(0,1)$ and $\lambda_j$ some non increasing sequence of positive numbers with $\sum_{j=1}^{...
- 119
0
votes
1
answer
460
views
Infinite-dimensional Gaussian measure vs finite-dimensional Wiener measure
I'm trying to figure out the connections between two contructions of Gaussian measure.
Let $(U, \langle\cdot,\cdot\rangle_U)$ be a seprable Hilbert space, and $\mathcal{B}(U)$ be the Borel sigma-...
- 227
1
vote
1
answer
82
views
Local inverse bound of Cameron Martin and Banach norms
Let $X$ be a Banach space with a centered Gaussian measure $\mu_0$. Let $E$ be the Cameron-Martin space of $X$. Let the respective norms be $\|\cdot \|_X$ and $\|\cdot \|_E$. It is well known (see ...
- 133
3
votes
2
answers
987
views
Regarding sample continuity of Gaussian Processes
Suppose we have a Gaussian Process $X_t$ on $\mathbb{R}^n$ with mean function $m(t)$ and covariance function $K(t,s)$. Then is $X_t$ being sample continuous (i.e. the sample paths of $X_t$ are almost ...
- 151
5
votes
1
answer
283
views
Malliavin derivative of stopped Brownian motion
Cross-posted from: "https://math.stackexchange.com/questions/3917971/malliavin-derivative-of-stopped-brownian-motion"
I have a small question concerning the Malliavin derivatives. It could ...
- 393
2
votes
0
answers
172
views
Non-integer conditional moment of exponential functional of Brownian motion
Let $B_t$ be a standard Brownian motion.
I want to solve the following:
$$
\mathbb{E}\left[\left(\int_0^1 e^{\sigma B_t}dt \right)^{1/(1-\beta) }\mid e^{\sigma B_1}=z \right],
$$
for some fixed $0<\...
- 284
3
votes
0
answers
569
views
Domain of the Generator of a Bessel process
Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$
\begin{align}
\rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t}
\end{align}
where $(W_{t})_{t\geq ...
4
votes
0
answers
322
views
Compactness of semigroups of one-dimensional diffusions
I have a question about semigroups of one-dimensional diffusions.
Let $X$ be the Ornstein-Uhlenbeck process on $\mathbb{R}$. The generator is expresses as
$$\frac{d^2}{dx^2}-x\frac{d}{dx}.$$
It is ...
- 721
1
vote
0
answers
134
views
Operator-valued stochastic integral and quadratic variation for operator-valued processes
Let $U$ be a separable $\mathbb R$-Hilbert space and $W$ be a $Q$-Wiener process on a complete and right-continuous filtered probability space. Let $H$ be a separable $\mathbb R$-Hilbert space and $X$ ...
- 167
2
votes
0
answers
169
views
How can we show that a $Q$-Wiener process on a Hilbert space $U$ takes values in $Q^{1/2}U$?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$
$U$ be an infinite-...
- 167
1
vote
0
answers
235
views
Associative law of the stochastic integral in Hilbert spaces
Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$T>0$
$I:=(0,T]$
$(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$
...
- 167
1
vote
0
answers
100
views
Convergence and boundedness in $L^\infty([0,T]\times \Omega)$ of Karhunen-Loeve expansion
Let $X:[0,T]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process in $L^2([0,T]\times\Omega)$. Consider the Karhunen-Loeve expansion of $X$:
$$ X(t,\omega)=\mu_X(t)+\sum_{n=1}^\infty \sqrt{\nu_n}\...
- 141
1
vote
1
answer
175
views
Stochastic operator on $\ell^1$ has dense range
Let $P:\ell^1(\mathbb{Z}^d) \rightarrow \ell^1(\mathbb{Z}^d)$
be given by
$$(Pz)(x)=\sum_{y \tilde \ x} \frac{1}{2d} z(y)$$
where the tilde indicates that $y$ is a neighboured vertex of $x.$
I ...
- 329
1
vote
0
answers
159
views
Construction of the quadratic variation process in infinite dimensions
Let
$H$ be a separable $\mathbb R$-Hilbert space
$(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a ...
- 167
3
votes
1
answer
281
views
Covariation of the stochastic integral and the Wiener process
Let$^1$
$T>0$
$U,H$ be separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ be nonnegative and self-adjoint operator with finite trace $\operatorname{tr}Q$
$(e^n)_{n\in\mathbb N}$ be an ...
- 167
6
votes
1
answer
386
views
Reference Request: Vector-Valued Ito Formula
I know that there exist Ito formulae to understand
$
f(X),
$
where $f: H\rightarrow \mathbb{R}$ is sufficiently nice, $H$ is a Hilbert space and $X$ is an $H$-valued semi-martingale.
However I'm ...
- 5,405
3
votes
0
answers
231
views
I've found a representation of the Itō-Stratonovich correction term and don't understand the used notion of a "trace"
Consider a Stratonovich SPDE $$X_t=X_0+\int_0^tb(s,X_s)\:{\rm d}s+\int_0^t\sigma(s,X_s)\circ{\rm d}W_s\tag 1$$ in a separable $\mathbb R$-Hilbert space $H$ with $W$ being a $Q$-Wiener process on a ...
- 167
3
votes
0
answers
78
views
Perscribed/Inverting Conditional Expectation
I'm having difficulty finding papers which deal with the following inversion problem.
Suppose I have a stochastic process $Y_t$ (which is described by a certain Hilbert-Space-valued SDE). I want to ...
- 5,405
4
votes
0
answers
414
views
Definition of the Stratonovich integral in Hilbert spaces
Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$
$B$ be a (standard, real-...
- 167
1
vote
1
answer
223
views
Stochastic integral is a continous or closed operator?
The Setup
Let $\xi_t$ be a process adapted to the filtration $\mathfrak{F_t}$ of the semi-martinagale $X_t$, such that both are square integrable. Then is the map
\begin{align}
F_T: L^2(\mathfrak{...
- 5,405
6
votes
2
answers
747
views
Does there exist a stochastic time derivative?
The Setup
Suppose I have a stochastic process $f(Z_t)$ where $Z_t$ solve the $d$-dimensional SDE
$$
dZ_t = \mu(t,Z_t)dt + \sigma(t,Z_t)dW_t
$$
and $f$ is a smooth function.
My Question
Is there a ...
- 5,405
1
vote
1
answer
164
views
Hilbert-Space Values SDE in terms of Basis
Suppose:
$$
dX_t = a(t,X_t)dt + b(t,X_t)dW^H_t
$$
is an SDE with values in a separable Hilbert Space $H$, and $W^H_t$ is an $H$-valued cylindrical Wiener process. Then can we write the dynamics for $...
- 5,405
1
vote
0
answers
135
views
infinite dimensional funtional ito calculus
I've been reading into functional Ito calculus and everything I've come across deals with processes generated by finite dimensional semimartingales. In Dupire's 2009 landmark paper he speaks about ...
- 5,405
6
votes
0
answers
774
views
Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term
Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs.
I'm reading Stochastic Differential Equations in ...
- 167
1
vote
0
answers
334
views
A problem on Markov chains and Dirichlet forms
Let $X$ be a countable set. Let $c:X\times X\to[0,+\infty)$ satisfy
$$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$
$$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$
$$c(x,x)=0\text{ for ...
- 369
1
vote
1
answer
294
views
A problem about the quotient space of an extended Dirichlet space
Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ${\...
- 369
3
votes
1
answer
199
views
Markov-semigroup Sobolev inequality
I have a question about the following definition:
A probability measure $\mu$, such that the Markov semigroup $e^{Lt} \in \mathcal{L}(L^2)$ exists and is symmetric, satisfies the Sobolev inequality ...
- 259
2
votes
1
answer
756
views
Functional representation of adapted jointly measurable stochastic processes
It seems like the question stated here in MSE has no answer yet and seems therefore for me to be not of a basic question type. For this reason I move it to MO.
Let $X_t : \Omega \to E, \ t \geq 0$ be ...
- 1,773
3
votes
0
answers
134
views
The distribution of maximum of fraction Brownian motion over finite time interval
Suppose that $\{B_t^H,\ t\geq 0\}$ is a fractional Brownian motion with Hurst exponent $H$, I wonder if there are explicit expressions for the joint distribution of
$(\sup_{0\leq t\leq T}B_t^H,B_T^H)$...
- 71
6
votes
1
answer
1k
views
How is Kolmogorov forward equation derived from the theory of semigroup of operators?
In Lamperti's Stochastic Processes, given
a time-homogeneous Markov process $X(t), t\geq 0$ with Markov transition kernel $p_t(x,E)$ and state space being a measurable space $(S, \mathcal{F})$,
a ...
- 357