All Questions
13 questions
4
votes
1
answer
181
views
Conditions for the SDE be transitive
This question was previously posted on MSE.
Let $f:\mathbb R^3 \to \mathbb R^3$ be a smooth Lipschitz function (bounded if needed), and $W_t$ a $3$-dimentional Brownian motion. Consider the SDE on $\...
- 333
3
votes
2
answers
2k
views
Kolmogorov continuity theorem and Holder norm
The Kolmogorov Continuity theorem (see for example the Wikipedia page) lets us prove that a stochastic process $X_t$ (on some complete metric space $(S,d)$) is Holder continuous almost surely provided ...
- 175
3
votes
2
answers
923
views
On representing a continuous time Markov chain by a stochastic integral of a Poisson random measure
Let $Q=(q_{ij})$ be the transition rate matrix of a continuous time Markov chain $\{ X_t \}$ with countable state space $M$. Let $q_i = -q_{ii}=\sum_{j \neq i}q_{ij}$, and let $\Gamma_{ij}$ be defined ...
- 31
2
votes
1
answer
159
views
Can we show that this transition semigroup preserves a certain Wasserstein space?
Let $E$ be a separable $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous, $$\rho(x,y):=\inf_{\substack{\gamma\:\in\:C^1([0,\:1],\:E)\\ \gamma(0)\:=\:x\\ \gamma(1)\:=\:y}}\int_0^1v\left(\gamma(...
- 167
2
votes
1
answer
469
views
Textbook definition for "path measure" or "probability measure over paths"
I need a formal definition for the path measure for stochastic differential equations.
Which textbook or paper should I consult?
2
votes
0
answers
80
views
Stability of Hölder constants of frozen Itô stochastic integrals
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- 835
2
votes
0
answers
66
views
Is $F: \mathbb T \times \mathbb R^d \times \Omega \to \mathbb R^d$ (constructed from Itô integral) Borel measurable in the product $\sigma$-algebra?
$
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- 835
2
votes
0
answers
155
views
Can a diffusion process admit an invariant measure with a non-differentiable density?
The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually,...
- 167
2
votes
0
answers
77
views
Extension of probability space problem: Hilbert space valued process V.S. random field
Maybe the question should be "Understanding the measurability: Hilbert space valued process V.S. random field"
Consider the SPDE $${\rm d}u+\cdots{\rm d}t=\sigma(t,u){\rm d}W.$$
Consider the ...
1
vote
1
answer
913
views
Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative
The problem:
Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in ...
- 247
0
votes
2
answers
182
views
Distribution of local martingale is absolutly continuous to that of the Brownian motion?
Let $B(t, \omega)$ be a Brownian motion defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, adapted to a filtration $\{\mathcal{F}_t\}$. Let $\phi(t, \omega)$ be a $\{\mathcal{F}_t\}$-...
- 227
0
votes
1
answer
461
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
0
votes
0
answers
53
views
Are the densities of a continuous stochastic process locally positive in time?
Let $X=(X_t)_{t\in I}$ ($I\subset\mathbb{R}$ a (non-degenerate) interval) be a stochastic process with continuous sample paths and such that $X_t$ admits a continuous Lebesgue density $\chi_t\in C(\...
- 463