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5 votes
1 answer
188 views

Girsanov's theorem for Gaussian measures as the Cameron-martin theorem with a random shift

Let $H \subset E$ be the Cameron-Martin space of a Gaussian measure $\mu$ on a separable Banach space $E$. The Cameron-Martin theorem states that for all $h \in E$ we have $h \in H$ if and only if $\...
Robert Wegner's user avatar
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 $\...
Matheus Manzatto's user avatar
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 ...
Peixue 's user avatar
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(...
0xbadf00d's user avatar
  • 167
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,...
0xbadf00d's user avatar
  • 167
1 vote
1 answer
83 views

Integral of $M^\text{*} - M$ with respect to $M^\text{*}$ is zero for $M^\text{*}$ the running maximum of $M$ a continuous local martingale

Given $M$ a continuous local martingale, and $M^\text{*} = \sup_{0 \leq s \leq t} M_s$ its running maximum, we consider the finite variation integral $$ I_T:= \int_0^T (M^\text{*}_s - M_s) \, \text{d}...
George's user avatar
  • 113
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 ...
BCLC's user avatar
  • 247
0 votes
1 answer
268 views

Tightness on a set $A$ implies tightness on a set $B$ where $A\subset B$?

From the book Billingsley - Convergence of probability measures, 1999, we have the following definitions of tightness and relative compactness and the Prohorov's theorem: Tightness: Let $\Pi$ be a ...
Mark's user avatar
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