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If $T$ is an infinite set and the Gaussian measure is non-degenerate, then the RKHS (Cameron Martin space, in Bogachev's language) is infinite dimensional; hence $\operatorname{Prob}(X \in \mathcal H( …

As formulated, there are some difficulties for the local Lipschitz case. For quadratic $f(x)=x^2$, consider the corresponding ODE for $x$ (i.e. the SDE with $\sigma = 0$). Then there exists a locally …

You are looking for the Lévy symbol of a 2-dimensional process $(N,D)$, so your symbol will be a function on $\mathbb R^2$. It will be $\phi_{N,D}(u_1, u_2) = \phi_N(u_1) + \phi_D(u_2)$, by the follow …

I don't know if continuity of the density functions gives you anything. For example, think about a pure jump process with smooth density function of the jump distribution. You may be familiar with Ko …

The key thing is that your diffusion is Feller under the stated conditions, so $x \mapsto \mathbf E_x f(X_t)$ is continuous. Therefore $x \mapsto \int_0^{\infty} \mathbf E_x f(X_t) e^{-rt} \ d t$ is c …

Let $(\Omega, (\mathcal F_t), \mathbb P)$ denote the usual Wiener space where $\Omega = C[0,\infty)$, etc., and where $(W_t)_{t \geq 0}$ denotes the Wiener process. Let $Z \in L^1(\mathbb P)$ with $Z …

I can now answer my own question. There exists no such counterexample. I can show that finite relative entropy in this setting implies $\mathbb E^{\mathbb Q} \int_0^{\infty} \theta_t^2 \ d t < \infty$ …

I think this holds in quite some generality by the following simple argument. Let $S$ be a Polish space, let's say. If $T(t)$ is a Markov-Feller semigroup on $C_b(S)$ with kernel $p_t(x,dy)$, then not …

This question is related to the topic of stochastic filtering theory. See e.g. the following monographs * Bucy, Joseph - Filtering for stochastic processes with applications to guidance * Bain, Crisan …

1) Explicit expressions for transition densities In the case of linear systems with additive noise of the form $d X(t) = (A X_t + b) \, d t + \sigma \, d B_t$ it is possible to obtain an explicit …

Suppose $P(x,dy)$ and $Q(x,dy)$ are two Markov transition kernels on a topological space $E$ equipped with Borel $\sigma$-algebra $\mathcal B(E)$. Suppose for every $x \in E$, $P(x,\cdot)$ and $Q(x, \ …

flawed, see Martin Hairer's comment below. Step I) Perform substitution $u_t(x) = \exp(\psi_t(x))$. The SPDE for $\psi$ becomes, after dividing by $u_t(x)$, \begin{equation} \frac{\partial}{\partial …

Let $0 < c < 1$. Consider the Markov chain $(X_i)$ on $\{0, 1, \dots, n\}$, with transition probabilities $$ P(k,k+1) = \left(1 - \tfrac {k}{n} \right)(1-c), \quad k = 0, \dots, n-1, $$ $$ P(k,k-1) …

Write $f(t) := \mathbb E \left[ \exp\left( - \int_0^t r_s \ d s \right) \right]$. Define stochastic processes $y_t = \exp \left( -\int_0^t r_s \ d s \right)$ and $z_t = r_t y_t$. Then $y_t = 1 - \int_ …