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New answers tagged stochastic-processes

1

For bounded and continuous $f$, the map $\omega\mapsto\int_0^T f(\omega(r))\,dr$, from $D([0,T]; \Bbb R^d)$ to $\Bbb R$ is continuous and bounded. See, for example, https://math.stackexchange.com/questions/271738/is-integration-a-continuous-functional-on-the-skorohod-space

0

Since $(X_t(\omega))_[0,T]$ is a cadlag process it is progressively measurable, in particular $(\omega,t) \to X_t(\omega)$ is measurable. Let $\tilde f(y,t) := f(y)$, then $\tilde f$ is bounded and continuous again. W.l.o.g. assume $T = 1$. Then $\mu_n \otimes \lambda \to \mu \otimes \lambda$ weakly. It immediately follows that $$\lim_{n \to \infty} \int \... 0 (i,ii) Not quite. A trivial counterexample is \ell(x, y) \equiv 0. A simple non-trivial counterexample would be something like$$\ell(x,y) = \begin{cases} 1 / (x^{1/2} y^{3/2}) & \text{if $0 < x \leqslant 1$, $y < x$} \\ 0 & \text{if $0 < x \leqslant 1$, $y \geqslant x$} \\ e^{-y} & \text{otherwise.} \end{cases} $$Then L_t started ... 6 Suppose the sidelengths of your hexagon are k, \ell, m. A standard way of looking at this is to turn the picture by 30 degrees to the left and to view the yellow / cyan tiles as m functions f_i \colon \{1,\ldots,k+\ell\} into \mathbb{Z} with the following constraints: f_i(1) = i, f_i(k+\ell) = i-k+\ell, f_{i+1}(n) \ge f_i(n)+1 and f_i(n+1)-f_i(... 1 Short answer: If the RKHS \mathcal{H}_C associated to the kernel of your GP is infinite dimensional, then sample paths are in \mathcal{H}_C with probability 0. Nevertheless, one can show that sample paths are contained in a "larger" RKHS, and Steinwart showed how to construct such an RKHS as a power of the original RKHS. Sections 4.2 and 4.3 of ... 0 Q1: I have not seen such a process, but I can easily imagine one: Let Z_t be any non-negative Lévy process with positive drift, and let X_t = \int_{t-\epsilon}^t Z_s ds. Q2: No, unless we know something more about X_t. Imagine the process X_t as above, with Z_t having frequent extremely large jumps. As a \to \infty, the random variable T_{a+b} -... 1 Q1: Have you seen an analogous process to X _t anywhere else?$$$$I have not. Are you sure they exist ? If \epsilon = 0 the increments must be from positive, infinitely divisible distribution, none of which has moments of all orders. The strictly monotone condition also bothers me. It seems to only work with b = 1. If b > 1 then X_t > t^b ... 0 (This is not a complete answer, rather an attempt to describe the distribution of the integral.) The integral \int_\rho^t W(t) dW(t) is simply \tfrac{1}{2} ((W(1))^2 - 1) - \tfrac{1}{2} ((W(\rho))^2 - \rho). Let us handle the other term:$$I_\rho = \int_\rho^1 W(t-\rho) dW(t).$$Let us approximate W(t-\rho) by an elementary process X_n(t), which is ... 1 A useful inequality is \begin{equation} \tag{1} \frac{1}{\sqrt{2 \pi}} \frac{x}{x^2+1} \mathrm{e}^{-x^2/2} \le \frac{1}{\sqrt{2\pi}} \int_x^\infty \mathrm{e}^{-y^2/2} \, \mathrm{d} y \le \frac{1}{\sqrt{2 \pi}} \frac{1}{x} \mathrm{e}^{-x^2/2} \end{equation} for x >0, which can be shown by elementary arguments (see below). For example, one can prove the ... 1 Let \tau be a Markov time, and define the usual \sigma-algebras:$$\mathcal F^{<\tau} = \sigma\{X_t^{-1}(E) \cap \{t < \tau\} : t \geq 0, \, E \text{ — Borel}\}$$and$$\begin{aligned} \mathcal F_{\geqslant\tau} & = \sigma\{X_{\tau + t}^{-1}(E) : t \geq 0, \, E \text{ — Borel}\} \\ & = \sigma\{X_t^{-1}(E) \cap \{t \geqslant \tau\} : t \geq ...

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This question is not research level and would be better suited on MathStackExchange. Positivity here just means $f\ge 0$ $\Rightarrow$ $Tf\ge 0$ (where $f\ge 0$ is defined as $f(x)\ge 0$ for each $x\in X$). The example is verified using Proposition 2.2: If $f(\eta)=\min f(X)$ then $g=f-f(\eta)1\ge 0$ and since $T(1)=1$ we get  (T-I)(f)(\eta)=T(f)(\eta)-...

1

Strong Markov property says that under the condition that $\forall i, y(\tau_i)=y_i$, on each $[\tau_i,\tau_{i-1}]$ the process $X_t$ is just a brownian motion starting at $(i,y_i)$ and ending at $(i-1,y_{i-1})$. This evolution only depend on the starting point and the ending point and not at all what append before $\tau_i$ or after $\tau_{i-1}$. Then $X_t$ ...

2

Let $X_i:=X^i$, $S_k:=\sum_1^n X_i$, $T_k:=S_k/\sqrt k$, $|\cdot|:=\|\cdot\|_2$, $n\in\{1,2,\dots\}$, and $m:=\lceil\log_2 n\rceil$, so that $2^m\ge n$. Then \begin{equation} \max_{k=1}^n|T_k|\le\max_{k=1}^{2^m}|T_k| =\max_{j=0}^m\max_{2^{j-1}<k\le 2^j}|T_k| \le\max_{j=0}^m 2^{-(j-1)/2}\max_{2^{j-1}<k \le 2^j}|S_k|. \end{equation} Hence, for any ...

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