# Questions tagged [stochastic-processes]

A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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### Correlation of stopping times for integral of Brownian motion increment

Let $\mu(x):=\int_{\epsilon}^{x}\exp\{B_{s+\epsilon}-B_{s-\epsilon}\}ds$, where $(B_{s})_{s\geq 0}$ is a Brownian motion (starting at $B_{0}=0$) and epsilon is small $0<\epsilon\ll 1$. Consider ...
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### Do Lyapunov functions imply exponential integrability of hitting times?

I have a question of some integrability of hitting times. Let $X=(\{X_t\}_{t \ge0},\{P_x\}_{x \in E})$ be a diffusion process on a locally compact separable metric space $E$. We assume that there ...
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### 2-d geometric Brownian motion hitting time distribution

I am trying to solve following problem: Given two independent geometric Brownian motions $\frac{d x_t}{x_t}=\mu_x dt + \sigma_x dw_t^x$ and $\frac{d y_t}{y_t}=\mu_y dt + \sigma_y dW_t^y$ and ...
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### Eigenspace of Gaussian Markov operator

Consider the (one-dimensional) Gaussian distribution $Q := N(\nu,\tau^2)$ and the (Gaussian) Markov operator \begin{equation*} \begin{array}{rccc} R : & L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) &...
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### Kolmogorov tightness criterion for stochastic processes

I am searching for the criterion stated above and also here: The question about Kolmogorov tightness criterion. It should state the following: If a sequence of stochastic processes $(X^n)$ fulfills: ...
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### Can the joint law $P \circ (X,Y)^{-1}$ of two random variables $X$ and $Y$ be written as $P \circ (X,\phi(X,U))^{-1}$ for $U$ uniform in $[0,1]$?

I want to know whether there is some general assumpitons we can make on two measurable spaces $E$ and $F$ (e.g. polish, complete, separable,...) such that we can ensure that the following "Theorem" ...
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### Stopping times about Brownian motion with draft

Assumet $M(t) = B(t) + \mu t$ where $B(t)$ is a standard Brownian Motion. Denote: $$T_a := \inf \{ t \geq 0, \, M(t) = a\}, \quad T_b := \inf \{ t \geq 0, \, M(t) = b\}$$ The question asks to ...
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### The weak convergence of finite dimensional distribution of Gaussian process does not imply the weak convergence in $C[0,1]$

In the study of weak convergence in $C[0,1]$, a common example is always being considered: $$X_{n}(t)=nt1_{[0,1/n]}(t)+(2-nt)1_{(1/n,2/n]}(t).$$ This example serves a counter-example to show that the ...
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### Find a conditional expectation of a difference of two independent Poisson process

Consider two independent Poisson processes $N,M$ with rate $\lambda$, and define $$X(t):=x+\dfrac{1}{\sqrt{n}}[N(t)-M(t)].$$ From this formula we know that $X(0)=x$. Now, I want to compute the ...
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### Calculate regime-switching correlation matrix without assumption on distribution

There's rich literature on Markov regime-switching dynamic correlation matrix. But most results seem to assume a certain kind of distribution and use MLE/EM. For example, some sort of multivariate ...
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### Orthonormal frame bundles on a manifold

Let $(\mathcal{M},g)$ be a torsion free compact Riemannian manifold of dimension $n$. Hence from the metric we know there is an associated horizontal sub-bundle $H_u F \mathcal{M}$ of the orthonormal ...
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### conditional expected value and in Stochastic differential equations

Let's suppose I have a bidimensional SDE of the form: \begin{equation} \label{eq:system} \begin{cases} dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\ X_0=x_0 \\ dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...
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### Markov semigroups and resolvents, difference of continuity

Let $(E,d)$ be a locally compact separable metric space. We have a Markov process $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in E})$ on $E$. For bounded measurable function $f$ on $E$, we define \begin{align*}...
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### Spitzer's condition, a slowly varying function and its behavior

Let $S$ denote a random walk that satisfies Spitzer's condition $$\frac{1}{n} \sum _{k=1}^n P (S_k > 0 ) \to \rho$$ for some $\rho \in (0,1)$. From the book Regular Variation (Bingham, Goldie, ...
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Let $\mathcal{H}$ be a separable Hilbert space. Assume we have some triangular array $W_{n,j}, j=1, \ldots ,n$ of $\mathcal{H}$-valued random elements with $\mathbb{E} \Vert W_{n,j} \Vert_{\mathcal{H}... 2answers 61 views ### Martingale optional stopping before a stopping time Here’s an easy one, I hope: Suppose$\tau$is a stopping time and$(M_t)$is a martingale which together satisfy the hypotheses of the optional stopping theorem so that$\mathbb{E}[M_\tau]= \mathbb{E}...
Let us consider a stochastic differential equation (SDE), $$dx_{t}=f\left( x_{t}\right) dt+\sigma\left( x_{t}\right) dW_{t}%$$ and a compact set $C\subset\mathbb{R}^{n}$. Given a stochastic ...