# Questions tagged [stochastic-processes]

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

1,775
questions

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**1**answer

44 views

### Bounding Brownian motion and an Ito process simultaneously

Let $(W_t)_{t\geq0}$ be a standard Brownian motion in $\mathbb{R}^n$ and $(A_t)_{t\geq0}$ be an adapted matrix-valued process such that $A_t$ is a positive symmetric matrix with bounded operator norm :...

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54 views

### Log Sobolev inequality for Wiener space

I am reading https://arxiv.org/pdf/1003.1649.pdf and saw equation 10.2.3 that said that on Wiener space
$$E\left[f^2\log\left[\frac{f^2}{E[f^2]}\right]\right]\leq 2 E[|\nabla f|_H^2],$$
where $\nabla$ ...

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52 views

### Probability that a $d$-dimensional Brownian bridge is greater than a given parameter

Let $(W_t)_{t\in[0,T]}$ be a Brownian bridge such that $W_0=a$ and $W_T=b$, the probability that $\forall t\in[0,T],W_t\geqslant x$ given the parameter $x\leqslant\min(a,b)$ is well known :
$$ \mathbb{...

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47 views

### Pulling random times out of conditional expectation (“Substitution rule”)

Problem
Let $G$ be a positive random variable (a random time) that is a.s. finite, $(X)_{t \geq 0}$ be a càdlàg process taking values in $\mathbb{R}^d$ and $g$ is some sufficiently nice real-valued ...

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22 views

### Convergence of the solution to ``some stochastic equation''

For every $\epsilon>0$, consider
\begin{equation*}
X^{\epsilon}_t = Z + W_t - 1 + \mathbb E\left[\exp\left(-\frac{1}{\epsilon} \int_0^t \max\left(-X_s^{\epsilon},0\right)ds\right)\right],\quad \...

**3**

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131 views

### Regularity of harmonic functions for a degenerate elliptic operator

This is a question on a degenerate elliptic operator.
Let $E$ be a closed unit ball in $\mathbb{R}^d$ centered at the origin. For a positive number $c>0$ and $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...

**4**

votes

**1**answer

134 views

### How to find the “natural scale function” for more general stochastic processes?

In stochastic analysis, for an Ito diffusion $X_t$ such that $dX_t=\mu(X_t)dt+\sigma(X_t)dB_t$, we can exlpicitly compute a "natural scale function"
$$S(x)=\int^x\exp\left(-\int^y\frac{2\mu(...

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35 views

### Inequality for reversed submartingales

For a submartingale $(X_k)_k,$ the following inequality holds, for all $\epsilon>0$:
$$\epsilon P(\max_{1 \leq k \leq r}X_k>\epsilon) \leq \int_{\{\max_{1 \leq k \leq r} X_k > \epsilon\}}X_r ...

**4**

votes

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262 views

+150

### Schwartz regularity for the density of a stochastic process

Let $B$ be a standard Brownian motion in $\mathbb R$. Define the variables
$$\begin{align*} X &= B_1, & Y &= \int_0^1B_s\mathrm ds, & Z&= \int_0^1B_s^2\mathrm ds. \end{align*}$$
It ...

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votes

**2**answers

162 views

### Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?

Given a potential function $U: \mathbb{R}^n \to \mathbb{R}$, Langevin diffusion is gradient descent plus a Brownian motion term: $X' = -\nabla U(X) + \sqrt{2} \text{ }dW$.
It happens that the ...

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**0**answers

23 views

### How to characterize the variance of a linear Gaussian system with switching?

Consider a random process described by the following linear dynamics:
$$
x_{k+1} = a x_k + n_k,
$$
where $|a|<1$ and $n_k$s are i.i.d. standard normal distributed.
It is quite easy to prove that $...

**2**

votes

**1**answer

59 views

### Generalized Fokker-Planck equation

Consider the diffusion process
$$
d X = \mu(X, t) dt + \sigma(X, t) dY.
$$
When $Y$ is a Brownian motion, we know that the density follows the Fokker-Planck equation. Here I'm considering the general ...

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34 views

### If $(\alpha_t)$ is $\mathbb{F}^X$-progressive for a continuous process $(X_t)$, can we write $\alpha_t = \tilde{\alpha}(t,X)$?

Let $X = (X_t)_{t \geq 0}$ be a continuous, real-valued process defined on some probability space $(\Omega,\mathcal{F},P)$, and let $\mathbb{F}^X = (\mathcal{F}_{t}^X)_{t \geq 0}$ be the filtration ...

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56 views

### Laplace Equation for Brownian Motion

So, I know that there is this theorem (taken from here):
For Laplace's equation $\Delta u = 0$ on a domain $D$ and $u=f$ on $\partial D$ (and some regularity conditions on $D$), we have
$$
u(x) = \...

**1**

vote

**1**answer

46 views

### Bound moments wrt. known initial and final moments

Let $X$ be an $L^p$ random variable, where $p\in (0,1)$ and $W_t$ usual Brownian motion (with $W_t$ independent from $X$). I'd like to bound
$$\mathbb E|X+W_t|^p$$
purely in terms of $\mathbb E|X|^p$ ...

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**0**answers

20 views

### What can be said about the coefficients of the expansion of a drift corresponding to a Girsanov measure?

Let $\mu_0$ be classical Wiener measure on $(C_0[0,T]), \mathcal B(\|\cdot\|_\infty))$. Let $\mu$ be another Borel measure so that $\mu\sim\mu_0$ are equivalent. Then by Girsanov there is a process $F(...

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53 views

### Mean square displacement of self-avoiding walk in dimension 5 or more

What would be the best detailed written resource to study Mean square displacement of self-avoiding walk in dimension 5 or more?

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36 views

### intuition about Gaussian processes over a finite space

In a paper that I am reading the authors defines $\mathbb P(n,q)$ the space of covariance tensors for $\mathbb R^q$-valued Gaussian processes on an abstract finite space $K=\{x_1,\dots,x_n\}$. In his ...

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48 views

### Brownian Motion excursion distribution

Consider a Brownian Motion $(W_t)_t$ and to some $x\in\mathbb T_2$, where $\mathbb T_2$ is a two-dimensional torus, the circles $\partial B_{r_i}(x)$ around $x$ with radii $r_i= R(\frac \varepsilon R)^...

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42 views

### Independence of variables predicted by the generator

Let $X$ be en compact metric set, and denote by $\mathcal{C}(X)$ the set of real continuous functions defined on $X$, endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega$ be the generator ...

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132 views

### Sticking martingales together

Let $(\Omega, \mathcal F, P)$ be a probability space with sigma algebras $\mathcal F_{n, t}$ for $n \in \mathbb N$, $t \in [0, 1]$, where for all $n$, $\mathcal F_{n, t} \subset \mathcal F_{n, h}\ $ ...

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**1**answer

66 views

### Probability to cross an envelopp for 1D random walk?

Imagine we have an evolving sequence composed of 1 and -1 (ex: -1-11-111...) where the probability to get -1 or 1 is 1/2. n is the lengh of my sequence.
I can make an analogy with random walk: let ...

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votes

**1**answer

140 views

### Perron-Frobenius and Markov chains on countable state space

The following question naturally arises in the theory of Markov chains with countable state space to which I would be curious to know the answer:
Let $A:\ell^1 \rightarrow \ell^1$ be a contraction, i....

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**2**answers

122 views

### Is a linear combination of Markov generator a Markov generator?

Let $X$ be a compact metric set and $\mathcal{C}(X)$ be the set on continuous real functions over $X$ endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega_1$ and $\Omega_2$ be two Markov ...

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**1**answer

76 views

### For stopping times $\tau_k,\mathcal{F}_{\sup_{k \in \mathbb{N}^*}\tau_k}=\sigma(\bigcup_{k \in \mathbb{N}^*}\mathcal{F}_{\tau_k})$?

$(\tau_k)_{k \in \mathbb{N}^*}$ is a sequence of stopping times (taking values in $\overline{\mathbb{N}}$) for the filtration $(\mathcal{F}_n)_{n \in \mathbb{N}^*}.$ Let $\tau=\sup_{k \in \mathbb{N}^*}...

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35 views

### Convergence in probability between step function, cutoff function and smoothed function

Let's say we have three deterministic functions
$$u_0(x)= \begin{cases}
u_l, x<0 \\[2ex]
u_r, x>0
\end{cases}$$
$$u_0^a(x):= \chi_{(-a,a)} \: u_0 (x) =\begin{cases}
0, x<-a \\[2ex]
...

**3**

votes

**1**answer

210 views

### Constructing the 'idealized white noise' stochastic process

There are some authors that define idealized 1-dimensional white noise as a generalized stochastic process $\{\dot W_t\}_{t\geq 0}$ with the following properties:
It is a mean zero Gaussian process.
...

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**3**answers

175 views

### English version on Dynkin's 1963 paper on stopping

I am looking for an English version of the following paper:
Е. Б. Дынкин, Оптимальный выбор момента остановки марковского процесса, Dokl. Akad. Nauk SSSR 150, 238-240 (1963).

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42 views

### Conservation of the Feller property after passage to the limit

Let $K\subset\mathbb{R}^d$ $(d\geq 1)$ be a compact set, $(D,\|\cdot\|_{\infty)}$ be the set of continuous functions form $[0,1]$ to $K$ endowed with the topology of the supremum norm, and $\mathcal{C}...

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35 views

### Distribution of total offspring of Poisson multitype branching process

The question I have is related to the question asked here: Total offspring of Poisson multitype branching process
Fix $d\in\mathbb{N}$ and let $Z_n\in\mathbb{N}^d$ be a multitype branching process, ...

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26 views

### Non-parametric estimator of continuous process's law from one sample path?

Motivation: If $X$ is a random-variable defined on some probability space $(\Omega,\Sigma,\mathbb{P})$ then Glivenko-Cantelli lemma guarantees that the empirical distribution $\frac1{N}\sum_{n=1}^N \...

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34 views

### Differentiable approximation of Brownian diffusion with unbounded volatility

Let $\{W_t\}_{t\in[0;T]}$ be a one-dimensional Brownian motion and let $\{\mathcal F_t\}_{t\in[0;T]}$ be the augmented filtration generated by this Brownian motion. Let $\{\sigma_t\}_{t\in[0;T]}$ be ...

**0**

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**1**answer

41 views

### Differentiable approximation of Brownian diffusion with bounded volatility

Let $\{W_t\}_{t\in[0;T]}$ be a one-dimensional Brownian motion and let $\{\mathcal F_t\}_{t\in[0;T]}$ be the augmented filtration generated by this Brownian motion. Let $\{\sigma_t\}_{t\in[0;T]}$ be ...

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40 views

### Occupation time of SDE

Let $b:\mathbb{R}^d\to\mathbb{R}^d$ be locally Lipschitz and assume that, for any $x\in\mathbb{R}^d$ and any $f\in C^{\infty}([0,1],\mathbb{R}^d)$, the equation
$$
X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...

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votes

**3**answers

509 views

### Convergence speed of a random dyadic rational generator

We are given a multiset $M$ of real numbers which initially is equal to $\{0,1\}$. In a sequential fashion, at each round $r\in\mathbb{N}$
two distinct instances $x_r$ and $y_r$ of $M$'s numbers are ...

**1**

vote

**1**answer

98 views

### Is the topology generated by the convergence of finite-dimensional distributions metrizable?

Let $\mathbf{D} := D([0,1]; \mathbb{R}^d)$ be the Skorokhod space (equipped with the Skorokhod metric) of càdlàg functions, and let $X = (X_t)_{t \geq 0}$ be its canonical process. The space of ...

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66 views

### Is my quadratic variation derivative bounded?

Let $\{W_t\}_{t\in[0;T]}$ be a Brownian motion, let $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ be continuous, bounded and Lipschitz continuous in the second argument, let $X$ be the unique ...

**7**

votes

**1**answer

133 views

### Regularity of law of conditional law of a Markov process equivalent to regularity of its paths

Let $(X_t^x)_{t\in [0,\infty),\,x\in \mathbb{R}^n}$ be a Markov process taking values in $\mathbb{R}^m$ and defined on some stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,\infty}), \...

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60 views

### Limit of a linear discrete-time stochastic process with uniform noise

I have posted this in the math and stats sites, but I am not sure where the proper forum for this question is. If it is not here, please go on and delete it.
Suppose we have a stochastic linear ...

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**1**answer

71 views

### Zeros of a non-degenerate bivariate Gaussian Process

Suppose $(X(t),Y(t))$ $t\in[0,1]$ is a bivariate Gaussian process. We can assume that each component is continuously differentiable, but not necessarily stationary, and that the covariance kernels of $...

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**1**answer

113 views

### Is it always possible to determine the distribution of a random variable given all its moments? [closed]

we we're asked about it, and I know that answer is "NO", and I haven't found an good enough answer yet
and would appreciate an explanation with examples.

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69 views

### Diffusion process, hitting times, and harmonic functions

Let $E$ be a locally compact metric space. We consider a diffusion process $X=(\{X_t\}_{t \ge0 },\{P_x\}_{x \in E})$ on $E$ whose lifetime $\zeta$ may be finite: $P_x(\zeta<\infty)>0$ for some $...

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**0**answers

128 views

### Continuous-time random walk on $\mathbb{R}$ that never stays still

Consider a walker on the real line $\mathbb{R}$ and two probability density functions $w$ and $j$ defined over $\mathbb{R}$.
A walker starts at $0$ and iterates the following: it samples a waiting ...

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69 views

### Translation of Dellacherie's Capacités et Processus Stochastiques

I have been studying the Strasbourg school's general theory of processes from Dellacherie and Meyer's Probabilities and Potential, and I really like it. I have heard very good reviews about another ...

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41 views

### Convolution of Wiener measure and measure on $W_0^{1,2}$

Let $F$ by a process adapted to the filtration of a standard Brownian motion. Suppose that the Doleans Dade martingale exists and is a martingale. $F$ is a measure on $W_0^{1,2}$, call it $\nu$. Let $\...

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**1**answer

59 views

### multi-time limit of a maximum of random walks

Suppose one has $N$ iid random walks $X^{(1)}_t,\ldots,X^{(N)}_t$ in discrete or continuous time $t$, let us say for example Poisson jump processes, and consider the stochastic process $Y^{(N)}_t = \...

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**1**answer

92 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-...

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71 views

### Continuity of the random variable defining the occupation measure of a continuous Gaussian process

Suppose $Z:\Omega \times [0,1] \to \mathbb{R}$ is a continuous Gaussian process with mean $\mu(t)$ and covariance kernel $C(t,s)$. Consider the random variable
$$
X_\alpha = \lambda( \{t \; : \; Z(t) &...

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**1**answer

100 views

### Probability to cross dynamic boundary for 1D-random walk?

context: Imagine we have an evolving bit sequence (ex: 001011...) where the probability to get 0 or 1 is 1/2. n is the lengh of my sequence (the number of bits)
I can make an analogy with random walk: ...

**1**

vote

**1**answer

59 views

### Local inverse bound of Cameron Martin and Banach norms

Let $X$ be a Banach space with a centered Gaussian measure $\mu_0$. Let $E$ be the Cameron-Martin space of $X$. Let the respective norms be $\|\cdot \|_X$ and $\|\cdot \|_E$. It is well known (see ...