# Questions tagged [brownian-motion]

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292
questions

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### L2-closure of absolutely continuous stochastic processes?

Assume we have a possibly multidimensional Brownian motion on a probability space $(\Omega,\mathcal F,\mathbb P)$ where $(\mathcal F_t)_{t\in[0;T]}$ is the Brownian standard filtration. Let $\Vert X\...

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118 views

+50

### 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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64 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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15 views

### Question on the choice of boundary in the CUSUM test when we make some resampling [migrated]

Question on the choice of boundary in the CUSUM test when we make some resampling
We are considering to make a CUSUM test for some economical time series $X = (x_1,..,x_n)$. Suppose $X$ contains many ...

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57 views

### Decay rate of transition density of a SDE system

Consider the following SDE system
$$dx_t = b(y_t)dt + dw^1_t, \quad dy_t = dw^2_t.$$
Here the drift $b(\cdot)$ is a smooth function that may decay slowly. For example, $|b(x)| \le C/|x|^\sigma$ for ...

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56 views

### Extension of the Kelvin transform

Suppose $B=B(y,r)$ is ball in $\mathbb{R}^m$ ($m\geq2$), and $u$ a superharmonic function on a neighborhood of the closure $\overline{B}$ of $B$. We know that the Kelvin transform of $u$ with respect ...

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60 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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57 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) = \...

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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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40 views

### Superharmonicity of the distance function

Suppose $V$ is a convex open proper subset of $\mathbb{R}^m$ ($m\geq2$). It is known that the function $u(x)=$dist$(x,\partial V)$ is superharmonic on $V$. Is there a similar result without $V$ being ...

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67 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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117 views

### A question on minimum principle

Suppose $D$ be an unbounded domain of $\mathbb{R}^m$ for $m\geq3$, and $u$ is superharmonic on $D$. We know that if $\liminf_{x\to y}u(x)\geq0$ for all $y$ in $\partial^\infty D$ (the boundary of $D$ ...

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48 views

### An open set whose complement is non-thin at infinity

Let $x^*$ designate the inverse of a point $x\in\mathbb{R}^m$ under the Kelvin transformation with respect to the circle of center 0 and radius 1. Recall that
$$x^*=|x|^{-2}x.$$
For a set $E$, we set $...

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35 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 ...

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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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41 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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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 ...

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120 views

### Superharmonicity at infinity

Some authors define superharmonicity at infinity in the following way. A function $u$ is superharmonic on an open set $V\subset\mathbb{R}^m\cup\{\infty\}$ (one point compactification), containing ...

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136 views

### Associativity rule for integration against fractional Brownian motion

In Itô calculus, it is easy to construct an associativity rule. Namely, if $B_t$ is a Brownian motion and $M_t = \int_0^t X_s dB_s$ for suitable $X_t$, then we have the following associativity rule: $...

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32 views

### Superharmonic extension 3

This question is related to the MO post
Superharmonic extension 2. Let $u$ be a superharmonic function on $\mathbb{R}^m$ ($m>2$) such that for some $\alpha\in\mathbb{R}$ and $\beta$, $R>0$,
$$u(...

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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: ...

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**1**answer

51 views

### Is a stopped Ito-integral integrable if the Ito integrand is only square-integrable on an open interval?

Assume a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\in[0;T)}, \mathbb P)$ with an $\mathbb R^n$-valued Brownian motion $\{W_t\}_{t\in[0;T)}$ and the filtration $\{\mathcal F_t\}_{t\in[0;T)...

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91 views

### Use stochastic process to express solution to Laplace equation in the whole space

Consider the Laplace equation in $\mathcal{R}^3$
\begin{equation}
\Delta u = f, ~~~\lim_{x\to \infty} u(x) = 0.
\end{equation}
Here we assume $f$ is a smooth, compactly supported function. Of course, $...

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32 views

### Quadratic variation of generalized stochastic integrals

My question is based on this paper: https://pdfs.semanticscholar.org/0b5a/e41096a3b16d0756a1d36da55143d861ed7c.pdf.
In summary, this talks about the generalization of stochastic integrals to a two ...

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141 views

### Local martingale but not martingale

For a 3-dimensional Brownian motion $B = (B_t, t ≥ 0)$ and $x ∈ \mathbb{R}^3 \backslash \{0\}$ define the process
$Y = (Y_t, t ≥ 0)$ via $Y_t =\frac{1}{|B_t+x|}$ how come this is a continuous local ...

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34 views

### 2d interpolation minimizing the integral of the norm of the Hessian

It is well known that cubic interpolation is the solution of the interpolation problem that minimizes the integral of the square of the second derivative:
$$ min_{f \text{ s.t. } f(x_i)=y_i} \int (f''(...

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265 views

### proof that the covariance function for a fractional Brownian motion / fractional Gaussian free field is well defined

Given $0 < t_1 < \dots < t_n$, we can show that the matrix $\Omega$ whose entries are defined by $M_{i,j} = min(t_i,t_j)$ is symmetric definite positive.
The proof is immediate once one ...

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72 views

### Malliavin derivative of stopped Brownian motion

Cross-posted from: "https://math.stackexchange.com/questions/3917971/malliavin-derivative-of-stopped-brownian-motion"
I have a small question concerning the Malliavin derivatives. It could ...

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85 views

### Does convergence of a sequence of subharmonic functions imply the vague convergence of their Riesz measures?

Suppose $D$ is a bounded domain of $\mathbb{R}^m$ for $m>1$ and $\{u_n\}_{n\geq1}$ is a sequence of subharmonic functions on $D$. Assume $u_n\to u_0$ pointwise on $D$ and $u_0$ is subharmonic on $D$...

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65 views

### The Itō isometry for Riemannian manifolds

If $\alpha$ is a real smooth $1$-form, and if $\mathcal C$ is the space of continuous functions $c : [0,1] \to \mathbb R^n$, endowed with the Wiener measure $w$, and if $I_\alpha : \mathcal C \to \...

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54 views

### Independent increments for the Brownian motion on a Riemannian manifold

In am not a probabilist, but I must do some stochastic-flavoured work on a connected Riemannian manifold $M$. A nice thing about the Brownian motion on $\mathbb R^n$ is that we may talk about its ...

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89 views

### A simple clarification on Riesz decomposition theorem

Let $D$ be a domain of $\mathbb{R}^{m}$ and let
$K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. According to Riesz decomposition theorem (Hayman and Kennedy, "subharmonic functions&...

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107 views

### Intersection of a Poisson bridge and a Brownian bridge

Take a Poisson process $N_t$, a Brownian motion $W_t$ and constants $T > 0$ and $a > 0$. Suppose $N$ and $W$ are independent. I'm interested in the probability that $W$ does not cross over $a + ...

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216 views

### Expectation of Brownian motion increment and exponent of it

While reading a proof of a theorem I stumbled upon the following derivation which I failed to replicate myself. Let $\mu$ be a constant and $B(t)$ be a standard Brownian motion with $t > s$. Show ...

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96 views

### Martingale derivation by direct calculation

I'm reading the proof of a theorem and stumbled across the following derivation which I cannot replicate myself.
Let $W(t)$ be a $Q$-martingale and be given by $W(t) = B(t) + \mu t$ with $B(t)$ a ...

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70 views

### Is the $\sqrt{{\rm time}}$ spread of a stochastic process about the global minima the ubiquitous phenomenon?

Given a function $f$ with a global minima at $x^*$, consider a stochastic process given as, $x_{t+1} = x_t - \nabla f(x_t) + \xi$ where $\xi$ is a random variable. Now we want to understand the ...

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81 views

### Large deviation for Brownian occupation time

I am asking for reference about the large deviation principle (LDP) for the occupation time of a Brownian motion/bridge. Let $f:\mathbb{R} \to \mathbb{R}$ be smooth and compactly supported. My ...

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33 views

### Backwards Regulated Branching Process with Browning Motion; duality

I am working on a problem which I have not well understood completely, so I can only give the intuition of it. Imagine that we have a population on the (unit) torus $\Bbb T\subseteq\Bbb R$ distributed ...

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73 views

### “Return map” for Brownian motion

Consider a Brownian motion $W$ reflected at the boundary of a domain $D$ in Euclidean space. I want to look at the process obtained by "restricting" it to the boundary.
I was thinking of ...

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198 views

### Maximum eigenvalue of a covariance matrix of Brownian motion

$$ A := \begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\frac{1}{2} & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\frac{1}{3} & \frac{...

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61 views

### Exit time for Brownian motion with stochastic barriers

I am interested in the expected exit time of a one-dimensional Brownian particle from a stochastically evolving interval as follows.
Context:
If $L_t$ and $R_t$ denote the distance to the left and ...

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35 views

### Heat Equation Boundary Value Problem - alternative expressions for solution

Let $B_t$ be a Brownian motion, with with density function $f(t,x)dx = P(B_t \in dx)$. Then $f$ solves the heat equation $\partial_t = \frac{1}{2} \partial_{xx}f(t,x)$. Let for a fixed $u > 0$, $\...

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34 views

### Weak convergence to fractional Brownian motion after transformation

Let $x_k$, $k = 1, 2, \cdots$, be a sequence of random vectors in $\mathbb{R}^l$ ($l > 1$), defined on the same probability space, and $f: \mathbb{R}^l \rightarrow \mathbb{R}$. I would like to ask ...

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273 views

### Stochastic integral with respect to a random field

I came across a generalized Black-Scholes equation formulation in this paper.
Let me highlight the basic idea below. Consider a random field $W(t,T)$ where for a fixed $T$, $W$ is a Brownian motion ...

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100 views

### Natural way to thicken Brownian motion to 2D?

If we have a smooth plane curve (Hausdorff dimension 1), we can thicken it by a small amount to get a 2D set (all points within distance $\epsilon$ to the curve).
What if we start with the graph of a ...

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228 views

### How to prove that a Brownian bridge $\mathbb{P}(M[0, 1/2]\geq s)\leq 2\mathbb{P}(B(1/2)\geq s/2)?$

Consider a Brownian bridge $B: [0,1]\to \mathbb{R}$ with $B(0)=B(1)=0$. Let $M[0, 1/2]=\max_{x\in[0,1/2]}B(x)$. How to prove that
$$\mathbb{P}(M[0, 1/2]\geq s)\leq 2\mathbb{P}(B(1/2)\geq s/2)?$$
...

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56 views

### Limit of an integral / Boundary behaviour of a Gaussian convolution / single layer potential

Let $k(t,x)$ be the transition density of Brownian motion $$ k(t,x) := \frac{1}{\sqrt{2 \pi t}} \exp \left\{ \frac{-x^2}{2t} \right\} , \quad t \geq 0, x \in {\mathbb R.}$$
Question
Let $0 < x &...

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113 views

### 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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103 views

### 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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239 views

### 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 ...