# Questions tagged [brownian-motion]

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251
questions

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votes

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49 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 ...

**-4**

votes

**0**answers

24 views

### . Let B be a Brownian motion. Compute the mean of the random variable [closed]

$$
\xi=e^{2 B_{T}} \int_{0}^{T} e^{2 B_{t}+t} d B_{t}
$$

**-1**

votes

**0**answers

30 views

### 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 ...

**2**

votes

**1**answer

118 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 ...

**0**

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35 views

### Existence of two stochastic processes

I am wondering if I can show that
For given $x,y\in \mathbb{R}$ there are two stochastic processes $S_t$ and $B_t$ such that $S_t$ and $B_t$ are two one dimensional Brownian motions starting at $x$ ...

**0**

votes

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16 views

### Bifractional Brownian motion admit a representation in the form of a stochastic integral?

good morning. You know the fractional Brownian motion, multifractional Brownian motion and sub-fractional Brownian motion, can be represented as a wiener integral ( moving average representation ). ...

**13**

votes

**2**answers

758 views

### How long for Brownian motion to “fill-out” a torus in d-dimensions?

I've been taken by the concise result1
that (roughly!), on a $2$-dimensional torus $\mathbb{T}^2$, the time it takes
to visit nearly every point (within $\epsilon$, as $\epsilon \to 0$) is: $\frac{2}{\...

**2**

votes

**0**answers

53 views

### Showing an “obviously-optimal” control is optimal (without smoothness assumptions)

Let $\mathcal{A}\subseteq\mathbb R$ be a compact interval, $T\in\mathbb R_+$ be a finite horizon, and $g:\mathbb R\to\mathbb R_+$ be a continuous function with $g\leq 1+|\cdot|$. Consider an optimal ...

**1**

vote

**1**answer

93 views

### Absolute value of a diffusion

Suppose $B_t$ is a standard Brownian motion on a filtered probability space $\langle \Omega, \mathcal F, \{\mathcal F_t\}_t, \mathbb P\rangle$. Consider two SDEs below.
Suppose, $X_0 = Y_0 = 0$
\...

**0**

votes

**0**answers

29 views

### Existence of an optimal control

I am looking for an existence result for the following control problem:
Fix a probability space $\langle \Omega, \mathcal F, \{\mathcal F_t\}_t, \mathbb{P}\rangle$ that satisfies the usual ...

**0**

votes

**1**answer

66 views

### Question about the exit time of a time-homogeneous Itô diffusion

Consider a one-dimensional Itô diffusion:
$$\mathrm{d} X_{t}=b\left(X_{t}\right) \mathrm{d} t+\sigma\left(X_{t}\right) \mathrm{d} B_{t}$$
where $X_0 = 0$ and $B_t$ is the standard Brownian Motion. ...

**1**

vote

**0**answers

47 views

### Local time as a measurable map from Wiener space

Let $B$ be a Brownian motion on $[0,1]$. The local time of $B$, which I will denote by $L$, is defined as the process on $\mathbb R$ such that
$$\int_0^1 F(B_t)~dt=\int_\mathbb R F(x)L(x)~dx,\qquad\...

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29 views

### Constructing uncountably many independent random variables with same distribution from Brownian motion?

It is well known one cannot construct uncountable many independent random variables on $([0, 1], \mathcal{B}[0, 1], \lambda)$. ($\lambda$ Lebesgue measure.)
Also, one can clearly construct infinitely ...

**0**

votes

**0**answers

41 views

### How to find the PDE for the following transition density

Suppose I have the following two stochastic differential equations ($t\geq 0$)
$$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t \ \ \text{ and } \ \ dZ_t =dt,$$
where $X = (X_t)$, $Z = (Z_t).$
Note that
$W=(...

**0**

votes

**2**answers

110 views

### Transience of 3-dimensional Brownian motion

I'm attempting Exercise 5.33 of Le Gall's Brownian motion, Martingales and Stochastic Calculus.
Let $B_t$ be a 3-dimensional Brownian motion starting from $x$.
Part 6 asks me to show that
$$|B_t| = |...

**0**

votes

**1**answer

53 views

### Prove that fractional Brownian motion is not a semimartingale using the p-variation

What follows, up to the horizontal line, is taken from Rogers "Arbitrage with fractional Brownian motion".
Consider an interval $[0,T]$ on which is defined the fractional Brownian motion $B$, and ...

**1**

vote

**2**answers

159 views

### Continuity of Brownian motion constructed from Kolmogorov extension theorem?

I'm trying to construct Brownian motion using the Kolmogorov extension theorem.
I am happy with the construction of a process with the required FDDs as (the canonical process associated with) a ...

**0**

votes

**1**answer

43 views

### Existence of strong couplings for Brownian motion

I have two different standard one-dimensional Brownian motions on different filtered spaces, $\langle\Omega,\mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P, (W_t)_{t\geq0}\rangle$ and $\langle\hat\...

**1**

vote

**0**answers

103 views

### Probability of m crossings of 0 before time 1 of a standard Brownian motion

Let $B$ be a standard Brownian motion. Could anyone show some hints and reference about how to compute the following probability?
Let $N(n) = \sum_{i=1}^n \mathbb{1}_{\{0 \in B[\frac{i-1}{n},\frac{i}{...

**1**

vote

**0**answers

132 views

### Extension of subharmonic function: can someone explain the details?

In this paper we have the following situation on page 60. $E$ is a compact subset of $\mathbb{R}^\tau\cup\{\infty\}$ (one point compactification) for $\tau\geq2$, $M_0$ is a point in the boundary of $...

**0**

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49 views

### Construct a random time such that the strong Markov property of Brownian motion fails

Let $\{B_t, \mathcal{F}_t; t\ge 0\}$ be a standard, one-dimensional Brownian motion. Can we construct a random time $S$ such that $P[0\le S < \infty] = 1$ and $W_t = B_{S+t} - B_S$ is not a ...

**2**

votes

**1**answer

44 views

### The converse of a Poincaré's result on regular boundary points

Let $V$ be a bounded open set in $\mathbb{R}^n$ with $n>1$. According to a well known result due to Poincaré, if $x$ is a point in the boundary $\partial V$ and there exists a ball $B$ such that $x\...

**1**

vote

**0**answers

74 views

### Probability of m crossings of 0 before time n of a Gaussian random walk

Let $S_n = \sum_{n=1}^n X_n$ be a Gaussian random walk where $X_n$ are i.i.d random variables with distribution $\mathcal{N} (0,1)$. Could anyone show some hints and reference about how to compute the ...

**0**

votes

**1**answer

91 views

### Extension of subharmonic functions at infinity

Let $W$ be the complement of a compact set $K$ in $\mathbb{R}^{n}$, and $u$ a subharmonic function on $W$. Can we find, under some conditions, a function $\tilde{u}$ that is subharmonic on $W\cup\{\...

**0**

votes

**1**answer

44 views

### A question on the problem of Dirichlet 2

Let $U$ be an open set in $\mathbb{R}^{n}$ with $n\geq2$ and $V$ an open set containing the boundary $\partial U$ of $U$. Suppose $u$ is subharmonic on $V$. We know that the generalized solution of ...

**1**

vote

**1**answer

133 views

### A question on the problem of Dirichlet

Suppose $U$ is an open set in $\mathbb{R}^{n}$ ($n\geq2$) whose complementary is not polar, and $f$ is a real-valued function defined at least on the boundary of $U$. We know that the generalized ...

**0**

votes

**1**answer

66 views

### Extension of superharmonic functions

Let $V$ be a bounded open set in $\mathbb{R}^{n}$ with $n\geq2$ and $W$ be an open neighborhood of the boundary $\partial V$ of $V$. If $u$ is superharmonic on $W$, is there a way to extend $u$ to a ...

**3**

votes

**1**answer

311 views

### White noise vs. black noise

In this excellent lecture ("2d Percolation Revisited") Stanislav Smirnov mentioned the connection of the theory of percolation with the notion of the so called black noise—see at 29:42 (the notion ...

**1**

vote

**1**answer

82 views

### Explicit densities for Brownian motion hitting times

I'm looking for functions $g: \mathbb{R}_+ \to \mathbb{R}$ such that the hitting time
$$\tau := \inf \{t \geq 0 : B_t \nleq g(t) \} $$
has an explicit density with respect to the Lebesgue measure, ...

**0**

votes

**0**answers

38 views

### A set of zero harmonic measure 2

Let 𝑉 be a bounded open set in $\mathbb{R}^{m}$, $m\geq 2$, and 𝑊 the interior of the closure of 𝑉. Let 𝐸 be a subset of ∂𝑉∩𝑊 (∂ means boundary) such that:
1) $E$ has positive ($m-$dimensional) ...

**2**

votes

**1**answer

95 views

### A set of zero harmonic measure

We know that a set may be of zero harmonic measure without its Lebesgue measure being zero (see Armitage and Gardiner, classical potential theory, pg 178).
Now, consider the following problem. Let $...

**0**

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35 views

### differential of posterior probability distribution over the mean drift of brownian motion

Let $W_t$ be the Weiner process, and let $X_t = W_t + \mu t$, where $\mu$ is either 0 or 1. We wish to get information about $\mu$ by looking at $X_t$. Let $q_t$ be the probability we assign to $\mu=1$...

**3**

votes

**1**answer

152 views

### Brownian level sets and continuous functions

Let $V_t$ and $W_t$ be independent standard Wiener processes ($t\ge 0$, $W_t,V_t\in\mathbb R$).
Let $C$ be the event that there is a continuous function $f$ such that for all $s$, $t$,
$$
W_t=W_s\iff ...

**1**

vote

**2**answers

220 views

### Thinness and polarity

Let $D$ be a bounded open set in $\mathbb{R}^{n}$ with $n\geq2$ and $E$ a subset of the boundary $\partial D$ of $D$. $D$ is said to be thin at a point $y\in D$ if there is a superharmonic function $u$...

**2**

votes

**1**answer

150 views

### Pathwise stochastic integral as a linear operator on continuous functions

Let $B$ be a Brownian motion. Definining a pathwise stochastic integral $I(f):=\int f~dB$ for certain classes of deterministic functions is straightforward: For instance if $f=\sum_ic_i1\{[t_i,t_{i+1})...

**-1**

votes

**1**answer

123 views

### Capacity and harmonic measure

Suppose $D$ is a bounded domain of $\mathbb{R}^{n}$ with $n>1$ and $E$ a subset of its boundary. We know that if $E$ has capacity zero I.e. it is a polar set , then the harmonic measure of $E$ ...

**2**

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**0**answers

58 views

### Sequence of harmonic measure

There is a well-known result stating that if $\mu_{n}$ is a sequence of uniformly bounded measures on a compact set $E$ of $\mathbb{R}^{m}$, then there is a subsequence $\mu_{n_{j}}$ that converges ...

**1**

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**1**answer

64 views

### A question about harmonic measure 2

Suppose $W$ is a bounded open subset of $\mathbb{R}^{n}$ and $n\geq2$. Let $V$ be the interior of the closure of $W$ and $E$ a subset of the boundary of $V$. If $\omega(x,W)(E)=0$ ($\omega(x,W)$ is ...

**0**

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**0**answers

50 views

### A question on harmonic measure

Suppose $W$ is a bounded open subset of $\mathbb{R}^{n}$ and $n\geq2$. Let $V$ be the interior of the closure of $W$ and $E\subset W$. If $\omega(x,V)(E)=0$ ($\omega(x,V)$ is the harmonic measure of $...

**0**

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93 views

### Wiener measure in the space of functions of two or more variables

Wiener measure in the space of continuous functions of two variables had been introduced by J. Yeh in the 1960 paper "Wiener Measure in a Space of Functions of Two Variables" (AMS free access)
(p. ...

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86 views

### Beta distribution and Wiener process

Suppose $W(t)$ is the standard Wiener process on $[0; 1]$ and $\{T_x\}_{x \in \mathbb{R}}$ is a collection of random variables defined by the following relation:
$$T_x = \mu(\{t \in [0;1] | W(t) > ...

**0**

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**0**answers

90 views

### Conditioned Brownian motion?

Let $U\subseteq (C_0[0,1];\mathbb{R})$ be an open subset of the Wiener space satisfying $0<\gamma(U)<1$; where $\gamma$ is the Wiener measure and let $W_t$ be the standard (Wiener) coordinate ...

**2**

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**2**answers

212 views

### Probability space with exactly one Brownian motion

Very recently, the following question was asked:
Often, we encounder the assumption that $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ is a stochastic base on which a Brownian motion is defined. ...

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40 views

### What is the Wiener measure of the set of curves with given Hölder constant on a Riemannian manifold?

Let $M$ be a connected Riemannian manifold and $x_0 \in M$. For $0 < \alpha < \frac 1 2$, let
$$H = \{ c : [0,1] \to M \mid c(0) = x_0 \text{ and } \exists C>0 \text { s.t. } d(c(s), c(t)) \...

**-2**

votes

**1**answer

112 views

### Brownian motion and Durret book [closed]

I have a problem to understand the following simple definition in Durrett book: Brownian motion and martingales in analysis. What does the following mean: $T = \inf \{t: B_t \in A\}$. It seems to ...

**0**

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**1**answer

268 views

### Ito integral and true martingale

Consider a twice diferentiable function $F$ on $R$ with bounded
first derivative $F'$ and a Brownian motion $W$. Show that $F(W_t)-\frac{1}{2} \int_{0}^{t} F'' (W_s)ds$ is a true martingale.
I tried ...

**0**

votes

**2**answers

954 views

### Quadratic covariation of two not independent Brownian motions

Given two not independent Brownian motions, $X$ and $Y$. I was wondering if we can say anything about the quadratic covariation of $X$ and $Y$, $\langle X,Y \rangle_t$. I know that for two independent ...

**4**

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**2**answers

111 views

### Density near at $0$ for the integral of the positive part of the Brownian motion

This question was asked recently on MO and then deleted by the owner, user Aalon. I think the question deserves to be answered, which is what I will try to do here. Aalon was reading this paper, where ...

**1**

vote

**0**answers

62 views

### Smoothness of expectation

Suppose that $X_t$ is a strong solution to the SDE,
$$dX_t = C_t \,dB_t$$ where $B_t$ is a standard Brownian motion and $C_t \ge 0$ is measurable with respect to the natural filtration generated by ...

**6**

votes

**0**answers

102 views

### Probabilistic characterization of first Neumann eigenvalue

In this MO post, a question has been asked (and answered) about the probabilistic interpretation of the first Dirichlet eigenvalue of the Laplacian in terms of boundary hitting times.
I wish to ask ...