Questions tagged [brownian-motion]

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2
votes
0answers
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
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0answers
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
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0answers
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
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1answer
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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0answers
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
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0answers
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
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2answers
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
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0answers
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
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1answer
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
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0answers
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
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1answer
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
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0answers
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\...
0
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0answers
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
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0answers
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
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2answers
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
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1answer
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
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2answers
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
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1answer
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
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0answers
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
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0answers
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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0answers
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
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1answer
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
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0answers
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
1answer
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
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1answer
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
1answer
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
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1answer
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
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1answer
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
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1answer
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
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0answers
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
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1answer
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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0answers
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
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1answer
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
2answers
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
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1answer
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
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1answer
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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0answers
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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1answer
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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0answers
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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0answers
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. ...
3
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0answers
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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0answers
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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2answers
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. ...
2
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0answers
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
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1answer
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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1answer
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
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2answers
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
votes
2answers
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
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0answers
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
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0answers
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 ...

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