Questions tagged [brownian-motion]
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251
questions
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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
vote
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
$$\int_0^1 F(B_t)~dt=\int_\mathbb R F(x)L(x)~dx,\qquad\...
0
votes
0answers
29 views
Constructing uncountably many independent random variables with same distribution from Brownian motion?
It is well known one cannot construct uncountable many independent random variables on $([0, 1], \mathcal{B}[0, 1], \lambda)$. ($\lambda$ Lebesgue measure.)
Also, one can clearly construct infinitely ...
0
votes
0answers
41 views
How to find the PDE for the following transition density
Suppose I have the following two stochastic differential equations ($t\geq 0$)
$$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t \ \ \text{ and } \ \ dZ_t =dt,$$
where $X = (X_t)$, $Z = (Z_t).$
Note that
$W=(...
0
votes
2answers
110 views
Transience of 3-dimensional Brownian motion
I'm attempting Exercise 5.33 of Le Gall's Brownian motion, Martingales and Stochastic Calculus.
Let $B_t$ be a 3-dimensional Brownian motion starting from $x$.
Part 6 asks me to show that
$$|B_t| = |...
0
votes
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
vote
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
votes
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
vote
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
vote
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
votes
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
votes
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
vote
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
$$H = \{ c : [0,1] \to M \mid c(0) = x_0 \text{ and } \exists C>0 \text { s.t. } d(c(s), c(t)) \...
-2
votes
1answer
112 views
Brownian motion and Durret book [closed]
I have a problem to understand the following simple definition in Durrett book: Brownian motion and martingales in analysis. What does the following mean: $T = \inf \{t: B_t \in A\}$. It seems to ...
0
votes
1answer
268 views
Ito integral and true martingale
Consider a twice diferentiable function $F$ on $R$ with bounded
first derivative $F'$ and a Brownian motion $W$. Show that $F(W_t)-\frac{1}{2} \int_{0}^{t} F'' (W_s)ds$ is a true martingale.
I tried ...
0
votes
2answers
954 views
Quadratic covariation of two not independent Brownian motions
Given two not independent Brownian motions, $X$ and $Y$. I was wondering if we can say anything about the quadratic covariation of $X$ and $Y$, $\langle X,Y \rangle_t$. I know that for two independent ...
4
votes
2answers
111 views
Density near at $0$ for the integral of the positive part of the Brownian motion
This question was asked recently on MO and then deleted by the owner, user Aalon. I think the question deserves to be answered, which is what I will try to do here. Aalon was reading this paper, where ...
1
vote
0answers
62 views
Smoothness of expectation
Suppose that $X_t$ is a strong solution to the SDE,
$$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
votes
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 ...
