All Questions
Tagged with stochastic-processes brownian-motion
219 questions
3
votes
1
answer
397
views
Fractional Brownian motion via Hilbert space
The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms:
Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, \...
- 2,703
6
votes
0
answers
220
views
Reference request: Stochastic integration and martingale theory on the whole real line
I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...
- 617
7
votes
1
answer
467
views
Properties of the algebraic self-difference set of Brownian motion zeros
As I was trying to exhibit new interesting(?) path transformations of Brownian motion, I became interested in
the (random) set of times $t$ such that $B(t)=B(t+1)=0$, where $B(t)$ denotes a standard ...
- 4,660
9
votes
3
answers
2k
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When is a continuous path stochastic process be representable as diffusion or Ito process?
When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
- 2,703
4
votes
0
answers
129
views
Tail for the integral of a diffusion process
I would like to compute the following tail,
$$
\mathbb{P}\left(\int_{0}^{T} f(X_t)\mathrm{dt}>x\right),
$$
assuming
$$
\mathbb{P}[f(X_t)>x] = x^{-\alpha} \log(x),
$$
and $X$ is a diffusion ...
- 5,670
2
votes
0
answers
646
views
Fundamental theorem of calculus for iterated stochastic integrals
I'm trying to find the rate (or a bound for it) with which an iterated integral of the type
$$\int_{-h}^0 \int_{-h}^{t} A_s d B_s A_t d B_t$$
converges to zero (in probability/distribution) for $h \...
- 105
1
vote
0
answers
160
views
number of times Brownian motion hits boundaries
Any experts here please direct me to some appropriate keywords that I can search for. Consider a Brownian motion constrained to an upper and lower boundaries. Let's say I want to know that how many ...
- 3,958
4
votes
2
answers
416
views
Probability of winding number of 2D Brownian Motion
Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau \...
CommunityBot
- 1
8
votes
1
answer
568
views
Escape Time of Fractional Brownian Motion
Let $B(t)$ be Brownian motion with $B(0)=x>0$ and let $A>x$. It is well known that the expected time for $B(t)$ to escape the interval $[0,A]$ is equal to $x(A-x)$.
Is the expected time known ...
- 2,703
1
vote
0
answers
1k
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What is the characteristic functional for Brownian motion on a sphere?
I'm a physicist, somewhat familiar with stochastic processes, but I'm a little unsure of what follows. What I basically have is a complicated quantity involving a vector that is equivalent to ...
- 111
4
votes
1
answer
2k
views
Expectation of the time t standard brownian motion stopped at itself's square
I have a one dimensional standard brownian motion $W$ defined under a stochastic basis with probability $\mathbf{Q}$ and filtration $\left(\mathscr{F}\right)_{t\in{\mathbf{R}}_{+}}$, and I want to ...
- 29.8k
10
votes
2
answers
1k
views
Does the strong law of Large Number hold for an infinite dimensional Brownian motion?
For finite-dimensional Brownian motion $W_t$, it is well known that
\begin{equation}
\lim_{t\to \infty}\frac{W_t}{t}=0,\text{ a.s. }\ \ \ \ \hspace{1cm} \langle 1\rangle
\end{equation}
Now suppose we ...
- 17k
10
votes
1
answer
553
views
Can one use Brownian motion to prove that two manifolds are not conformally equivalent?
Let me start by a very simple example; consider the following question:
"Let D1 be a square and D2 a rectangle (boundary included). View them
as subsets of the complex plane. Does there exist a ...
CommunityBot
- 1
3
votes
2
answers
325
views
Ito Diffusions with low regularity?
I would like to have an Itô Diffusion
$$ X_t = \int_0^t b(s) \mathrm{d}s + \int_0^t \sigma(s) \mathrm{d}B_s.$$
where the (vector- and matrix-valued, respectively) functions $b$ and $\sigma$ have lower ...
- 3,069
1
vote
1
answer
233
views
Can we express a one-dimensional raised Bessel Bridge as a function of a single Brownian Motion?
A Bessel Bridge is a Brownian Motion, conditioned such that $B(0) = B(1) = 0$ and $B([0, 1]) \ge 0$. A raised Bessel Bridge is a generalization of this: it's a Brownian Motion conditioned such that $...
- 4,010
2
votes
0
answers
392
views
Differentiability of integral w.r.t. hitting time of Brownian Motion
I have been trying to prove the following conjecture for a while, but so far to no avail. Would be very grateful for some tips!
(I edited the entire thing to make it clearer)
The conjecture is the ...
- 21
4
votes
1
answer
773
views
SDE-removal of the diffusion coefficients
from math.stackexchange
I'm currently looking at stochastic differential equations with irregular coefficients such as $W^{1,p}_{loc}$. If I have
\begin{align}
dX_t=b(X_t)dt+\sigma dW_t,
\end{align}
...
- 931
6
votes
0
answers
646
views
Integrating a Bessel Bridge
Preliminaries
An order-3 Bessel Process is the one-dimensional stochastic process $X$ described by $X(t) = \sqrt{W_1(t)^2 + W_2(t)^2 + W_3(t)^2}$, where each $W_k$ is an independent Brownian Motion. ...
- 693
2
votes
2
answers
571
views
Family of Brownian Motions
I am trying to show the following statement
Let $D\subset \mathbb{R}^2$ be an open and bounded subset. $\Pi=(P^x : x \in D )$ a Family of standard Brownian Motions started at $x \in D$. Then $\Pi$ ...
- 279