Questions tagged [brownian-motion]
Questions related to Brownian motion, a continuous stochastic process denoted by $W_t$, $t\geq 0$, with independent increments, such that $W(t)-W(s)$ is normally distributed, with $0$ mean and variance $t-s$.
4,396
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PDE on the probability that the brownian motion stays in [a,b]
This was part of an exam question.
Let $a<b$, $(X_t)$ a brownian motion and $$\forall x \in \mathbb R, t\ge 0, \quad \pi(x,t):=P(\forall s \in [0,t], x+X_t \in [a,b]).$$
Given that $\pi$ is $C^2$ (...
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Brownian motion construction on the unit sphere in Oksendal text book
In example 8.5.8 of Oksendal text book, the Brownian motion is constructed on the unit sphere.
I don't understand how he defines the time change $Z_t(\omega) = Y_{a(t,\omega)}(\omega)$. He defines the ...
- 67
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13
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Intuitive explanation for Feller processes
We know that a Markov process is a stochastic process in which the future evolution of the system depends solely on its current state and is independent of its past states given the present state. ...
- 31
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9
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Why A Lévy process may be viewed as the continuous-time analog of a random walk
It seems that paths of the Brownian motion are always continuous (please correct me if i am wrong).
Levy process is the generalization of Brownian motion by allowing for jumps at random times.
So why ...
- 31
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12
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Power Spectrum Density of cosine (or sine) of stochastic process related to Bronian Motion
Suppose $W(t)$ is a standard Bronian Motion (or a Wiener Process), thus for $0\le s<t$, we have $$W(t)-W(s)\sim N(0,\sigma^2(t-s)), W(0)=0$$
Now define another stochastic process $A(t)$ as
$$A(t)=...
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0
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18
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A question on placing an upper bound on the probability of a standard brownian motion exiting an interval in a given time interval
I'm supposed to show that there exists a $c > 0 $ such that, for each $P\left(|B_t| \leq \varepsilon, \forall t \in [0, 1]\right) \leq e^{-\frac{c}{{\varepsilon}^2}}$. We're given the hint: $B_{k\...
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Is pre-Brownian motion unique?
I am trying to understand the notion of pre-Brownian motion using the Book "Brownian Motion, Martingales, and Stochastic Calculus" by J. Le Gall.
He first define the Gaussian white noise to ...
- 181
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1
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There exists a probability space on which standard Brownian motion exists
I wanted to learn basics of the Brownian motion, and I started with the introductory book "Continuous Time Markov Processes: An Introduction" by Ligget. I have difficulty understanding ...
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53
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Ito's Lemma for Process given Generator
I am going through a paper called "The KPP Equation as a Scaling Limit of Locally Interacting Brownian Particles", https://arxiv.org/abs/2101.01031v2. We construct a process by defining a ...
- 301
-1
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1
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How to understand brownian-motion is not monotonous in ant interval?
Just like the title, I really can not understand how brownian-motion can be nonmonotonous. I thinck the probability of a brownian-motion moving in same direction in a really short interval can be a ...
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31
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Solution manual for "Continuous Time Markov Processes" by Liggett
Is there a solution manual for "Continuous Time Markov Processes: An Introduction" by Thomas M. Liggett?
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How to compute $E[B_{1}^2B_{2}^2]$ where $B_{t}$ is a standard Brownian motion starting at 0 [closed]
We are given that $E[X^4] = 3\sigma^2$ for gaussian X, and I've tried solving for $E[(B_1^2+B_2^2)^2]$ to use this fact, but I really am not clear on what to do
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Harmonic Measure is absolutly continuous with respect to Lebesgue measure.
currently I am reading Greg Lawler's book conformally invariant processes and I am already having some experience with harmonic measures and calculating explicit Poisson kernels and related exit ...
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1
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29
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Understanding correlation in the context of a time series, simulation and brownian motion
I have the doubt of calulating and meaning of correlation. I know it is from my incapacity to grasp a concept, specially regarding time series but would appreciate any comments on it.
I think I ...
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15
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Continuity of stationary measure of reflected Brownian motion in higher dimensions
For Brownian motion contained in a closed region (specifically a polyhedron), is it true in general that the stationary density on the boundary is continuous with the value on the interior, in a sense ...
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Brownian motion and Holder-$\frac{1}{2}$-continuity
Let B be a Brownian motion. For every $K>0$, we have
$$
P[\inf \left \{ t>0: B_t\geq K t^{1/2} \right \} =0]=1 \quad\quad\quad(1)
$$
To prove this in Example 21.16 of Probability Theory (3rd ...
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43
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Path continuity of Time Inversion of a Brownian motion: proof doubt
Let $(B_t)_{t\geq 0}$ be a Brownian motion. Define $X_t=tB_{1/t}$ if $t>0$ and $X_t=0$ if $t=0$.
In proving that $X$ is a Brownian motion, A. Klenke in Theorem 21.14 of his book Probability Theory (...
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Donsker's theorem for multivariate Brownian motion [closed]
Let $B = (B^1, \dots, B^n)$ be an $n$-dimensional Brownian motion (i.e. $B = (B_t)_{t \geq 0} \in \Omega \rightarrow (\mathbb{R}^n)^{[0,\infty)})$.
Do we have something as Donsker's theorem to show ...
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Probability of the behavior of a Wiener process : $\mathbb{P}\{W_1>0, W_5 <0\}$
I want to find the following probability :
$\mathbb{P}\{W_1>0, W_5 <0\}$
Where $W_t$ is a Wiener process, so it follows the law $\mathcal{N}(0, t)$.
My question is : can say that $\{W_1>0\}$ ...
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1
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37
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Zero Set of the Minimum of Brownian Motions
We know that the zero set of a Brownian motion $(B(t))$, $T:=\{s\in [0,1]:B(s)=0\}$, is almost surely homeomorphic to the Cantor set. I would like to prove that the zero set of the minimum the ...
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Meaning of the terms in the infinitesimal generator formula
In the following formula
$$Af(x) := \lim_{t \downarrow 0} \frac{\mathbb{E}^x(f(B_t))-f(x)}{t} $$
If $B_t$ is the Brownian motion, what are $f(B_t)$ and $f(x)$? here some explanation was given. Can ...
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34
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Independence of increments of Wiener processes
Let $W_t$ be a Wiener process.
In the demonstration provided below (which is right), it is said, knowing $s<t$, that $W_s$ and $W_t - W_s$ are independent because increments of Wiener processes are ...
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1
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58
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sde for brownian bridge
Let $B(t)$ be a standard Brownian motion and
\begin{align*}
Y(t)=B(t)-tB(1) &&
Z(t) = \left\{
\begin{array}{ll}
Z(t)=(1-t)B\left(\frac{t}{1-t}\right)& t\in [0,1)\\
0 & t=1 \\
\end{...
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1
answer
75
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Hitting time of Brownian motion past a given point in time
The random variable whose distribution I am interested in is defined as follows:
$$\tau := \inf\{u > 1: W_u = 0\}$$
where $W$ is Brownian motion.
I derive the distribution below but it doesn't ...
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1
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77
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Infinitesimal generator of Brownian motion on the unit sphere
The infinitesimal generator of a standard Brownian motion (as Markovian process) in $\mathbb R$ can be computed with
$$Af(x) := \lim_{t \downarrow 0} \frac{\mathbb{E}^x(f(X_t))-f(x)}{t} = \lim_{t \...
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59
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Show that $Z(t)=\exp{\sigma X(t)-(\sigma \mu + \frac{1}{2} \sigma^2)t}$ is a martingale
I'm trying to show that $Z(t)=\exp{(\sigma X(t)-(\sigma \mu + \frac{1}{2} \sigma^2)t)}$ is a martingale.
Attempt:
I want to show that $E[Z(t)|\mathcal{F}(s)] = Z(s)$
$E[Z(t)|\mathcal{F}(s)] = E[Z(t)/ ...
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1
answer
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Why $\limsup_{t\rightarrow s} \frac{B_t-B_s}{f(t-s)}$ is not $\mathcal F_s$ measurable?
Let $(B_t)_{t\geq 0}$ be a Brownian motion equipped with its natural filtraton $\mathcal F_t = \sigma(B_s, 0 \leq s\leq t)$ and let $f$ be a measurable function such that $f(x) >0$ for all $x >0$...
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Distribution of AR(1) process with unit roots - why does one need the Functional Central Limit Theorem?
I was reading Chapter 17 of Hamilton's "Time Series Analysis" about univariate processes with unit roots. In particular, I am looking at the AR(1) process $y(t) = y(t-1) + \epsilon(t)$ with $...
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11
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Why I can numerically use signatures on rough paths?
In the paper deep signature transforms ({bonnier, kidger, perez, salvi, tlyons}) they use the signatures on Brownian motion and they invert it with the inversion method defined in (The insertion ...
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Infinitesimal generator of brownian motion [closed]
It the literature we see that the infinitesimal generator of Brownian motion is $\frac{1}{2} \Delta$.
However, when we search about the generator of the Brownian motion on the (unit) sphere, we see ...
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1
answer
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Expected value $\mathbb{E}\left(\int_0^t e^{B_s}ds\right)$
I am asked to solve this problem.
Let $B$ be a BM and $t>0$ a real number. What is the value of :
$$ \mathbb{E}\left(\int_0^t e^{B_s}ds\right) \text{ ?}$$
I have tried to use Ito formula to have ...
- 305
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23
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From Large Deviations to Finite Time Probability Tails
Let $(B_t)$ be a standard $d$-dimensional Brownian motion. It is well-known that
$$\mathbb P(\sup_{s\in[0,t]}|B_s|\ge \alpha) \le 4de^{-\alpha^2/2dt}.$$
One possibility to obtain such a result is ...
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Prove that Brownian Motions have normal distribution using central limit theorem
In the book Brownian Motion, 3rd edition by Rene Schilling, he defines a $d$-dimensional Brownian motion $B = (B_t)_{t\geq0}$ indexed by $[0,\infty)$ taking values in $\mathbb R^d$ as a process that ...
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1
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Fractional Brownian Motion is not a semimartingale
In some sources (for example here: Why is a fractional Brownian motion not a semi-martingale?) you can find a proof why the fractional Brownian motion is not a semimartingale via the Bichteler-...
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Proving an estimate of SDE process under local Lipschitz continuity while the linear growth [duplicate]
$X_t$ is a process defined by an SDE:
$d X_t=\mu\left(t, X_t\right) d t+\sigma\left(t, X_t\right) d B_t$
Under the global Lipschitz continuity and linear growth conditions, suppose that $E\left[\|\xi\|...
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52
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conditional expectation of a stochastic process including Brownian motion
Let $B(t)$ be a standard Brownian motion and $X(t)=t^2B((t+1)^2)+\mu \cdot f(t)$. How can I calculate $E[X(t)|X(t_0)=x_0]$?
I started with $E[X(t)|X(t_0)]=t^2E[B((t+1)^2)+\frac{\mu f(t)}{t^2}|B((t_0+1)...
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2
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Brownian Motion Conditional Characteristic function
I am trying to prove that a one dimensional continuous process $B=(B_t)_{t \geq 0}$ with $B_0 =0$ is a standard Brownian motion if and only if for $\epsilon \in \mathbb{R}$ and $t > s$:
$$
\mathbb{...
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49
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Expectation and Variance of $Y_t = \int_0^t sdW_s$ + Martingale property
Denote the process $Y_t = \int_0^t sdW_s$. I want to answer the following question:
Q) Calculate the expectation and variance of $Y_T$. Is $Y_T$ a martingale?
A) I have read somewhere that it was ...
1
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0
answers
16
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what is the dimension of the intersection of the planar brownian range with a line
We consider two dimensional standard brownian motion $B: t\mapsto (B_{1}(t),B_{2}(t))$. Let $D$ be its range (that is the image of $[0,+\infty[$ by $B$ i.e, $B([0,+\infty[)$).
Is there some known ...
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68
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Probability of reaching ruin time with Brownian motion with positive drift
Question
You are about to start a new job. As your new job is quite well paid, and you are generally responsible with money, you are confident that your income will exceed your outgoings and that your ...
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2
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42
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$\mathbb{E}_x[f(B_t)]$ is strongly continuous contraction semi group, $B_t$ Brownian Motion, where is my mistake?
For $t \geq 0$, we define a family of operators $P_t f(x) = \mathbb{E}_x[f(B_t)]$, where $B_t$ is one-dimensional Brownian Motion and $f \in C_0(\mathbb{R})$, i.e. continuous $\mathbb{R} \rightarrow\...
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0
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12
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Von Neumann stability analysis for radial diffusion in a sphere
I have to solve a diffusion probleme, radial diffusion into or out of a sphere. So far so good, I have the explicit solution via difference quotient.
But I have trouble evaluation the stability ...
- 31
2
votes
1
answer
54
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class of processes guaranteed to exceed any thresholds
Is it correct to say that any martingale (e.g., a GBM or anything similar), taking values in some real state space and not converging to a degenerate random variable, has the property that, given a ...
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30
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How do Brownian/Wiener processes involve randomness?
My financial mathematics course notes have
A Brownian motion is a family of random variables $\{B_t|t\geq0\}$ on some probability space $(\Omega,\mathcal{F},P)$ such that: \begin{align}
(1) \; & ...
- 2,051
2
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69
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Non-negativity properties of SDE with singular drift at zero (with Bessel process as special case)
Consider the following family of SDEs:
$$
dX_t = \frac{1}{X_t^p} dt + dW_t
$$
with $X_0 > 0$.
For $p=1$, this is the 3-dimensional Bessel process, which stays positive for all time almost-surely.
...
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1
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1
answer
33
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Returns of an asset in risk-neutral measure and its PDE
I am a bit confused regarding how an asset returns in a risk-neutral measure (say $\mathbb{Q}$), and subsequently its Black-Scholes-esque PDE.
In class, we learned about the approach to take when ...
0
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0
answers
20
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Wiener Process Textbook or Reference Specifically Containing Ramp Intersection (or Exit Time) Analysis
I am looking for a reference (preferably a textbook so that additional preparation material is handy) that calculates the exit time of a Wiener process from a region bounded by sloped lines.
Thank you,...
- 1
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0
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52
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Probability of a stopping time to be lower than another stopping time
I was wondering on how to compute the probability of a stopping time being lower than another stopping time.
More in details, just consider a drifted Brownian motion process $X_t$, and consider two ...
- 53
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0
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16
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What is the density of log-normal geometric Brownian motion $x=log\left(\frac{S}{K}\right)$
If $S_t$ is a GBM with $$S_t \sim LN\left(S_0 e^{\mu t + \frac{1}{2} \sigma^2 t} , \quad S_0e^{2 \mu t + \frac{1}{2} \sigma^2 t}\left(e^{\sigma^2 t} -1\right)\right) $$
Then if $x=log\left(\frac{S}{K}...
1
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1
answer
79
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Combining Convergence in Distribution and Almost Surely
Suppose I know that $\frac{1}{\sqrt{m}}X(mt)\xrightarrow[m\to\infty]{d} B(t)$ where $B$ is Brownian motion, and that $\alpha(mt)/m \xrightarrow{a.s.} 0$. Then I am trying to prove the following ...
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