Brownian motion Definition and 99 Threads

Hi! Probably I am just confused, but why for the exact solution of the geometric brownian motion dX_t = \mu X_t dt+\sigma X_t dW_t we have to apply Ito's lemma and manipulate the expression obtained with dlogX_t? Couldn't we directly use the espression dX_t / X_t = dlogX_t in the equation dX_t /...

Can someone please explain how the standard deviation is relevant when we are talking Brownian motion? And why is statistics important when we want to understand the movement of a brownian particle?

Homework Statement A grain of pollen shows Brownian motion in a solvent, such that the position x(t) on the x-axis varies with time. The displacement during one second, x(t + 1) - x(t), is measured many times and found to have a Gaussian distribution with an average of 0 and standard devation...

I need to show that (Wt)2 is a brownian motion So let Vt = (Wt)2 I need to first show that Vt+s - Vs ~ N(0,t) Vt+s - Vs = (Wt+s)2 - (Ws)2 = (Wt+s + Ws)(Wt+s - Ws) (Wt+s - Ws) ~ N(0,t) But is (Wt+s + Ws) ~ N(0,t)? If it is what happens when I multiply two RV's that are normally...

Hi guys, It's been a while since high school, and now I'm faced with a problem I need to solve in a few days (attached). Would someone please help me through that? I would really appreciate support.

1) For each $t$, find $P(B_t\neq 1)$ . 2) For any $T>0$, prove $V_t=B_{t+T}-B_T$ is a Weiner process. ... For 2) should I be looking at something like this: Let: $\Delta (t+T)=\frac{t}{N}$ for large $N$ $\Rightarrow...

I need to answer these questions, but I don't have a clue what they mean. Could anybody shed some light? Find: (a) $E({B_1^4})$ (b) $E({B_1^6})$ (c) $E(e^{B_1})$ (e) $E({5B_1}^4+{6B_1}^2+{5B_1}^3)$ (e) $E(B_2 B_3)$ (f.) $E(e^{B_2+B_3})$

Hello, Given a Brownian Motion process B(t) for 0≤t≤T, we can write it more explicitly as B(t,ω) where ω\inΩ, where Ω is the underlying sample space. My question is: what is the cardinality of Ω. I.e. what is |Ω|? My thoughts are that it is an uncountable set, based on the observation...

I am reading "Quantum Mechanics and Path Integrals" (by Feynman and Hibbs) and working out some of the problems... as a hobby of sorts. I have run into a problem in section 12-6 Brownian Motion. On page 339 (of the emended edition), the authors demonstrate, by example, a method for...

Hi all, I feel like there's a missing link in my understanding of brownian motion. I'm comfortable with the "method of http://fraden.brandeis.edu/courses/phys39/simulations/Uhlenbeck%20Brownian%20Motion%20Rev%20Mod%20Phys%201945.pdf" where the signal is written as a Fourier series, and with...

http://xanadu.math.utah.edu/java/brownianmotion/1/ I can't imagine

Problem: Let M(t) = max X(s), 0<=s<=t Show that P{ M(t)>a | M(t)=X(t)} = exp[-a^2/(2t)] Attempt at solution: It seems this should equal P(|X(t)| > a), but evaluating the normal distribution from a to infinity cannot be expressed in closed form as seen in the solution (unless this is...

Problem: Let X(t), t>0 denote the birth and death process that is allowed to go negative and that has constant birth and death rates Ln = L, un = u (n is integer). Define u and c as functions of L in such a way that cX(t), t>u converges to Brownian motion as L approaches infinity. Attempt...

Let B_t be Brownian motion in \mathbb R beginning at zero. I am trying to find expressions for things like E[(B^n_s - B^n_t)^m] for m,n\in \mathbb N. So, for example, I'd like to know E[(B^2_s - B^2_t)^2] and E[(B_s - B_t)^4]. Here are the only things I know: E[B_t^{2k}] = \frac{(2k)!}{2^k...

Homework Statement Suppose B_t is a Brownian motion. I want to show that if you fix t_0 \geq 0, then the process W_t = B_{t_0+t} - B_{t_0} is also a Brownian motion.Homework Equations Apparently, a stochastic process X_t is a Brownian motion on \mathbb R^d beginning at x\in \mathbb R^d if it...

Let me start off by saying that I very well know that PM is impossible. Thermodynamics aren't just good ideas-they're the law. :) I have heard that Brownian Motion will go on infinitely, but you can't harness it and it is useless perpetual motion. Is this true?

Is there a theory regarding why particles move in random paths. My high school physics teacher said it's energy left over from the big bang, but that doesn't explain why they move in random paths, they could just as easily obey Newton's Laws of Motion and still have energy left over from the...

Hello everybody! I'm working on a Stochastic Processes course based on Ross' Introduction to Probability Models. I hope you can help me work through this problem. Homework Statement {B(t), t >= 0} is a standard Brownian motion process Compute E[B(t1)B(t2)B(t3)] for t1 < t2 < t3...

Dear list, Imagine a table of upper surface area S sitting in an open field, with nothing on it. We know that S is subjected to a downward atmospheric pressure P due to a cylindrical column of air of volume V extending vertically from S to the end of the terrestrial atmosphere. Assume this...

Homework Statement No specific problem to solve, just looking for a better explanation of the implications of the quadratic variation not being zero in Brownian motion. Why is this so important in the study of stochastic calculus and Brownian motion? I understand that quadratic variation in...

I watched a show on Fractals and it sort of remind me of Brownian motion. So my question is has anyone ever used fractals to explain Brownian motion?

for a brownian motion W(t) W(t_i+1)-W(t_i) is normal distribution with mean 0 and variance t_i+1-t_i so this means var(W(t_i+1)-W(t_i))=var(W(t_i+1))-var(W(t_i))=t_i+1-t_i I don't think the above equation satisfies because W(t_i+1) and W(t_i) are not independent. Any comment? thanks

Hi: I want to know the quadratic covariation of Brownian motion B(t) and poisson process N(t).Is it B(t)? Thanks !

I'm trying to write a fokker Planck equation for a particular SDE, but I'm caught up on an aside by the author I'm following. He has a SDE with drift b \in \mathbb{R}^n, a dispersion matrix \sigma \in \mathbb{R}^{n\times k}, and k-dimensional brownian motion W_t, resulting in something like this...

Using Brownian Motion to solve for 4 things PLZ HELP! Brownian motion. Molecular motion is invisible in itself. When a small particle is suspended in a fluid, bombardment by molecules makes the particle jitter about at random. Robert Brown discovered this motion in 1827 while studying plant...

I am quite well versed with the random walk problem and am interested in finding out more about Brownian motion. Does anyone have any suggestions for books that explain Brownian motion in detail? I suspect these will be books on statistical mechanics.

If Brownian motion is continuous, why then is it not inherently deterministic? Are the events that Brownian motion covers based on previous states and causal factors? So, unpredictable (too many variable at play), yet causal? What am I missing here?

Hi, I am studying brownian motion and the Black-Scholes formula. Our problem assumes that 1. Stock returns follow a normal distribution 2. Based on #1 the stock price follows a lognormal distribution because y = exp(X) is lognormal if X is normally distributed. Here the stock prices...

Homework Statement Let Bt be a standard Brownian motion. Let s<t: a) Compute P(\sigma B_{t}+\mu t|B_{s}=c) b) Compute E(B_{t}-t|B_{s}=c) Homework Equations Defition of brownian motion: B(t) is a (one-dim) brownian motion with variance \sigma^{2}if it satisfies the following conditions: (a)...

Hi, I'm trying to prove that X=(X_{t})_{t\geq0} is a Brownian Motion, where X_{t} = tB_{1/t} for t\neq0 and X_{0} = 0. I don't want to use the fact that it's a Gaussian process. So far I am stuck in proving: \[ X_{t}-X_{s}=X_{t-s} \quad \forall \quad 0\leq s<t \] Anyone has any ideas?

This is an exerxise from G. Lawler's book on conformally invariant processes. Show there exist constants 0<a,c<inf such that the probability p for a planar Brownian Motion B_t starting at x makes a loop around the origin before hitting the unit circle is at least 1-c xa. I have not really an...

Homework Statement Show that \frac{X ( a^2t) }{a} is a brownian motion.Homework Equations http://img168.imageshack.us/img168/8453/83818601fz4.png The Attempt at a Solution I found this in my lecture notes but isn't the proof just replacing (t-s) by a^2(t-s) and s by a^2 s and dividing...

I'm not sure where this belongs but I figure that this is the right place for it. Homework Statement The equation is m*v'(t)+\mu*v(t)=f(t), where m is mass, \mu is the drag coefficient, and f(t) is some random function. I am asked to find the values for v(t), <v2>, and <x2> Homework...

Homework Statement Let \{ B(t) \}_{t \geq 0} be a standard Brownian motion and U \sim U[0,1] and {Y(t)}_t\geq0 be defined by Y(t) = B(t) + I_{t=U}. Verify that Y(t) is a Gaussian process and state its mean and covariance functions. Is Y(t) a standard Brownian motion?Homework Equations The...

The brownian motion setup using smoke and air particles represents and allow us to conclude that gaseous particles move randomly (in any direction). Is there a setup using other particles and another fluid instead of smoke and air to represent the movement of liquid particles? 1) Can we...

Hi all. My teacher briefly mentioned brownian motion a few days ago but didn't really go in depth. I am planning to do my final paper on this topic and I just have a few questions. Hopefully, someone can point me in the right direction. 1. Let's suppose I have a container of water and some...

I'm reading an old, maybe outdated, paper by Karl Popper about the 2. law of thermodynamics, brownian motion and perpetual motion. Popper writes: Before that, Popper has described Planck's law as: So, my question is: Is brownian motion considered to be a violation to the 2. law of...

http://arxiv.org/abs/0710.1904 Relativistic generalization of Brownian Motion Authors: T. Koide, T. Kodama 11 pages (Submitted on 10 Oct 2007) "The relativistic generalization of the Brownian motion is discussed. We show that the transformation property of the noise term is determined by...

HI, i would need some help to solve the Brownian motion given by the Weiner Integral(over paths): \int \mathcal D [x_{t}]exp(-\int dt (m/2(\dot x)^{2}-V(x)) for the case V(x)=\delta (x) +\delta (x-1)+\delta (x-2) any help would be appreciated, thanks

Hi all, I need help with a question. Let B(t), t>= 0 be a standard Brownian motion and let u, v, w > 0. Calculate E[B(u) B(u+v) B(u+v+w)], using the fact that for a zero mean normal random variable Z, E[Z^3] = 0. I tried to do this question by breaking up the brownian motions, i.e...

In one of my homework problems it is a problem under the section of Brownian motion. It asks me to compute the average velocity of particles! here is the exact problem: The average speed of hydrogen molecules at 0 degrees C' is 1694 m/s. Compute the average speed of colloidal particles of...

I have read the next article and i want to realice the same calculation but i have some doubt www.chemengr.ucsb.edu/people/faculty/squires/public_html/laugasquires05.pdf In the section II. Image systems near a partial slip surface subsection a. Set up and boundary conditions A doubt it's...

Brownian motion... Hello ..since it has several application to physics i would like to hear about Brownian motion..in fact i think you can approach it by means of a functional integral of the form: \int D[x] e^{-a \int_{a}^{b} L(x,\dot x, t)} and that from this you derive the "difussion"...

hey there, i'm curious as to why they call it fractional Brownian motion. please don't say its Brownian motion that is fractional :-p many thanks

I know that this site is not for speculation, but can someone help me in this doubt ?:rolleyes: I know that perpetual motion of secod kind is considerated impossible. So I would like to know why what are descripted under doesn't work. I've a small permanent magnet that can remain in...

Describe the process of simulating a brownian motion with drift of 4 units and diffusion of 2 units. write a program in any application to imulate such a brownian motion. Anyone knows where should i start first if i use excel to do it. I don't know what equation to use.

Consider the density probability function following the diffusion equation with diffusion parameter D, with the initial condition f(x,t=0)=\delta(x) : f(x,t)=\frac{e^{-\frac{x^2}{4Dt}}}{\sqrt{4\pi Dt}} From this : if t=0, then the particle is at x=0. Consider a very small t>0...then...

Let's say we restrict 6 coin tosses to a period t so that each toss will take \frac{t}{6} . The size of the bet is \sqrt{\frac{t}{6}} Then why does \sum^n_{j=1} (S_{j}-S_{j-1})^{2} = 6 \times(\sqrt{\frac{t}{6}}) = t . Or more generally why does: \sum^n_{j=1}(S_{j}-S_{j-1})^{2} =...

A while back, I pointed out a paper in the Relativity section of PF that claimed to have a description of Special Relativity that is meant for "school-going children". It seems that such activity is a popular one for researchers at the Tata Institute of Fundamental Research in Mumbai, India...