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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 $t-s$.

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448 views

The probability that a linear Brownian motion will hit a curve

Summary I am trying to estimate the probability that a standard linear Brownian motion will hit some curve. To make things a bit simple, I can assume that the curve is a graph of a function, that is ...
10
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2k views

How to prove Brownian motion is Gaussian Process?

I'm reading Bernt Oksendal's "Stochastic Differential Equations" and this is one of the proof that I'm totally lost. This is from Ch2.2, page 12-13 (sixth edition). First, Brownian motion is ...
8
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189 views

Brownian Motion in Confined space, any results?

I am searching for work regarding Brownian motion in a confined space, like a sphere or a cylinder, where the wall will serve as reflection boundary. I am wondering if it is possible to derive results ...
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128 views

How the Brownian motion escapes the minimum? A question based on the Lecture notes from Zeitouni on RWRE

In the "Lectures on Probability Theory and Statistics Ecole d’Eté de Probabilités de Saint-Flour XXXI - 2001" in page 249 one reads $$A_n^{J,\delta}=\left\{\begin{array}{ll}\omega\in\Omega:\!\!\! &...
7
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446 views

Relationship of SDE and Feynman-Kac PDE

I am struggling with this problem: Given a stochastic differential equation $$ dX_t = b(X_t) dt + \sigma (X_t) \,dW_t $$ where $W$ is a Brownian motion and the functions $b$ and $\sigma$ are ...
7
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475 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
6
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357 views

Schilling's proof of the Feynman-Kac Formula for Brownian motion

This is part of a proof to the Feynman-Kac formula from Schilling's Brownian motion. I need some help understanding the proof to this theorem. Theorem (Kac 1949). Let $(B_t)_{t\ge 0}$ be a $d$-...
6
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45 views

Brownian motion proofs using Itos Formula

Using Ito’s formula, write an expression for $\int_0^1(B(s))^2dB(s)$ Not sure exactly if I did this right. Was hoping for feedback. I let $f(x)=x^3$. Then by definition, Itos formula states that $...
6
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575 views

Brownian Motion Third Power Martingale using Ito Integral

Let $(B_t)_{t \geq 0}$ be a standard Brownian motion and $M_t = B_t^2 - t$. According to this and this posts we know that \begin{align} [M] = [B^2] = 2 \int_0^t B_s^2\ ds. \end{align} Now, without ...
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80 views

Accelerated Eigenfunction Expansions of Random Functions

I am interested in eigenfunction expansions of random functions. We know that the autocorrelation of brownian motion, $\{ B_t \}_{t \geq 0}$, is given by $$ E[B_t B_s] = \min\{s,t \}, $$ which can ...
5
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91 views

Prove Brownian Bridge is Sampling Without Replacement

Let's say we have a bowl containing $n$ many $+1$'s and $n$ many $-1$'s. You sample numbers from the bowl randomly without replacing. Let $k_1, k_2, ..., k_{2n}$ denote the random sequence of numbers ...
5
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183 views

Can any stochastic differential equation be mapped onto a (generalized) Langevin equation?

There might be different definitions of what a generalized Langevin equation is, but let us consider the following expression: $$ \dot{x}_i = \frac{dx_i}{dt} = f_i(\mathbf{x}) + \sum\limits_{m=1}^{n} ...
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705 views

Using Girsanov theorem to prove density of stopping time

Let $B$ be a standard Brownian motion and for $a>0$ and $b>0$, and set $$\sigma_{a,b} = \inf\{t\,:\, B_t + bt = a\}.$$ There are at least two ways to solve the following problem (the other one ...
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0answers
332 views

What is the distribution of the area between a Brownian Bridge and the x-axis?

Lets say that we have a Standard Brownian Bridge ($\sigma=1$) with endpoints $(0,0),(1,0)$ Is there a way to derive the distribution of the area between a sample path of this bridge and the x-axis?? ...
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0answers
2k views

Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
5
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0answers
444 views

Intuition for the optimality of bold play

There is a standard result (I think originally by Dubins and Savage) that if one wants to maximise the probability of winning a certain amount in an unfair game of chance then an optimal strategy is “...
5
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169 views

Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion

Let $ 0 < r < 1$, fix $x > 1$ and consider the integral $$ I_{r}(x) = \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}.$$ In the investigation of ...
4
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58 views

Using Galmarino's test

I am wondering if somebody has an example of use of Galmarino's test. The Galmarino test says that for $X=(X_t)_{t\in T}$ a continuous stochastic process with $\mathcal{F}$ the natural filtration, a ...
4
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0answers
49 views

independence of 1-dimensional brownian motions proof (Oksendal SDE Exercise 2.11)

I am having some difficulty proving the following statement taken from Oksendals SDE's exercise 2.11. If $B_{t} = (B^{(1)}_{t},...,B^{(n)}_t)$ is an $n$-dimensional Brownian motion, then the $1$-...
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177 views

Maximum likelihood for the Brownian Motion with drift

Given the Brownian Motion with drift $$ dX(t) = \mu dt + \sigma dW(t) $$ It is well known that its distribution has the following form $$ f_t(x) = \frac{1}{\sqrt(2 \pi \sigma^2 t)} e^{-\frac{(x -\...
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106 views

How to empirically verify convergence results with stochastic differential equations (with fractional Brownian motions)

Let $ \frac{1}{2} < H < 1$ and let $B^H_t$ be a fractional Brownian motion with Hurst parameter $H$. Then the following stochastic differential equation $$\mathrm{d}Y_t = 5Y_t\mathrm{d}B^H_t, \...
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133 views

Girsanov theorem and filtrations

Let $\{W_t\}$ be a standard Wiener process on a probability space $(\Omega, \mathcal{F},P)$. Let $\mathcal{F}^W$ be the natural filtration generated by $\{W_t\}$. Let $\{\theta_t\}$ be an $\mathcal{...
4
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108 views

If $W$ is a standard Brownian motion, does $W(1)$ take every real number?

Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $W:[0,\infty)\times \Omega\rightarrow\mathbb{R}$ be a standard Brownian motion. Is it true that for all $x\in\mathbb{R}$ there exists $\omega\...
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222 views

Conditional expectation and stopping times

I think I found a way to prove a result but I have doubts about it. I may have made a mistake. ($X$ is a geometric Brownian motion and I do not want to use the strong Markov property) I have proved ...
4
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0answers
145 views

Show that semimartingale related to Brownian motion is local martingale

I got stuck on another problem regarding whether certain process is a local martingale. Set $T_\epsilon=\inf\{ t\ge 0:B_t=\epsilon\}$, $\lambda >0$ and $\alpha\neq 0$ Show that the process $$ ...
4
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0answers
124 views

Exponential of Brownian motion on Lie algebra?

Let $G \subset GL_n(\mathbb{R})$ be a compact Lie group. Let $\mathfrak{g}$ be its Lie algebra, with inner product given by $\operatorname{tr} (A A^t)$. Isometrically identify $\mathfrak{g} \subset ...
4
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0answers
436 views

Brownian motion on sphere proof?

proving the brownian motion on the sphere equation the stratonovich form differential equation $$\partial X=n(X)\times \partial B$$ the equation in ito's form becomes $$dX=n(X)\times dB+H(X)n(X)...
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0answers
665 views

Clarification on the Augmented Filtration

Consider the following definition. Definition. Let $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ be a probability space and $W$ a Brownian motion. Let $\mathcal{F}^W_t=\sigma\left(\left\{W_s\mid s\...
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0answers
314 views

Is $X_t = tW\left(\frac{1}{t}\right)$ a Martingale?If not, how could it be a Brownian Motion?

As is proved, $X_t = tW\left(\frac{1}{t}\right)$ is a Brownian motion. For example see Theorem 4.2 in this paper http://math.uchicago.edu/~may/REU2012/REUPapers/Leiner.pdf I'm just confused because ...
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346 views

What can you tell me about backward Brownian motion?

I'm trying to understand "backward Brownian motion" and how it relates to standard Brownian motion. In this paper, they construct a solution to Burgers Equation (transformed via Cole-Hopf) with ...
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0answers
1k views

Integral of Brownian Motion with respect to an independent Brownian motion

I have this seemingly simple problem which I haven't been able to solve. I have two standard Brownian motions, $B$ and $W$, on the same probability space and under the same filtration (I am not so ...
4
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0answers
327 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
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77 views

2-D exponential functional brownian motion

I'm looking for the distribution of $X = \int_0^T e^{-W_t} dt \int_0^T e^{W_t}dt$ and $Y = \frac{\int_0^T e^{-W_t} dt}{ \int_0^T e^{W_t}dt}$ (where $W_t$ is a standard brownian motion) On most ...
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0answers
520 views

Time scaling of Brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion and $A_t$ be an increasing continuous process adapted to the filtration generated by the Brownian Motion and $A_0 = 0$. I am trying to prove that $(...
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0answers
208 views

First hitting time for a brownian motion with two exponential boundaries

I asked a previous related question here: First hitting time for a brownian motion with a exponential boundary Now Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first ...
4
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0answers
330 views

Integral representation of fractional Brownian motion

Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $${1\over C(H)}\int_\mathbb{R}\left((t-s)_+^{H-{1\over2}}-(-s)_+^{H-{1\over2}}\right)dB(s)$$ ...
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0answers
866 views

Quadratic variation of a Brownian motion up to time $T$ converges to $T$ in $L^2$?

In Stochastic Calculus for Finance II: Continuous-time Models by Steve Shreve, Theorem 3.4.3. Let $W$ be a Brownian motion. Then $[W, W](T) = T$ for all $T > 0$ almost surely. where $[W, W](T)...
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42 views

Geometric Brownian Motion Price Processes in high Dimensions

This is my first post so I am open for an suggestions in formating improvement. For some reason I can not find suitable literature for the following problem What I want to do is calculate the option ...
3
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0answers
43 views

Ito Isometry for Brownian Bridge

I have been looking at the following expression, and I'm a bit stuck: $$\left(\int_{0}^{t}\frac{dW_{u}}{1-u}(B_{1} - B_{u})\right)^{2}$$ where $\{W_{t}\}_{t\in[0,1]}$ is a standard Weiner Process and $...
3
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0answers
71 views

“Straightforward” estimate on intersection-probabilities of Brownian Motion

I am currently working on a paper and there appears this so called "straightforward estimate": $$\mathbf{P}\{B[0,1] \cap B[3,n] = \emptyset\} \leq \frac{c}{\ln(n)} \quad \text{where}\ c<\infty\ \...
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0answers
59 views

Bounding 2D Brownian Motion expected barrier hitting time of a square

I have been doing a computation but I am making a mistake somewhere and cannot figure out where. The question I have is: where am I making my mistake? I will run through my reasoning: Consider ...
3
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0answers
115 views

Will simple random walk on $n$-cycle converges to Brownian motion on $S^1$?

I know that, by Donsker's theorem, simple random walk on $\mathbb{Z}$ will converge to Brownian motion on $\mathbb{R}$. Here, simple random walk means that the Markov chain with probability from $n$ ...
3
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0answers
157 views

Equality in law of two stochastic processes

Let $\lambda(t)$ be a CIR process, i.e. the strong solution of the SDE $$ \mathrm{d}\lambda(t)=\kappa(\theta-\lambda)\mathrm{d}t+\sigma\sqrt{\lambda(t)}\mathrm{d}W^1(t) $$ The integrated CIR is ...
3
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0answers
62 views

Law of supremum of time-scaled Brownian motion

I would like to know if there is a formula for the law of $$ \sup_{l \leq t \leq u} \frac{B_t}{\sqrt{t}} $$ where $B$ is a standard Brownian motion, and $0 < l < u < 1$ are constants? The ...
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0answers
109 views

How to solve Fokker-Planck PDE for Brownian particle in square potential driven by periodic time-dependent force

Problem statement: We have a Brownian particle in harmonic potential with additional time-dependent force. Langevin equation(mass taken to be unit): $\ddot{x} + \gamma \dot{x} + \omega_0^2 x = \...
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0answers
42 views

For a stopping time $T$ respect Brownian motion, $T_{2}=T(B_{2})+T$ is stopping time,where $B_{2}(t)=B(T+t)-B(T)$

I'm trying to prove the next: Let $\{B(t)\}_{t\geq 0}$ be a standard Brownian motion on the line, and $T$ be a stopping time with $E(T)<\infty.$ Define $T_{1}=T$ and $T_{n}=T(B_{n})+T_{n-1}$ where ...
3
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0answers
102 views

Questions concerning the canonical construction of a Brownian motion.

I have some questions concerning the canonical construction of a Brownian motion and would be very happy if someone can help me. In my probability lecture I have seen the following definitions: Let ...
3
votes
0answers
206 views

Seriously struggling with Ito's lemma, and understanding what this paper does!!!

So, this is a theorem that I need to apply on a simpler system, and understanding it thus far has been the bane of my existence. I was wondering how exactly both Ito's formula and the Burkholder-...
3
votes
0answers
109 views

Independent brownian motions with different $B_1$

I am asked to compare two Brownian Motions which both start at 0, where one is conditioned on the event $B_1 \leq 5$ and one conditioned on $B_1 \geq 5$. After a short amount of time, on, say $[0,dt]$,...
3
votes
0answers
36 views

Pushing the Iterated Logarithm Rule to the limits.

Given a standard Brownian motion, the Iterated Logarithm Rule says that with probability one, $$\frac{|w(t)|}{\sqrt{t \log\log t}},\ (1\le t)$$ has $\limsup$ $\sqrt{2}$ as $t \to\infty$. But what is ...