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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237 views

Defining Brownian motion through Kolmogorov's extension theorem

In section 2.2. of Oksendal's book on Stochasic differential equations, he defines Brownian motion by specifying a family of probability measures $\nu_{t_1, \ldots, t_k}(F_1, \ldots, F_k)$ that ...
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43 views

fractional Brownian motion is not a semimartingale. How to apply Ergodic theorem in the proof of this theorem?

Here is the proof of the theorem. I couldn't understand how to apply Ergodic theorem in this proof. Let $X=(X_t)_{t\geq0}$ be a fractional Brownian motion with self-similar parameter $H\in(0,1)$. We ...
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29 views

Optional Sampling Theorem Application

Let x, y > 0. Define the first passage time of a Brownian motion $W_t$ as $\tau_a$ = min{t $\ge$ 0: $W_t$ = a}. I need to show that $$E[e^{-u\tau_x}1_{\tau_x < \tau_{-y}}] = ...
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19 views

Level sets of a Wiener process

Assume we have a Wiener process $W$ starting at $W_0=0$. What can one tell about the Lebesgue measure of "level sets" $A_y = \{t>0; W_t=y\}, y \in \mathbb{R}$? I actually need to estimate these ...
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32 views

Expectation of Exponential of Stochastic Integral

Let $z$ be the standard Brownian motion, $\omega$ an element of the sample space. Is it true that $$ \mathbf E\bigg[\exp\Big(\int_0^t f(\omega,s)\,\mathrm dz(s)\Big)\bigg] = \mathbf ...
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33 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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28 views

Cameron Martin Theorem

I am struggling with two versions of the Cameron Martin Theorem. 1) We define the measure spaces $(\Omega,\mathcal{F},P)$ and $(C[0,1],\mathcal{C},\mathbb{L}_0)$, where $\mathcal{C}:=\sigma(f\mapsto ...
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57 views

Expectation of absolute value of Brownian motion

I'm working on this problem that I can't seem to figure out. The problem involves a 1-dimensional Brownian motion, $B_t$, where the subscript denotes the time, and it asks me to show that the ...
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37 views

Ito's integral from the definition

I am doing Oksendal's book exercises one by one. I got stuck in 3.2. I need to prove, from the definition that $$\int_{0}^{t}B_s^2\text{d}B_s=\frac{B_s^3}{3}-\int_{0}^{t}B_s\text{d}s,$$ where ...
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44 views

$E[e^{\lambda X_t}|\mathcal{F_s}]$, where $X_t=\int_0^t(W_s-\frac{s}{t}W_t )ds$

I was trying to compute $E[e^{\lambda X_t}|\mathcal{F_s}]$, where $X_t=\int_0^t(W_s-\frac{s}{t}W_t) ds$, $\mathcal{F}$ is associated to $W$. I tried the following. 1) Splitting the integral ...
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54 views

Hyperbolic vs Euclidean Brownian Motion

In this article, page 4 of the linked pdf file, Lalley and Sellke claim that a hyperbolic Brownian motion can be obtained by time-changing a 2-dimensional Euclidean Brownian motion, conditioned to ...
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19 views

Stopped supremum of the Brownian local time still $L^p$ bounded in space?

Let $B_t$ be a standard Brownian motion and $L_t^x$ its local time in $x$ at time $t$. For fixed $t$ and $p>1$, it holds that $$ \sup_{x \in \mathbb{R}} \operatorname{E} [ (L_t^x)^p ] < ...
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43 views

Is there a modern iteration of Einstein's Brownian motion theory?

I was arguing with my friend that Brownian motion, in the sense of a pollen moving in the fluid, could be explained by physics laws (such as $F=ma$) and statistics laws. To check it out I found ...
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39 views

exercise 1.21 of chapter 1 of Revuz and Yor's

This is the exercise 1.21 of chapter 1 of Revuz and Yor's: Let $X=B^+$ or $|B|$ where $B$ is the standard linear BM, $p$ be a real number $>1$ and $q$ its conjugate number ($q^{-1}+p^{-1}=1$). ...
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44 views

Law and Brownian Bridge

Let $Z_{t}= W_{t}-tW_{1}$ and $Y_{1}=\sup_{0\leq t\leq 1}Z_{t}$, $(W_t, t \geq 0)$ standard Brownian motion Find the law of $Y_{1}$ I know that $\textbf{P}(\sup_{0\leq t\leq 1}W_{t}\geq x , ...
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69 views

The completed natural filtration of brownian motion is right-continuous, proof?

I have a question concerning a claim in J.F. LeGall's book Mouvement brownien, martingales et calcul stochastique. Let $(\mathcal{F_{t}})$ be the canonical completed filtration on $\Omega$ of a real ...
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38 views

Probability of hitting a Borel set by transient Brownian motion ($d\geq 3$)

I am looking for references/progress made in estimating the hitting probability for Borel sets. For spheres we have $P_{x}(T_{B_{r}(0)}<\infty)=(\frac{|r|}{|x|})^{d-2}$, where $x=B_{0}$ for ...
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26 views

Branching Brownian Motion and KPP equation

I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...
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30 views

Reference: Computing Martin Capacity

For Borel set $A$ the Martin Capacity is defined as: $\mathrm{Cap}_{M}(A)=[\inf\{\int \int \frac{G(x,y)}{G(0,y)}d\mu(x)d\mu(y):\mu \mbox{ probability measure on }A \}]^{-1}$ and Green's function ...
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19 views

Stability of simulation of brownian noise

As I understand, Brownian noise can be simulated by the process $$x_{n+1}=x_n+R_n$$ where $R\sim U[-a,a]$. The expected value for $x_n$ is then $x_0$. But $\text{Var} x_n\to\infty$ as $n\to\infty$ ...
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41 views

Brownian motion starts fresh variant

It is a standard result that if $W_t$ is a Brownian Motion and $S$ is a stopping time of the standard filtration $F_t$ then we have that $B_t = W_{S+t} - W_S$ is a Brownian Motion. I quote the ...
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20 views

Convergence of sampling from Brownian motion

For a standard linear Brownian motion $\{B(t)\mid\ 0\le t\le 1\}$, for natural $n\ge 0$ and natural $1\le k\le 2^n$, let $d(n,k)=B\left(k2^{-n}\right)-B\left((k-1)2^{-n}\right)$ be the differences of ...
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18 views

A conformal image of a Brownian motion is a time changed Brownian motion

I have read a paper which has stated the following: A conformal image of a Brownian motion is a time changed Brownian motion. The paper cites R. Durret, Brownian motion and martingales in ...
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25 views

Expression for $B_1$

I think that it is indeed the case that $$ B_1 = \int_0^1 \frac{B_1 - B_t}{1-t} dt, $$ where $B$ is a standard one-dimensional Brownian motion. Am I right? If so, how you we prove it? Thanks a lot ...
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24 views

Product of Geometric Brownian motions

Let $S,P$ be geometric BMs: $$dS_t=S_t(\mu dt + \sigma dW_t^1)$$ $$dP_t=P_t(\tau dt + \beta (\rho dW_t^1+ \sqrt{1-\rho^2}dW_t^2)$$ Where $W^1$ and $W^2$ are independent standard BM. I want to show ...
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62 views

Variance of Integrated Geometric Brownian Motion

I'm just asking for verification that my derivation is correct, as I can't seem to find this result elsewhere. I'd like to calculate $Var(\int_0^T X(t) dt)$ where $X(t) = X_0e^{(\mu - ...
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76 views

Donsker for randomly stopped processes

A question regarding Donsker's invariance principle. Donsker states that if $X_1, X_2, ...$ are independent and identically distributed with mean $0$ and variance $\sigma^2$ and if $S_t^n$ is the ...
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60 views

Ito's Lemma and Geometric Brownian Motion With Jumps

I have a price process: \begin{equation} dF_t = d\Pi_t - \mu_\pi \sigma_t F_t \gamma \, dt + \sigma_t F_t \, dz \end{equation} And wish to simulate the process $x_t = \ln(F_t)$ by Euler method, ...
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91 views

Tail of hitting times for Brownian motion on the circle

For $y\in \mathbb R/\mathbb Z$ and $\varphi\in C([0,\infty);\mathbb R/\mathbb Z)$ let $T_{y}(\varphi) \ := \ \inf\{t>0: \varphi_t = y \} \ \ \ $ (first time the path $\varphi$ hits $y$) ...
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34 views

Hitting time for a planar diffusion

Let $A$ be an open subset of $\Bbb R^2$, and let us consider a diffusion $\mathrm dX_t = f(X_t)\mathrm dt + g(X_t)\mathrm dW_t$ where $f$ and $g$ are globally Lipschitz continuous maps. Suppose I am ...
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73 views

Expectation of the infimum of a GBM

does somebody know a reference, where I can find the value of the expectation of the running infimum of a geometric Brownian motion, namely: Given a filtered probability space ...
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125 views

An application of the Dambis-Dubins-Schwarz theorem. Is my argument correct?

I attended a lecture today, in which the professor went through an example with a lot of tedious calculations to show something which I'd think would follow directly from the Dambis-Dubins-Schwarz ...
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57 views

Probability that the value at time T from one geometric Brownian motion process is greater than the value from another GBM

I am having a competition between $n$ people (starts at time $t$=0), each who accumulates points on a daily basis, which I assume is a geometric Brownian motion process with parameters $\mu_i$, ...
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364 views

Running maximum for Geometric Brownian Motion

Can anyone provide the expression and source for the running maximum $M_t$ for geometric Brownian motion $X_t$ as a function of the initial value $X_0$, drift $\mu$ and diffusion $\sigma$? $X_t$ ...
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92 views

Verifying a standard Brownian Motion?

Let $\{X_t, t\ge 0\}$ be a standard Brownian motion process. For a fixed positive number s and all $t\ge 0$, we define $Y_t = X_{t+s} - X_s$. Is $\{Y_t, t\ge0\}$ a standard Brownian motion? Attempt: ...
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27 views

Brownian motion minimisation problem

Let $B_t$ be a Brownian motion, let $\sigma > 0$ be fixed and let $X_t$ be a process with fixed beginning value $x_0$ that satisfies $dXt = u_tdt + \sigma X_tdB_t.$ Solve ...
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29 views

Probability of winding number in 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 ...
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40 views

Conditional covariance

Just a simple question when we have $v_{st} = \operatorname{cov}(B_s, B_t\mid Z)$, where $B_t$ is a brownian motion. I know that the answer is $\min(s, t) - E[B_s Z]E[B_t Z]/E[Z^2]$ but i don't know ...
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27 views

Finding asymptotic behaviour

I have a problem in finding the asymptotic behavior of this sum: $$\sum_{i=0}^{n-1} \bigl|B^2 (t_{i+1})-B^2 (t_i)\bigr|$$ over $[0,T]$ when $h= t_{i+1}-t_i \to 0$ and $B$ is Brownian motion. The ...
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68 views

Bounding the expectation of monotone function of stopping times of Brownian motion

Let $X_t$ be a standard Brownian motion and let $Y_t:=X_t + \epsilon B_t$ where $B_t$ is an independent standard Brownian motion and $\epsilon>0$ is small. Let f be a monotone increasing function. ...
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30 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 ...
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146 views

Brownian motion hitting probability

Let $B_t$ be a brownian motion and $g(t)$ a function of the time $t$. $B_0=0$. Let $\Phi$ be the c.d.f. of a normal distribution. At time $t$, the probability that $B_t > g(t)$ equals ...
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54 views

Bounded Brownian Bridge

I am trying to calculate the following expectation value of $$ E[exp(-\int_{t_0}^{t_1} X_s ds)] $$ in which Xs is a bounded Brownian bridge, which means $X(t_0)=a$, $X(t_1)=b$ and $A<X(t)<B$. ...
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168 views

Brownian motion with drift

I need help with the following problem: Let us denote the water level in a dam at time $t$ by $X(t)$, where $t$ is measured in months. We will assume that, at least until the first time that the dam ...
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119 views

Conditional expectation with three random variables

We have $N_1, N_2, N_3$ normally distributed random variables with $µ_i =E[N_i]$, $σ_{ij}=Cov(N_i,N_j)$. We also have $\tilde{µ}_i=E[N_i|N_2 = x] $, $\tilde{σ}_{ij}=Cov(N_i,N_j|N_2 = x)$ and $v^2 ...
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48 views

Contradiction on equality with stochastic integrals

I want to compute $E[∫_0^tB_u \, du ∫_0^sB_u \, du]$ and I know from another source that should be equal to $ts^2/2$. But when I try to compute it like: $$\begin{align} & E\left[(tB_t- \int_0^tu ...
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133 views

Brownian motion, rate of large events

Given the most simple brownian motion: $$ \dot x(t) = \sigma \eta(t)$$ where $\langle \eta(t)\eta(t')\rangle=\delta(t-t')$, I define as large event in a time-frame $\tau$ a portion of the trace ...
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96 views

Probability a geometric Brownian motion stays within an interval.

Let $X_s$ be a $(\mu,\sigma)$ geometric Brownian motion with $X_0 = x$. For some positive numbers $c < x < d$ and time $t$, what is the probability $X_s \in [c,d]$ for all $s \in [0,t]$? In ...
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110 views

Leibniz Rule applied to Brownian integral

I am looking to take the partial derivative of an integral with respect to brownian motion. For Simplicity I will make it the same integral as in this post (don't have enough reputation to comment): ...
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673 views

Integrating deterministic function with respect to Brownian motion

I have looked everywhere for a satisfactory answer to this, including Shreve's textbooks, but I can't find one. If I want to integrate a some deterministic function f(t) with respect to brownian ...