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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$dX_t=-\mu X_tdt + \sigma dW_t$. Prove that $X_t = e^{-\mu t}X_0 + \sigma \int_0^t e^{-\mu(t-u)}dW_u $

So the solution says use Ito-s formula, taking $Y_t:= e^{\mu t}X_t$ to obtain $dY_t = [\mu e^{\mu t}X_t - e^{\mu t}\mu X_t + e^\mu t \sigma dW_t] $. As far as I can see though, Ito's formula says ...
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92 views

Is this true about Brownian Motion?

I have the following in my notes and I'm not sure if it's true or not. Any help would be highly appreciated. If $\{W_t\}_{t\geq0}$ is a standard Brownian motion stochastic process, $\Delta>0$ and ...
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124 views

Brownian Motion Conditional Probability Question

If $X_t$ is a standard Brownian motion, how does one calculate a conditional probability. Specifically, $P(X_2\gt 0 |X_1\gt0)$. I am thinking that the two are independent so I can just calculate ...
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1answer
114 views

Integrating brownian motion times exponential function

I am trying to calculate $$\int_0^tB_se^{\lambda s}ds$$ but I am unsure of how to start the computation. The motivation behind this is that I read (and am now trying to prove) that ...
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148 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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61 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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86 views

Probability calculations with integral of Geometric Brownian Motion

I am looking for the probability $Prob[A_t<A_0]$, where $A_t$$=$$A_0$$+$$\int_0^t$$S_0$$e^{(\mu-(1/2)\sigma^2)s+\sigma W_s}$$ds$$-$$c$$t$ The expression $S_0$$e^{(\mu-(1/2)\sigma^2)t+\sigma ...
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230 views

Cubed Brownian motion

I have to do the following exercise: Let $(W_t)$ be a Brownian motion. (a) Does X given by $X_t:=W_t^3$ have constant expectation? (b) Is it a martingale? (c) Does it have independent increments? ...
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1answer
92 views

An application of the strong Markov property in the proof of the connection between Brownian motion and harmonic functions

Let $U$ be an open, connected set in $\mathbb{R}^n$ and let $(B(t))_{t \geq 0}$ be an $n$-dimensional Brownian motion with start at $x \in U$ and let $\overline{B_x(\delta)}$ be the closed ball about ...
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52 views

A Lemma in the book “ Mathematical Method for financial markets” (Chapter 5, Section 5.7)

In page 307, Section 5.7, Chapter 5 of the book "mathematical methods for financial markets" by Jeanblanc, Yor and Chesney, Lemma 5.7.1 is given as follows: Lemma 5.7.1.1 Let $W$ be a Brownian ...
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270 views

A planar Brownian motion has area zero

I'm looking for proofs of Paul Lévy's theorem that a planar Brownian motion has Lebesgue measure $0$. I know of only two proofs: one is in Lévy's original paper (Théorème 12, p. 532) and the other is ...
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0answers
18 views

2 2-dimensional Brownian motions are close to each other

Suppose $B^1$ is a standard 2 dimensional Brownian motion and $B^2$ is a 2 dimensional Brownian motion with mean zero and covariance matrix $\Gamma = \begin{pmatrix} a & b \\ b & a \\ ...
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2answers
99 views

Does a Brownian motion remain in any given open set for a given interval of time with positive probability?

Let $B$ be a standard $d$-dimensional Brownian motion. Given $b>a>0$ and an open ball $U$ in $\mathbb{R}^d$, I want to be able to comment on the probability that $B$ remains in $U$ during the ...
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0answers
66 views

Why is a brownian motion conditioned to stay positive a Bessel-3

I am told this result long ago but I still don't know how to prove it. Is it because that this conditioning can be turned into a Girsanov probability change? Or is there any simpler ways to see it?
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48 views

Conditional expectation and coupled set of ODEs

How to find a coupled set of ODEs and initial conditions for the deterministic functions $a$ and $b$ such that $$\mathbb{E}\left[e^{-\int_{t}^{T} W^2(u)du} | \mathcal{F(t)}\right] = e^{-a(T-t) - ...
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0answers
90 views

Geometric Brownian motion - Volatility Interpretation

A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ...
3
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1answer
463 views

Proving the reflection principle of Brownian motion

The reflection principle of Brownian motion states that Brownian motion reflected at some stopping time $\tau$ is still a Brownian motion. The proof found in Mörters & Peres (as well as in ...
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2answers
35 views

Probability of Position of Brownian motion at hitting time

this might be a stupid question but I am a bit stuck here. let $B$ be a standard Brownian motion and $H_a$ the first hitting time of level $a$. I now want to find the probability $\mathbb{P}(B_{H_a} ...
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80 views

The 1-dimensional Hausdorff measure of a curve in the plane

For a set $X\subseteq\mathbb{R}^2$, let $H^1(X)$ be its 1-dimensional Hausdorff measure. Suppose $X$ is a regular curve (say, a graph of a continuous function $f:\mathbb{R}\to\mathbb{R}$). In that ...
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1answer
48 views

Clarification about a very simple stochastic integral

I'm studying stochastic integrals right now and I feel like this question is incredibly easy but I'm not sure. I want to evaluate $\int_0^t sdB_s$. Using Ito's formula I get $tB_t$ by setting ...
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25 views

How to understand this equation for brownian motion

I am reading this article from the notes 'an intro to SDE'. Here I dont know why in (1) he take that integral from - infinity to infinity. I mean why we do that? I just dont know what the physics or ...
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1answer
49 views

Expectation of a product of (many) 1-dimensional Brownian motions.

Let $0=t_0<t_1<t_2<\ldots$ be a sequence of positive reals. Denote by $B(t)$ the 1-dimensional Brownian motion with time $t$. It is easy to show the the expectation of the product of two ...
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1answer
42 views

Integrating the difference of brownian motion

I'm reading the solutions to an exercise where it is stated that $$\int_t^T\Big(W(u) - W(t)\Big)du = \int_t^T (T-u)dW(u).$$ But can someone enlighten me to what theorem/rule can be used to show this? ...
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0answers
92 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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1answer
106 views

Brownian Motion with Optional Stopping Theorem (OST)

Let $(B_t)_{t \geq 0}$ be a standard Brownian Motion and let $T:=\inf\{t \geq 0: B_t=at-b\}$ for some positive constant $a,b>0$. Calculate $\mathbb{E}[T]$. How do i begin it?
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1answer
102 views

Invariance of Brownian motion under orthogonal transformations

Let $\left(B_t\right)_{t \in [0,\infty)}$ be an $n$-dimensional Brownian motion with start at $x \in \mathbb{R}^n$, and let $A$ be an orthogonal $n \times n$ real matrix. I'm trying to show that $AB$ ...
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0answers
35 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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20 views

Show that $ \text{ess-sup}_\Omega g (x+ B_T) = \sup_{y \in \mathbb R ^d }g(y)$

Show that $$ \text{ess-sup}_\Omega g (x+ B_T) = \sup_{y \in \mathbb R ^d }g(y)$$ where $B$ is a d-dimentional brownian motion , $x \in \mathbb R ^d $ and g a Lipschitz bounded function of $\mathbb R ...
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0answers
74 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 ...
2
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1answer
71 views

The probability of a Brownian motion's tail event is unaffected by the starting point

Consider the measurable space $\left(\mathbf{C}\left[0,\infty\right), \mathcal{B}\left(\mathbf{C}\left[0,\infty\right)\right)\right)$ and the stochastic process $\left(X_t\right)_{t \in ...
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1answer
83 views

Diffusion processes

I am trying to work out a problem to which I have not found similar solutions on the website. Perhaps you can help me out. Let $X = (X_t)_{t\geq0}$ be a non-negative diffusion process which solves ...
2
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1answer
69 views

Expectation of Integrals of Brownian Motion

Hello I am not a native english speaker so please let me know if something does not make sense. I am interested in computing the following: $$E\int_0^T(B_s(\omega,t))^4dt$$ Or at least showing it is ...
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1answer
55 views

A few questions about Stochastic Processes and Numerical Methods

I am having a few problems understanding the Ornstein Uhlenbeck solutions, on wikipedia under solution (http://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process) it described using variation of ...
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2answers
49 views

Simple Brownian Motion Proof

I've been given the following question and solution: Let $W_t$ be a standard Brownian Motion w.r.t. ($\mathbf{P},\mathcal{F}_t)$. Prove that \begin{align} E[|W_t|] < \infty, \forall \text{ } t ...
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0answers
48 views

Local time of fractional Brownian motion

For BM, there is a downcrossing representation of the local time at 0. Namely, $L_t(0)=\lim_2 (b_i-a_i)D(a_i,b_i,t)$, where $D$ is the number of downcrossing between level $b_i$ and $a_i$. I am ...
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2answers
65 views

$4^{Brownian(t)}$ martingale proof

Let $B(t)$ a Brownian motion. I like to prove that $4^{B(t)}$ = martingale I rewrote the expression into an exponential form (like $\exp(\ln(4) B)$), but then I don't know how to proceed.
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120 views

Estimating the maximum of a Brownian motion over the unit interval

Let $\left(B_t\right)_{t \in \left[0,\infty\right)}$ be a standard Brownian motion over the probability space $\left(\Omega, \mathcal{A}, P\right)$. For each $x \in \left(0, \infty\right)$, give an ...
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1answer
62 views

First hitting time in the one-dimensional case by solving a boundary value problem

If have a question about section 3.1 in the paper Kramers' law: Validity, derivations and generalisations by Nils Berglund. (See http://arxiv.org/abs/1106.5799 page 7 - 9) On page 8 it says, that ...
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2answers
584 views

What is the importance of the infinitesimal generator of Brownian motion?

I have read that the infinitesimal generator of Brownian motion is $\frac{1}{2}\small\triangle$. Unfortunately, I have no background in semigroup theory, and the expositions of semigroup theory I have ...
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1answer
92 views

How does the natural filtration of a Brownian motion look like?

I am trying to understand how the natural filtration for a Brownian motion might look like. Definitions: I will start with the definitions for reference. The definition of a natural filtration is ...
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1answer
123 views

Conditional expectation brownian motion

Somebody has an idea on how to tackle this quantity $$\mathbb{E}_{W_T}\left[ \frac{\int_0^T e^{\alpha W_t} dt}{\int_0^T e^{-\alpha W_t} dt + \int_0^T e^{\alpha W_t} dt} \right]$$ For $\alpha \in ...
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2answers
68 views

$\mathbb{E}[B^4(t)]$ with $B$= brownian motion

Can anyone help me to find: $\mathbb{E}[B^4(t)]$ where $B$ is a brownian motion? I thought using this density function: $f_{B_t}(x) = \frac{1}{\sqrt{2 \pi t}} e^{-\frac{x^2}{2t}}$, but I don't know ...
2
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0answers
172 views

Brownian motion conditional probability

If $B$ is the standard brownian motion and $a,b >0$ I want to show, using the reflection principle $$\mathbb{P}\left(B_t\geq a-b | \inf_{s\leq t} B_s \geq -b\right) = \frac{\mathbb P(|B_t+x|\leq ...
2
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1answer
288 views

Computing cross variation of independent brownian motions

I am familiar with computing the quadratic variation of Brownian motion, but was confused when the text I'm working through introduced cross variation of independent Brownian motions. the notation is ...
2
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2answers
129 views

A Brownian motion $B$ that is discontinuous at an independent, uniformly distributed random variable $U(0,1)$

Suppose that $\left\{B\left(t\right): t \geq 0\right\}$ is a Brownian motion and $U$ is an independent random variable, which is uniformly distributed on $\left[0,1\right]$. Then the process ...
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1answer
97 views

Strong Markov property of Brownian motion

I was able to understand Brownian Motion $\{B(t):t\geq0\}$ has Strong Markov Property i.e. For any stopping time $\tau$, $P(B(t+\tau)\leq y | \mathcal{F}_{\tau})=P(B(t+\tau)\leq y|B(\tau))$ a.s. , $y ...
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0answers
207 views

Is the absolute value of Brownian motion a super martingale?Is it a sub martingale? Is it a Markov process?

I've just started to study random processes and I'm trying to solve the following problem: Let $W(t)$ be a Brownian motion with filtration $F(t)$ generated by $ W(t)$ (i.e., $F(t)=\sigma \left( ...
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1answer
56 views

Differentiability of paths of brownian motion

On a book I'm reading (Stochastic processes by Bass. R.F.) after he proves the law of iterated algorithm for a brownian motion $W$, namely that $$\limsup_{t\rightarrow \infty} ...
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2answers
59 views

two brownian motions in $ \mathbb{Z}^2 $

I was wondering what is the probability for 2 brownian walkers coming from 2 different initial positions to be at the same position at time t. I consider that at each step, each point can ...
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0answers
41 views

Expected Value of the minimum stock price where stock price is an exponential brownian process

Hi I am trying to figure out what would be the solution to the following equation: $\tilde{E}[S_{min}]$ where $S_{min}$ is the minimum stock price and the stock price is of the form ...