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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4
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1answer
125 views

maximum of a brownian motion and its integral

Let $W_{t}$ be a brownian motion and $$ W^{*}_{t} = \max_{s<t} W_{s} $$ Then can you please explain why we have this: $$ (W^{*}_{t} - W_{t})dW^{*}_{t} = 0 $$
3
votes
1answer
362 views

minimum of hitting time of a brownian motion

Let $Y$ be an exponential random variable with rate parameter $\lambda$. Let $T_{a}$ be the first hitting time of a Brownian Motion. I want to find $$ P(\min(T_{a}, T_{-a}) < Y) $$ In order to ...
1
vote
1answer
491 views

Convergence of the exponential martingale

How can we show that this martingale $$ e^{aW_{t} - \frac{1}{2}a^2t}$$ converges to $0$ as $ t \rightarrow \infty$ using law of iterated logarithm, for $a \neq 0$.
5
votes
3answers
1k views

expected value of brownian motion

How can you find this expected value? $$ \mathbb{E}[|W_{t}^2 - t|] $$ where $W_{t}$ is a brownian motion.
10
votes
2answers
270 views

Problem about partial sum of exponential random variable

Let $X_1, X_2, \dots,X_n, X_{n+1}$ be independent random variable of exponential distribution, and the mean is 1. Let $S_i = X_1 + \dots + X_i$ I want to know ...
2
votes
1answer
137 views

Interpretation of Notation of Laplacian and Brownian Motion

Let $f: \mathbb{R}^d \to \mathbb{R}$ be a twice differentiable function. In particular, $\Delta f$ is well defined. Let $W := (W_t)_{t \geq 0}$ be a $d$-dimensional standard Brownian Motion. ...
2
votes
1answer
452 views

Difference between a Brownian Motion and the root of its square

Let $W_{t}$ be a Wiener Process (a Brownian Motion starting at $W_{0} = 0$). What is the difference between $W_{t}$ and $\sqrt{W_{t}^{2}}$? Using the Ito formula (in differential notation), ...
2
votes
1answer
145 views

Length of Wiener Sausage

I am deriving a formula for a volume of Wiener sausage in one dimension. $$\mathbb{E}[\operatorname{vol}(W(t))] = 2r+\sqrt{\frac{8t}{\pi}}$$ where $W(t) = \bigcup_{s\leq ...
2
votes
1answer
588 views

Absolute value of Brownian motion

I need to show that $$R_t=\frac{1}{|B_t|}$$ is bounded in $\mathcal{L^2}$ for $(t \ge 1)$, where $B_t$ is a 3-dimensional standard Brownian motion. I am trying to find a bound for ...
0
votes
0answers
335 views

Brownian Bridge. Law of a process

Let $(B_t , 0 ≤ t ≤ 1)$ be a standard Brownian motion in 1 dimension. We let $(Z_t^y = yt + (B_t − tB_1 ), 0 ≤ t ≤ 1)$ for any $y \in R$ and call it the Brownian bridge from $0$ to $y$. Let $W_0^y$ be ...
0
votes
0answers
364 views

Min and Max of Geometric Brownian motion

I am trying to derive the distribution of $M_X(t) = \max\limits_{0\leq s\leq t}X(s)$ and $m_X(t) = \min\limits_{0\leq s\leq t}X(s)$, where $dX(t)=\mu X(t) dt+\sigma X(t)dB(t)$ and $B(t)$ is standard ...
2
votes
1answer
309 views

Expectation of an integral of the minimum of a Brownian motion and a constant

I would like to compute the expectation of the following expectation $\mathbb{E}[\int_a^\infty e^{-rt}\min(x_t,c)\,dt]\,$ where a, r, c are constants, $dx_t = \mu x_t dt + \sigma x_t dW_t$ is a ...
4
votes
0answers
159 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 ...
3
votes
1answer
100 views

$\varlimsup_{t\rightarrow\infty} \frac{B_t}{\sqrt{t}}>0$

I am trying to prove the following statement about the standard Brownian Motion: $\varlimsup_{t\rightarrow\infty} \frac{B_t}{\sqrt{t}}>0$. I know that it is trivial to prove the above statement by ...
5
votes
1answer
856 views

Quadratic Variation of Brownian Motion

Quadratic Variation of a Brownian motion $B$ over the interval $[0,t]$ is defined as the limit in probability of any sequence of partitions $\Pi_n([0,t])=\{0=t^n_0<\cdots<t^n_{k(n)}=t\}$ of the ...
1
vote
1answer
684 views

confusion about the multi-dimensional Brownian motion

I am confused on the multi-dimensional Brownian motion. $B_t$ is a standard Brownian motion based on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})$ if ...
4
votes
1answer
336 views

Stochastic integral inequality

Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that ...
2
votes
0answers
111 views

Ruin probability

Let $X_t$ be a solution of the stochastic differential equation $$ dX_t= -\frac{c-1}{2 X_t}dt+ dB_t, \, \qquad X_0=x_0$$ where $c$ is a real constant and $B_t$ is a Brownian motion. Can you give me ...
8
votes
2answers
544 views

Sobolev meets Wiener

Even though the Wiener process (Brownian motion) is continuous, it has no derivative at any point. Does it at least have weak derivatives?
4
votes
1answer
1k views

How to derive the Ornstein-Uhlenbeck Stochastic Integral Equation?

I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-a\int^t_0 X_s\,ds +B_t$. B is the Brownian motion. And its analytic ...
4
votes
1answer
378 views

Solutions to stochastic differential equations

I'm in a bit of trouble with my homework and was wondering if anyone could help me find the solutions to these two stochastic differential equations. Would really appreciate it! Thanks in advance! :) ...
4
votes
3answers
575 views

Wiener Process $dB^2=dt$

Why is $dB^2=dt$? Every online source I've come across lists this as an exercise or just states it, but why isn't this ever explicitly proved? I know that $dB=\sqrt{dt}Z$, but I don't know what ...
3
votes
1answer
259 views

Are hitting times of Brownian motion independent?

Suppose that $B_t$ is a standard Brownian motion. And $T_a$, $T_b$ are the hitting time whereas $a<0$, $b>0$. Then are these two random variables independent?
3
votes
1answer
511 views

Proving that a process is a Brownian motion

Let $B$ be a Brownian motion with natural filtration $(\mathcal{F}_t)_{t\geq 0}$ and let $\mathcal{H}_t$ be the $\sigma$-algebra generated by $\mathcal{F}_t$ and $B_1$. Define $$A_t = ...
4
votes
2answers
291 views

Brownian motion introduction

I didn't get any answers to my previous question; so I am trying a different tack. I am familiar with a first course in probability theory using measure theory, to the extent of proving the Central ...
5
votes
1answer
846 views

Brownian motion - characteristic function

Let me remind first the construction of Brownian motion. Fix a vector $x \in \mathbb{R}^n$ and define $p(t,x,y) := (2\pi t )^{-\frac{n}{2}} \cdot \exp{\left( - \frac{|x-y|^2}{2t} \right)},$ for $y ...
9
votes
1answer
1k views

Quadratic variation of Brownian motion and almost-sure convergence

Say that $W(t)$ is a Brownian motion. The quadratic variation $[W,W](t)$ is defined in terms of a partition $\Pi = \{0 = t_0 < t_1 < \cdots < t_n = t\}$ by $$ \begin{split} [W,W](t) &= ...
4
votes
1answer
602 views

Brownian hitting time of a _very_ simple linear boundary

I realize that general results on the hitting times of a curve are practically nonexistant, but I am hoping that someone can string together a sequence of tricks to tell me what $$ \Pr\left( ...
2
votes
0answers
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 ...
5
votes
1answer
364 views

Hausdorff dimension of graphs of one-dimensional Brownian motion

First question here, my apologies if it is a duplicate or inappropriate. There is a page on Wikipedia listing fractals by Hausdorff dimension and it includes the graph of a "regular Brownian ...
3
votes
1answer
236 views

a question about Wiener process

I don't quite understand a property of the Wiener process. Such process has the property that $W(t) - W(s) \sim \mathcal{N}(0, t-s)$ where $t > s > 0$. What I don't understand is this. As the ...
9
votes
4answers
1k views

Showing that Brownian motion is bounded with non-zero probability

How do you show, that for every bound $\epsilon$, there is a non-zero probability that the motion is bounded on a finite interval. i.e. $$\mathbb{P} (\sup_{t\in[0,1]} |B(t)| < \epsilon) > 0$$ I ...
1
vote
1answer
489 views

The infimum of a drifting Brownian motion

Using the reflection principle, it's easily shown that the infimum $\displaystyle \inf_{u \in [0, T]} B_u$ of a non-drifting Brownian motion $B_t$ has the same distribution as $-| B_T |$. The ...
2
votes
1answer
912 views

Distribution of Brownian motion

How would I go about finding the distribution of $B(u) + B(u+v)$ where $u+v > u$? I know that both $B(u)$ and $B(u+v)$ are normal random variables. The sum of two normal random variables is also ...
9
votes
2answers
611 views

Joint moments of Brownian motion

My approach to this SE question uses the following joint moments of Brownian motion. For $n=1,2$ they are obvious and well-known, the others are not terribly hard to work out. Is there a reference ...
13
votes
3answers
2k views

The Laplace transform of the first hitting time of Brownian motion

Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is ...
6
votes
2answers
343 views

Confidence band for Brownian Motion with uniformly distributed hitting position

Let $(B_t)$ denote the standard Brownian motion on the interval $[0,1]$. For a given confidence level $\alpha \in (0,1)$ a confidence band on $[0,1]$ is any function $u$ with the property that $$ ...