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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7
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145 views

Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
6
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0answers
168 views

Does this random variable have a density?

I have a persistent problem, which I'm almost certain can be answered using elementary probabilistic arguments, but for some reason I've been stuck for some time. Here is the problem. Let $(B_s, s ...
6
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293 views

Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
5
votes
0answers
26 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 ...
5
votes
0answers
137 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 ...
4
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0answers
125 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 ...
4
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0answers
47 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 ...
4
votes
0answers
106 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 ...
4
votes
0answers
206 views

Question about an exercise in Revuz/Yor

I'm solving exercise 2.28 in Revuz/Yor. I was able to prove 1). Unfortunately at 2) I got stuck. I have to show: Let $B$ be a d-dimensional Brownian motion and $A\in \mathcal{A}:=\cap_t ...
4
votes
0answers
158 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
0answers
58 views

Law of iterated logarithms for BM

The law of iterated logarithms for the standard Brownian motion asserts that $(\ast) \limsup\limits_{h \downarrow 0} \frac{B(h)}{\sqrt{2h\log\log(\frac{1}{h})}} = 1$ I'm trying to prove the ...
3
votes
0answers
44 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) - ...
3
votes
0answers
41 views

When is a continuous path stochastic process be representable as diffusion or Ito process?

When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
3
votes
0answers
43 views

Upper bounds on the sum in a Martingale process

My question is related the hitting time of not a random walk, but a more general martingale process. Suppose we start with an arbitrary $x_0=x$ with $0\leq x\leq 1$. We compute $x_{t+1}$ from $x_t$ ...
3
votes
0answers
193 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 ...
3
votes
0answers
120 views

infinitesimal generator of reflecting Brownian motion

Suppose $f\in C_0^{\infty}([0,\infty))$ and $f'(0)=0$. I'm having trouble proving that $$\frac{1}{t}E_x[f(|W_t|)-f(x)]\to\frac{1}{2}f''(x)$$ uniformly on $[0,\infty)$ as $t\downarrow0$. Showing the ...
3
votes
0answers
159 views

Show that $O_t$ is a Gaussian Process

Let $B_t$ be a Brownian motion process. Let $$O_t = e^{-\alpha t} \int^t_0 e^{\alpha s} dB_s$$ Find $\mathsf{E}[O_t]$ and show that $O_t$ is a Gaussian process. I think ...
3
votes
0answers
113 views

A question regarding the strong Markov property

In our lecture on Brownian motion & stochastic calculus we proved: If $ X $ is a canonical RCLL process having the strong Markov property and $ \tau $ is a stopping time with $ \tau < + \infty, ...
3
votes
0answers
193 views

Expected time spent in the set

An exercise 2.14 from Bernt Øksendal's "Stochastic Differential Equations": Let $B_t$ be $n$-dimensional Brownian motion and let $K\subset \mathbb R^n$ have zero $n$-dimensional Lebesgue measure. ...
2
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0answers
31 views

Maps that preserve Brownian motion law

I am looking for a list of maps that take Brownian motion to Brownian motion: Here are some: Any rigid transformation ...
2
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0answers
26 views

Finding a pre-visible process

Question: Let $W_t$ be a standard brownian motion under P with filtration $\mathscr F_t$. Let: $$ M_t=\mathbb E[W_T^2|\mathscr F_t] $$ Show that $M$ is a P martingale. This is simple enough using ...
2
votes
0answers
57 views

Defining the scale function of a diffusion process

My question has to do with correctly calculating the scale function of a diffusion process, but ultimately might only have to do with calculus. I'll briefly set-up my calculations, so you can quickly ...
2
votes
0answers
39 views

Hitting time of a maximum of random walk converges to that of Brownian motion

Suppose $S_n$ is a simple random walk; formally, $S_n=\sum_{i=1}^n X_i$ for $X_i\sim\mathcal{U}(-1,1)$, i.i.d.. Denote by $M_n$ the maximum of the random walk on $n$ steps; formally, $M_n=\max_{0\le ...
2
votes
0answers
50 views

Stopping times, open sets and Brownian Motion

Let $B_t$ be a brownian motion started at 0. I am trying to prove that $\tau$, defined as: $$ \tau = \inf\{t > 0 \mbox{ }|\mbox{ } \left|B_t\right| \geq \frac{1}{1+t} \} $$ is a stopping time with ...
2
votes
0answers
40 views

A right-inverse of Brownian motion local time at zero has stationary independent increments

Let $L_0^t$ be the local time for a standard Brownian motion at $0$ and define $$X_t=\sup\{s\ge0:L_0^s\le t\}, t\ge0. $$ I would like to show that $(X_t)$ has stationary independent increments. That ...
2
votes
0answers
28 views

Eigenfunctions of a 2D fractional Brownian motion covariance

The fractional Brownian motion is a centered Gaussian process with the following covariance function (covariogram): $E[B(t)B(s)]=C(\Vert t \Vert ^{2H}+\Vert s\Vert^{2H}-\Vert t-s\Vert^{2H})$ ...
2
votes
0answers
45 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 ...
2
votes
0answers
52 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?
2
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0answers
60 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 ...
2
votes
0answers
41 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 ...
2
votes
0answers
145 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( ...
2
votes
0answers
100 views

Almost sure non differentiability of Brownian Motion

Problem: Let $t>0$, show that the standard Brownian motion is almost surely not differentiable a $t$ Now, through a Borel Cantelli argument I proved that, almost surely $$\limsup_{\epsilon ...
2
votes
0answers
49 views

Can anyone explain me this proof about a Brownian Motion?

Prove that the process $W_t=(1+t)U_{t/(1+t)}$ on $[0,\infty)$ is a Brownian motion. $\text{(b)}$ Clearly $Y_0=U_0=0$, and inherits continuity of sample paths from $U_t$ (and hence from $W_t$). ...
2
votes
0answers
58 views

first hitting time probability for a Brownian motion with variable diffusion

I am looking for the first hitting time probability of the following Brownian motion: $dX=\mu X dt+ \sigma (X) X dW$ assuming $X(0)=X_0$ and $\sigma(X)= \sigma_1$ if $X>X_1$ and ...
2
votes
0answers
45 views

Exercise in brownian motion

Consider a system of n particles moving in three dimensional space under the action of an external force with $C^1$ potential V and coupled to a heat bath causing an external random effect. Then we ...
2
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0answers
106 views

Quadratic variation process of $G$–Brownian motion

I would like to prove the inequality $$\hat{\mathbb{E}}\left[\left(\int^T_0 \eta_t d \langle B \rangle_t \right)^2\right] \leq C \hat{\mathbb{E}}\left[ \int^T_0 \eta^2_t dt \right],$$ where $\langle B ...
2
votes
0answers
70 views

Negative moments of a functional of Wiener process

At the moment I am reading D. Nualart's The Malliavin Calculus and Related Topics. The problem I am trying to solve is the following: Show that the random variable $\int_0^1 s^2\arctan W_s\, ds$ ...
2
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0answers
105 views

Independence of Brownian motion-related stopping times

Let $(B_t,\mathcal{F}_t)_{t \geq 0}$ a Brownian motion on a probability space $(\Omega,\mathcal{A},\mathbb{P})$. For $a \in \mathbb{R}$ define a stopping time $\tau_a$ by $$\tau_a := \tau(a) := ...
2
votes
0answers
117 views

Independence of Brownian Motion with respect to a stopping time

Let $B_t$ be a brownian motion, $B_0=0$, and $\gamma \in \mathbb{R}$. Now, let's build the following stopping time: \begin{equation} T = \inf \{ t \geq 0 : |B_t + \gamma t| = 1 \}. \end{equation} If ...
2
votes
0answers
96 views

A problem with regard to Wiener process

Let $W$ be a Wiener process and $U_x$ is the amount of time spent below $x$ during time interval $(0,1)$. Hence $U_x=\int\limits_0^1I_{\{W(t)<x\}}dt$. My question is: what is the probability ...
2
votes
0answers
130 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 $${1\over C(H)}\int_\mathbb{R}\left((t-s)_+^{H-{1\over2}}-(-s)_+^{H-{1\over2}}\right)dB(s)$$ ...
2
votes
0answers
160 views

Correlated diffusion processes and covariance matrix

I'm really noob in maths topics so I hope you will excuse me if I use terms which aren't correct. I would like to simulate $n$ dimensional diffusion processes with $n$ noises. Each process has its ...
2
votes
0answers
45 views

prove that two r.v.s share the same law

I have a question in my homework about Brownian motion. Does someone have a idea about the following question? Let $X=B^+$ or $|B|$ where $B$ is a standard BM, $p>1$ be a real number and $q$ its ...
2
votes
0answers
108 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 ...
1
vote
0answers
29 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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0answers
26 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 ...
1
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0answers
9 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 ...
1
vote
0answers
26 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 ...
1
vote
0answers
18 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 $n\to\infty$ ...
1
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0answers
34 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 ...