2
votes
0answers
28 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 ...
1
vote
0answers
27 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 ...
0
votes
0answers
24 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 ...
0
votes
1answer
38 views

Property of Wiener process sample path

What is a mean of time, when the trajectory of wiener process $W_t$ is over the line $y=t$? We need to find $\mathbb{E}\tau$, where $\tau=\sum\limits_{a,b:\forall t\in(a,b) ; ...
0
votes
1answer
15 views

Expectations of certain Brownian motion equations

$B_t$ is Brownian motion. It is assumed that motion starts at $0$. I do not understand how the highlighted equalities hold true. Is the first one equivalent to ...
1
vote
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 ...
4
votes
2answers
88 views

Distribution related to brownian bridge

Let $B(t)$ be a Brownian Bridge and $U$ is uniformly distributed on $(0,1)$. I wish to know the distribution function $B(U)$. Is it possible? As we know, $B(t)\sim N(0,t(1-t))$. But, I haven't a clue ...
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 ...
0
votes
0answers
25 views

Computing equilibrium measure for Borel sets eg. Ball

I am asking for methods to compute equilibrium measures. The more the better. Here is the definition of equilibrium measure in the Brownian motion setting: Let $\gamma=\sup\{t\in [0,T]: B_{t}\in ...
2
votes
1answer
32 views

Scaled integrated Brownian motion has limit

Let $B$ be a standard Brownian motion and put $$X(t)=\frac{1}{\sqrt{t}}\int_{0}^{t}f(B(s))ds,$$ where $f \in L_1(\mathbb{R}^{1})$ and $\int f(x)dx=1$. Show that $$ \lim_{t \rightarrow \infty} EX(t) ...
0
votes
0answers
15 views

Given $T=\inf\{t\geq 0: B(t)=a \text{ or }b\}$, identifying $T^{2}$ and $E[B(T^{2})]$.

For stopping time $T=\inf\{t\geq 0: B(t)=a \text{ or } b\}$ , standard Brownian motion $B(t)$, and $a<0<b$, do we get $T^2=\inf\{t\geq 0: B(t^{1/2})=a \text{ or } b\}$? Also, can we say sth ...
1
vote
1answer
48 views

(Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$

The purpose of this question is to complete my personal exposition on the rigorous proof of Ito's lemma. I have consulted more than half a dozen mathematical finance texts and not a single one, for ...
1
vote
1answer
94 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
1
vote
1answer
55 views

SDE and Stochastic calculus

$W_t$ is 1 dimension Brownian morion. $X_t=(cosW_t,sinW_t)$ Write SDE about $X_t$ I thought that $f(t,x)=(cosx,sinx)$, but I can't how "$t$" is expressed. I heard that the hint of this question is ...
1
vote
0answers
41 views

conditional expectation of the Brownian motion [duplicate]

$(B_t)$ is a Brownian motion and i assume that $s<t<u$ we have $$E[B_t |\sigma(B_s,B_u)] = G(B_s,B_u)$$ Does anyone knows the explicit expression of $G$ ? (the calculus is easy but ...
1
vote
0answers
24 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 ...
0
votes
1answer
36 views

How big a Brownian bridge can get? Confidence band.

If we know the endpoints of the Brownian path, is there any theorem telling us if it can be contained within a ball a.s. (with probability one)? For example contained in two big enough balls (call it ...
2
votes
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 ...
1
vote
2answers
51 views

MGF of stochastic integral

Question: Let: $$ Y_t=\int_0^t\alpha_s \, dW_s $$ where $\alpha_t$ is a deterministic, continuous integrand and $W_t$ is a P BM. Calculate the moment generating function of Y. I can solve this ...
4
votes
1answer
160 views

Lookback option with floating strike: boundary condition

I am trying to make sense of one of the boundary conditions of a look-back option with floating strike. Some notation first: let $v(t,x,y)$ denote the price at time $t$ of the option under the ...
1
vote
2answers
61 views

Uniformly integrable martingale

I have the following martingale. $M_n=\exp\left(aB_n-\frac{1}{2}a^2n\right)$ for $n\geq0$ and $a\neq0$, $B_n$ is a BM. I have to show that for $a>0$, $M_n\rightarrow0$ in probability. Is $M_n$ ...
0
votes
1answer
35 views

Brownian Motion with drift (stupid question)

How do you prove that $$ \lim_{t\to +\infty} (B_t+ct)=+\infty $$ almost surely? $(B_t)_t$ is the standard Brownian Motion starting from $0$.
0
votes
2answers
74 views

Doob's decomposition of a brownian motion.

Let $B_n$ be a discrete Brownian motion. I need to find the Doob decomposition for ($B_n^2$). Can someone help me please. Thank you in advance.
2
votes
2answers
118 views

conditional expectation of brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion in $\mathbb{R}^d$. It is intuitive that, for fixed $s<t<u$ $$\mathbb{E}[B_t\mid \sigma(B_s,B_u)]=B_s+\frac{t-s}{u-s}(B_u-B_s).$$ However, I ...
1
vote
1answer
33 views

Property of Brownian Motion's paths

We are considering a Brownian Motion $(B_t)_t$ with values in $\mathbb{R} $ starting from $x$ defined on the stochastic basis: $$(\Omega,\mathcal{E},(\mathcal{F}_t)_t,\mathbb{P}^x)$$ Then, let's ...
0
votes
2answers
64 views

Is the following Itô-Integral not zero?

is the following statement true: $$\int_0^T t \, dW(t) \neq 0$$ I need it for a counter-example, that one can not change the order of integration between $dW$ and $dP(\omega)$. I thought of taking ...
4
votes
1answer
46 views

Expectation of $e^{-4B_\tau}$, where $\tau$ is an extended stopping time

This is an specific example so with a bit of luck I can get some general methodology from your answers. I have this stopping time: $$ \tau = \inf\{t \geq 0; B_t < t-2 \} $$ This is a clear ...
2
votes
1answer
42 views

The law of the iterated logarithm for BM and boundedness of stopping times

My question is regarding the usefulness of the law of the iterated logarithm, and its connection to stopping times. In many answers of this forum, I understand that some people often claim that some ...
5
votes
2answers
124 views

Submartingale example: proof

I am trying to prove if the process $M_t = e^{W_t^2-t}$ is a submartingale ($W_t$ is the Wiener Process). The proof becomes a bit difficult, to the point where I am unsure how to move forward. Let ...
5
votes
1answer
90 views

Is $t^{-\frac{1}{2}}B_{t^2}$ a Brownian Motion?

I think the title says it all. Let $X_t = t^{-\frac{1}{2}}B_{t^2}$, with $B_t$ being a brownian motion started at $0$. I think I have proved continuity at $0$ by doing the following: $$ X_t = ...
2
votes
1answer
56 views

Hitting time for Brownian Motion Surplus Process

I'm trying to solve this question for a continuous surplus process. The surplus process is $$U_s=U_0+s-B_s$$ where $B_t$ is a Brownian motion representing payouts, $U_0$ is starting capital, $s$ is ...
2
votes
0answers
56 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 ...
4
votes
1answer
93 views

Ornstein-Uhlenbeck process: increments

I'm new to the forum so I hope this first question goes well. Let the Ornstein-Uhlenbeck process be defined as: $$ dV_t = - \beta V_t dt + \sigma dW_t $$ with $V_0 = v$, where $W_t$ is a Wiener ...
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 ...
0
votes
1answer
26 views

Probabilistic solution Poisson problem

Let us consider the Poisson problem \begin{cases} \frac{1}{2}u''=-f\qquad\text{in}\,\,(a,b)\\u(a)=u(b)=0 \end{cases} where $f:(a,b)\to\mathbb{R}$ is continuous and bounded. We have obtained ...
0
votes
0answers
27 views

Strong law of large numbers for a Brownian motion

I have a question in regard to the strong law of large numbers. It is well known that, if $B_t$ is a Brownian motion, then $\displaystyle\lim_{t\to\infty}\frac{B_t}{t}=0$. This had me wondering... ...
1
vote
1answer
26 views

Reflection Principle interpretation

Given a standard Brownian motion $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P},(B_t)_t)$ (the standard filtration $(\mathcal{F}_t)_t$), we define $$\forall t\ge 0: M_t:=\max_{0\le s\le t} B_s$$ ...
1
vote
1answer
67 views

Mean and Variance of Gaussian Process

Let $B = (B_t : t \geq 0)$ be a standard Brownian Motion. Fix $0 \leq s \leq t$. How can I prove that, conditionally on $\{B_s = x, B_t = z\}$, the intermediate value $$B_{\frac{t+s}{2}}$$ has ...
0
votes
1answer
74 views

Ornstein-Uhlenbeck process and Markov property

There isn't a similar question in the forum, so here it goes. Firstly, let the O-U velocity process be defined as $$ dV_t = -\beta V_t dt + \sigma dB_t $$ with $V_0 = v$, and $B = (B_t), t \geq 0$ a ...
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 ...
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 ...
1
vote
1answer
89 views

Expectation and variance of correlated exponential brownian motions

What is the expectation and variance of correlated exponential Brownian motions for the random variable $F$, where $A$ is real constant, $\sigma$ is a real constant and $\rho$ is the correlation. $$F ...
1
vote
0answers
70 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 ...
2
votes
1answer
81 views

Is the graph of a Brownian motion over an interval measurable?

Let $n \in \mathbb{N}_1 := \{1, 2, \dots\}$ and let $B:\Omega \times [0, \infty) \rightarrow \mathbb{R}^n$ be a standard, $n$-dimensional Brownian motion over the probability space $(\Omega, ...
1
vote
1answer
44 views

Convergence to integral: $\sum_{k=1}^{k_n}f\left(B_{t_{k-1}^{(n)}}\right)\left(B_{t_{k-1}^{(n)}}-B_{t_{k}^{(n)}}\right)^2 \to_p \int_0^Tf(B_t)dt$

The problem goes: Let $(B_t)$ be a standard Brownian motion, and $f:\mathbb{R}\to\mathbb{R}$ be continuous. Show that if $T>0$ and $(P_n)$ is a sequence of partitions of $[0,T]$: ...
3
votes
1answer
82 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 ...
4
votes
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 ...
0
votes
0answers
23 views

Reflected borel sets are worse traps for Brownian paths.

This is for a research project. I am trying to prove that given a borel set, it's reflected version will have a lower Wiener measure of brownian paths intersecting it. In this paper they elaborate ...
1
vote
1answer
67 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 ...
8
votes
2answers
213 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 ...