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Questions tagged [brownian-motion]

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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Running Supremum of standard Brownian motion and probability distribution

I am reading Optimal Stopping and Free-Boundary Problems by Peskir and Shiryaev and noticed a result on page 151 as follows: $\mathbb{P} (\sup_{t \geq 0} (B_t - \alpha t) \geq \beta) = \exp(-2\alpha \...
Harry Wang's user avatar
2 votes
1 answer
48 views

Expected hitting time of 2-d brownian motion with $B_0 = (x_0, y_0)$

I wonder if it is possible to calculate the expected time of hitting a ball of radius $R$ while the initial position of the 2-d Brownian motion is not at the origin but $(x_0,y_0)$ inside the ball. I ...
LOREY CHU's user avatar
2 votes
0 answers
44 views

Probability not to fall after $n$ steps

Let's say I stand back to a pit. Every second a (fair) coin is tossed and its result dictates if I step forwards or backwards. Of course stepping backwards at the first second means falling into the ...
Anne Aunyme's user avatar
1 vote
0 answers
38 views

Finding the law of a Brownian motion triple

Let $B=(B_t)_{t\geq 0}$ be a standard Brownian motion and let $S_{s,t} = \sup_{s \leq r \leq t} B_r$. It is well known from a classic result of Lévy that $S_{0,t} - B_t$ realises a Brownian motion ...
user82832's user avatar
2 votes
1 answer
36 views

Hitting time by the minimum of two Brownian motions

Consider the hitting time, $\tau_0$, of $0$ by a Brownian motion $B_t$, started from $B_0=1$. It is well known that $\mathbb{E}\tau_0 = \infty$. However we also know that the minimum of two random ...
user1598's user avatar
  • 395
2 votes
1 answer
51 views

How to show Geometric Brownian motion is not a Gaussian process?

Let's consider Geometric Brownian motion: $$ X_t = e^{\mu t + \sigma B_t} $$ where $B_t$ is Brownian motion. Question: How to prove that this process is not Gaussian? I understand that $B_t$ itself is ...
poiug07's user avatar
  • 41
1 vote
0 answers
24 views

General Solution of Dirichlet Problem on a Jordan Domain

I am trying to find a proof (or related materials) for the Theorem 3.4 in this paper by Mervin E. Muller. The theorem is stated as follows: Let $D$ be a Jordan domain, with boundary $\Gamma$. Let $f(\...
ARessegetes Stery's user avatar
4 votes
1 answer
53 views

What is the typical probability space when we talk about a Brownian motion?

What is the typical probability space $(\Omega, \mathcal{F}, P)$ when we talk about a Brownian motion $(B_t)$? I found some papers used $\Omega = C[0,\infty), \mathcal{F}=\mathcal{B}(C[0,\infty))$, ...
Mingzhou Liu's user avatar
1 vote
1 answer
50 views

Questions in proving $\mathbb{P}\left(T_a<\infty\right)=1$ with $T_a:=\inf \{t>0: B_t \ge a\}$

Let $\left(B_t, t \geq 0\right)$ be a one-dimensional Brownian motion starting from the origin (i.e, $\left.B_0=0\right)$. Let $\mathcal{F}_t:=\sigma\left(B_s: s \leq t\right)$ be the filtration ...
Ho-Oh's user avatar
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Quadratic Variation and Scaling Property Of Brownian Motion [closed]

Given smooth functions $\psi$ and $\phi$, where $\psi$, $\phi$ are non-negative and the latter is also strictly increasing, I have calculated that $\langle \psi B_{\phi(\cdot)} \rangle_t = \int_0^t \...
AlexAsks's user avatar
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0 answers
42 views

Multiplying Brownian Motion by Imaginary Unit

Let $B$ be a $d$-dimensional Brownian motion defined over forward and backward time, i.e. the whole real line is the time domain. It is a well-known result that if we multiply by a linear ...
AlexAsks's user avatar
0 votes
0 answers
36 views

Time derivative of Ito Integral does not exist [closed]

Consider a continuous, square-integrable, real-valued function $f$ over $[0, \infty)$. It is known from standard stochastic analysis that $$\int_0^{\cdot} f(s) d B_s$$ is a closed martingale in $L^2$, ...
AlexAsks's user avatar
1 vote
0 answers
19 views

Constant in front of Riemann-Liouville process and terminology

Consider a Riemann-Liouville process of the form: $$ B^H_t = c_H \int_0^t (t-s)^{-H-1/2} dB_s$$ Why can the constant $c_H$ be changed without affecting the properties of the process (as it will change ...
gb2718's user avatar
  • 21
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0 answers
73 views

Rescaled time Ito Integral

I am wondering what can be said about the differential $$ dB_{\phi(t)} $$ for some continuously differentiable function $\phi: [0, \infty) \to [0, \infty)$ with $\phi'(t) >0$. Meaning: Is there a ...
AlexAsks's user avatar
2 votes
0 answers
36 views

Is there a closed form solution to the distribution of the running maximum for Gaussian processes?

Let $X=(X_t)_{t\geq0}$ be a zero mean Gaussian process with variance $\sigma(t)=\mathbb{E}X_tX_t^\top$ and define $$ S_T=\sup_{t\leq T}X_t\quad T\in[0,\infty) $$ the running maximum of $X$. Question: ...
Daan's user avatar
  • 362
0 votes
1 answer
55 views

Why is Brownian motion normally distributed?

I'm reading the book Probability Theory: A Concise Course (Rozanov), and in the chapter where the normal distribution is first discussed, Brownian motion was given as an example. Here are the ...
Alex's user avatar
  • 142
2 votes
0 answers
95 views

Brownian Motion / heat flow generated by Hodge Laplacian

Let $\square_M = - (dd^* + d^*d)$ be the Hodge Laplacian on the differential forms $\Omega(M)$ (or if you wish, on a fixed $\Omega^k(M)$). What is the stochastic process generated by this operator? ...
Alex's user avatar
  • 637
-1 votes
1 answer
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Geometric Brownian Motion inequality. [closed]

I have $\int_0^\infty e^{-\beta t}(1+\mathbb E[|Y_t|]+\mathbb E[Z_t])dt \le \int_0^\infty e^{-\beta t}[1+\mathbb E[|Y_t|^2]+z\mu t]^{\frac{1}{2}}dt$- Here $Z_t = ze^{\mu t-\frac{\sigma^2}{2}t+\sigma ...
oxedex's user avatar
  • 27
1 vote
1 answer
46 views

Prove that solution to geometric brownian motion is correct (plug into SDE)

As many previous questions/answers point out, the solution to the geometric brownian motion stochastic differential equation (GBM SDE) $$ \left\{ \begin{array}{ll} dX_t &=& \mu X_t \,dt + \...
SebaGM's user avatar
  • 13
0 votes
1 answer
45 views

$X\in L^{\infty,p},Y\in L^{2,q},prove \int_{0}^{t} XY_{s}dB_{s}$ is martingale

$L^{\infty,p}$ is X is progressively measurable and $E[\max_{0\le t\le T} \left | X_{t} \right |^{p} ]<\infty$ $L^{2,q}$ is Y is progressively measurable and $E[(\int_{0}^{T}\left | Y_{t} \right |^{...
Yu GongLian's user avatar
0 votes
1 answer
95 views

Show this process is Brownian motion

Let $U(t)=W(t)-\int_0^t\frac{W(s)}{s}ds$. Show that $U(t)$ is a Brownian motion. I am struggling to prove that $\text{cov}(U(t),U(v))=\min\{t,v\}$. If anyone can help it would be greatly appreciated. ...
Cyno Benette's user avatar
3 votes
1 answer
76 views

Reverse engineering a property for Geometric Brownian Motion

During a lecture on financial mathematics we were given a side (non-homework) question to hone the SDE solving skills. Setting $\left(\Omega, \mathcal{F}, \mathbb{P},\left(\mathcal{F}_t\right)_{t \in[...
markovian's user avatar
  • 157
0 votes
1 answer
24 views

Find $P(B_2 > 0, B_8 > 0)$

We need to compute $P(B_2 > 0, B_8 > 0)$ where $B_t$ is brownian motion Now, $P(B_2 > 0, B_8 > 0) = P(B_2 > 0, B_8 - B_2 > -B_2) = P(Z_1 > 0, Z_2 > -Z_1)$ where $Z_1 \sim N(0, ...
Harsh's user avatar
  • 378
2 votes
1 answer
47 views

Is $\int_{\tau_0}^{\tau_1}f(W_t)\mathrm{d}t$ independent of $\mathcal{F}_{\tau_0}$ for some stopng times $\tau_0, \tau_1$ and Lebesgue integrable $f$?

Let $W=(W_t)_{t\geq 0}$ be the standard Brownian motion, let $X_t=W_t+X_0$ with initial distribution $X_0\sim\delta_x$ for $x\in\mathbb{R}$, let $\tau_1^1=\tau^1=\inf\{t>0:X_t=1\}$, write $\tau_0^1=...
Daan's user avatar
  • 362
2 votes
0 answers
26 views

Conditional Expected Stopping Time for Brownian Motion with Drift

I previously asked the following question: Expected stopping time for Brownian motion conditional on lower barrier being hit first. I attempted to extend the result to a Brownian motion with drift, ...
Andrew Ferdowsian's user avatar
2 votes
0 answers
33 views

Brownian motion runs through a circle

Let $(B_t)$ be standard Brownian motion starting from 0, and $\mathbb{R} \to \mathbb{R}/2\pi\mathbb{Z} = S^1$ be a canonical projection. I want to calculate the expectation of the stopping time $T$, ...
nessy's user avatar
  • 582
1 vote
0 answers
23 views

A question on the proof of the ratio limit theorem for Brownian motions.

I am trying the prove the following ratio limit theorem, and I think I got it but I am not entirely sure that the mathematical logic in the last part of the proof is correct, and so I am asking if ...
Daan's user avatar
  • 362
2 votes
2 answers
62 views

Is $\int_s^t f(W_u)\mathrm{d}u$ independent to $\mathcal{F}_s$ for all (Lebesgue) measurable $f$?

I am trying to prove (or disprove) whether $$ I[s,t]=\int_s^t f(W_u)\mathrm{d}u $$ is independent of the $\sigma$-algebra $\mathcal{F}_s$ for any $t,s$ with $t\geq s$, for $W=(W_t)_{t\geq 0}$ the ...
Daan's user avatar
  • 362
1 vote
0 answers
27 views

Brownian motion $B_t$ and $\int_0^t\mathrm{sgn}(B_s)dB_s$ equal in distribution

I am reading a paper about local time and the running maximum of Brownian motions. Let $B$ be a Brownian motion and $L^x_t$ its local time. By Tanaka's formula, we have $$|B_t|=\int_0^t\mathrm{sgn}(...
Schrödinger's cat's user avatar
3 votes
2 answers
58 views

What is the transition semigroup of toroidally wrapped Brownian motion?

More generally, let $(B_t)_{t\ge0}$ be an $\mathbb R^d$-valued Lévy process and $W:=\iota(B)$, where $$\iota:\mathbb R^d\to[0,1)^d\;,\;\;\;x\mapsto x-\lfloor x\rfloor$$ (the floor function is applied ...
0xbadf00d's user avatar
  • 13.9k
1 vote
1 answer
35 views

What is the probability that Brownian motion hits $\mathbb{Q}^d$?

Let $B$ be a standard $d$-dimensional Brownian motion. What is $\mathbb{P}(\exists t > 0 : B_t \in \mathbb{Q}^d)$? For $d = 1$, this is clearly $1$. However, intuitively, it seems that it should be ...
J. S.'s user avatar
  • 412
4 votes
1 answer
102 views

Quadratic variation of the square of Brownian motion

Let $B_t$ be the Brownian Motion. Find the quadratic variation of a martingale $ M_t = B_t^2-t$. My solution: By Ito's formula for $f(t, x) = x^2-t$, we know $$d(B_t^2-t) = 2B_t dB_t$$ thus $\langle M ...
nessy's user avatar
  • 582
3 votes
1 answer
96 views

every neighborhood of every zero of Brownian motion takes positive and negative values almost surely

How can I prove that almost surely every neighborhood of every zero of Brownian motion takes positive and negative values? (*) I can already prove that for every zero of Brownian motion, Brownian ...
romperextremeabuser's user avatar
2 votes
1 answer
50 views

Sum of hitting times and hitting times of sum of Brownian Motion

I have came across a question that asks to prove that for a brownian motion $B$, the first hitting time of $T_a=\inf\{t≥0 : B_t=a\}$ has a $1/2-$stable distribution, in that if we have $n$ independent ...
R.V.N.'s user avatar
  • 767
6 votes
1 answer
79 views

Simulate a Brownian motion by exponential time stepping

Let $(B_t)_{t\ge0}$ be a Brownian motion. We can simulate a path of $(B_t)_{t\ge0}$ using the Euler-Maruyama discretization scheme. Now, in the paper Efficient Numerical Solution of Stochastic ...
0xbadf00d's user avatar
  • 13.9k
0 votes
1 answer
24 views

Proving the Conditional Expectation of a Uniformly Distributed Random Variable Given the Sum of Two i.i.d. Uniform Random Variables [closed]

Consider two independent and identically distributed (i.i.d.) random variables (X) and (Y), each having a uniform distribution over the interval ([0, 1]). Define a new random variable (Z) as the sum ...
AshCode002x's user avatar
1 vote
1 answer
56 views

May the sum of Wiener processes be a Wiener process?

May $X_t = W_t + \tilde{W}_t$ be a Wiener process, if $W_t, \tilde{W}_t$ are Wiener processes? I know that: $X_t$ may be not a Wiener process, e.g. in case $W_t = \tilde{W}_t$. If $W_t$ and $\tilde{...
Sergei Nikolaev's user avatar
1 vote
1 answer
76 views

Showing the stationary distribution of Langevin Diffusion without Fokker Plank (Rosenthal 15.6.9)

Given a standard diffusion: $dX_t = \sigma(X_t)dB_t + \mu(X_t)dt$ with $\sigma(X_t)=1$ and $\mu(X_t) = \frac{1}{2}\frac{\pi^\prime(X_t)}{\pi(X_t)}$ for suitably smooth $\pi$ such that $\int \pi(x)dx = ...
Oxderp's user avatar
  • 61
3 votes
1 answer
54 views

Let $(x_t: t\geq 0) $ be the Brownian motion. $E(f| x_t)=0$ implies $f=0$?

Let $(x_t: t\geq 0)$ be the standard 1-dimensional Brownian motion on the Wiener space $(\Omega, \Sigma, d\mu)$. We assume $\Sigma$ is the (complete) $\sigma$-algebra generated by all $(x_t: t\geq 0)$....
Landau's user avatar
  • 1,963
2 votes
1 answer
56 views

Convergence of weighted sum to Brownian Motion

Let $\{\varepsilon_t\}_{t = 1}^T$ be a sequence of iid random variables such that $\varepsilon_t \sim N(0, \sigma^2)$ and $\sigma^2 > 0$. Then it is known that (see 17.3.6 in James Hamilton's Time ...
Wittgenstein's Poker's user avatar
0 votes
0 answers
25 views

Conditional Expectation - Geometric Brownian Motion

Given a stock dynamics following Geometric Brownian Motion (GBM) such that $$dS_t = rS_tdt+\sigma S_tdW_t.$$ Then, the stock price at $t+\Delta t$ can be expressed as $$S_{t+\Delta t} = S_{t}\exp\left(...
Xuan's user avatar
  • 81
1 vote
1 answer
25 views

Probability that one Brownian motion hits 2 before the other hits 1

Let $B_t$ and $W_t$ be independent brownian motions. What is the probability that the first one hits 2 before the second one hits 1? Express the answer as an integral. I am quite stuck on this. I have ...
I_cosine_this's user avatar
0 votes
0 answers
18 views

Unproved proposition about constant probability concerning Wiener processes

At Brzezniak, Zastawniak, Basic Stochastic Processes, p.159 the hint of exercise 6.30 says: [Given that $V_c(t)=\frac 1c W(c^2t)$ is a Wiener process, where $W(t)$ is a Wiener process too] Deduce that ...
SK_'s user avatar
  • 653
1 vote
2 answers
130 views

Prove that $(t+1) X_{\frac{t}{t+1}}$ is a Brownian Motion using Levy's characterisation where $X$ is a Brownian Bridge

The following result is well documented and is a result of the Stochastic Processes course I am following. Below the result, I present the standard proof presented in my course (Method 1) and my ...
FD_bfa's user avatar
  • 4,343
3 votes
2 answers
51 views

Completion of the Brownian motion filtration: is this right-continuous?

I am self-studying Durrett's probability book. Let $\{B_t^x\}_{t \geq 0}$ be the Brownian motion starting from $x\in \mathbb{R}$. Here, the sample space is the set of continuous functions $\{ \omega| \...
J. Doe's user avatar
  • 1,075
3 votes
1 answer
50 views

Expected stopping time for Brownian motion conditional on lower barrier being hit first

Suppose we have a Brownian motion $B(t)$, with $B(0) = 0$, and boundaries $a < 0 < b$. Define $\tau_a = \min\{t : B(t) = a\}$ and $\tau_b = \min\{t : B(t) = b\}.$ Conditional on the fact that $a$...
Andrew Ferdowsian's user avatar
10 votes
2 answers
721 views

How to apply Ito's Formula to show that this is a martingale?

In the book Brownian Motion, Martingales and Stochastic Calculus by J.F. Le Gall, in order to give an alternatice derivation of the distribution of $L_{U_{a}}^{0}(B)$ where $L^{0}_{t}(B)$ is the Local-...
Dovahkiin's user avatar
  • 1,285
1 vote
0 answers
29 views

Writing $\frac{dM}{dt}(t)$ for $M(t)$ being a stochastic process.

In the paper, the equation 2.35 is an implicit form of a stochastic differential equation. Here is the excerpt from the paper: The implicit form of a stochastic differential equation is given as $$\...
MonteNero's user avatar
  • 337
2 votes
1 answer
36 views

Exercise 9.1 in Introduction to stochastic processes by Lawler

This is not for any assignment or homework. I am studying Chapter 9, Stochastic Integration, in Introduction to Stochastic Processes by Lawler. Exercise 9.1 states Since I am self-studying, I'm ...
toronto hrb's user avatar
2 votes
0 answers
26 views

Proof of Theorem 5.38 in Morters and Peres' Brownian Motions book

In the proof of Theorem 5.38 which states that if $\tau_{a}$ is the hitting time of $a>0$ and if $\sigma_{a}=\inf\{t\geq 0:B(t)\geq |a|\}$, then $\int_{0}^{\tau_{a}}\mathbf{1}_{\{0\leq B(t)\leq a\}}...
Dovahkiin's user avatar
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