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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Probability of Brownian Motion hitting -2 before 1?

Why is the probability of Brownian Motion hitting -2 before 1 is equal to 1/3? This is an interview question asked for Quant roles. I found a similar question was previously asked: Brownian motion ...
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1answer
92 views

What is the distribution of minimum of Brownian motion on arbitrary interval?

We know that $P(\min_{0 \leq s\leq t} B_t \leq x)=2P(B_t\leq x)$. This can be found in any standard stochastic calculus textbook. However I am curious about instead of the interval $[0,t]$ if we ...
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0answers
14 views

Is there a statistical test to test whether a time series behaves like a Brownian motion?

The observed data that I am dealing with is pictured above. It is generated by some mathematical algorithm (simulations) described in section 4.1.(c) in my article, here (data available in spreadsheet ...
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12 views

Understanding a statement about Brownian motion processes

A book I'm reading says, "an important result about Brownian motion is that, conditional on the value of the process at time $t$, the joint distribution of the process values up to time $t$ does not ...
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2answers
46 views

Why does proving $\lim_{h\to 0} (X(t + h) - X(t)) = 0$ prove continuity?

Why does proving $\lim_{h\to 0} (X(t + h) - X(t)) = 0$ prove continuity? I don't think it matters, but more specifically, $X(t)$ is a Brownian motion. I thought that a function $f$ is continuous at ...
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31 views

Covariance of Gaussian Processes based on Brownian Motion

I know that the covariance of two Brownian Motions (Bs,Bt) is min(s,t), but I don't know how to use that to find the covariances of Gaussian processes that are based on Brownian Motion t>0. Yt := e^(...
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1answer
151 views

Black-Scholes model with time-dependent volatility

We consider the Black-Scholes model with time-dependent volatility $\sigma(t)$: $$ dS_{1}(t)=rS_{1}(t)dt+\sigma(t)S_{1}(t)dW(t) $$ The question: what constant $\hat{\sigma}$ one needs to apply such ...
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1answer
215 views

Probability that Brownian motion is negative in $[1, 2]$, given endpoints are positive

Let $B_t$ be a Brownian motion. Compute $P(\inf_{t \in [1,2]} B_t < 0 \mid B_1 > 0, B_2 > 0)$. This is a practice interview question I found here. My attempts are below, and I would ...
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0answers
36 views

Proving that occupation measure of Brownian Motion is absolutely continuous almost surely

I am reading the section on occupation measures from Morters and Peres. I need some help with the following. $\{B(t):t\geq0\}$ denotes the standard Brownian Motion on the probability space $(\Omega,\...
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1answer
49 views

Natural filtration of a Brownian motion and Wiener measure

I have a problem with understanding independence of a process with respect to say a given r.v $\tau$. $B$ and $\tau$ are independent by definition iff $P(B_{t_1} \in A_1, \dots ,B_{t_m} \in A_m, \...
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1answer
56 views

Conditional distribution of process $W$ given $\{W_1 = y\}$ is Gaussian.

Suppose that $X=(X_t)_{t \in [0,1]}$ is a continuous Gaussian process, for which $\mathbb{E}(X_t) = 0$ for all $ t \in [0,1]$ and $Cov(X_s,X_t) = s(1-t)$ for all $0 \leq s \leq t \leq 1 $. Let $Y \sim ...
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1answer
39 views

Constructing Correlated Wiener Processes

Construction Hello. I'm reading the attached paper about the construction of correlated processes given a correlation matrix. But I am stuck on equation (2.23) -- surely it should say $c_{ik} . c_{...
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2answers
72 views

Proving $\omega\mapsto B(\cdot, \omega)$ is measurable.

Let $(B_t)_{t \geq 0}$ be a Brownian motion on the probability space $(\Omega, \mathcal{A},P)$, and $\pi_t$ the canonical projection at time $t$, and $C(\mathbb{R}^+_0,\mathbb{R}^d)$ is the set of ...
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1answer
1k views

Explaining Ito's Lemma

Find $$\int_{0}^{T}W(t)dW(t)$$ using Ito's Lemma. Now, I know that the answer to that question is: $\int_{0}^{T}W(t)dW(t)= \frac{W^2(T)}{2}-\frac{T}{2}$ but can somebody explain the idea behind the ...
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24 views

Construct Correlated Wiener Processes

Construction Hello.. I am trying to better understand how one can correlate independent Wiener processes given a correlation matrix. Please see the attached notes. This method uses the Cholesky ...
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1answer
29 views

Questions about Quadratic Variation given by Brownian Motion

We know that for a submartingle $A(t)$, $A(t)-\langle A\rangle_t$ is a martingale where $\langle A\rangle_t$ is its quadratic variation. For processes like $W^3(t)$ ($W(t)$ being standard Brownian ...
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1answer
46 views

Prove that $\lim_{L\rightarrow\infty} P\left(\sup_{0\leq s\leq t}|B(s)|>L\right)=0$, for each $t\geq0$, where $B$ standard Brownian motion.

Let $B(t)$, $t\geq0$, be a standard Brownian motion. I would like to prove that $$\lim_{L\rightarrow\infty} P\left(\sup_{0\leq s\leq t}|B(s)|>L\right)=0,$$ for each $t\geq0$. In my class notes, ...
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1answer
295 views

Supremum of absolute value of Brownian Motion

I know that by the reflection principle, $$ P\left[\sup_{0 < s < t} B_s > a \right] = 2P[B_t> a] $$ where $B_t$ is a Brownian Motion. But what is $P\left[\sup_{0 < s < t} |B_s|> a ...
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1answer
75 views

Recurrence of Brownian Motion

I am reading a proof of recurrence of Brownian Motion from the book of Morters and Peres. I have a question about a particular step in the proof of neighborhood recurrence for Brownian Motion in ...
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1answer
22 views

Image of a symmetric law

Assume I have a probability space $(\Omega, \mathcal{F}, P)$ that is mapped by a measurable function $X$ into $(E,\mathcal{E})$, moreover $P(X \in U)=P(-X \in U)$, now $Y$ maps this measurable space ...
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0answers
28 views

Reflected Brownian Motion is a Markov process

Let $B=(B_{t},t\geq 0)$ be a Brownian Motion and $M=(M_{t}, t \geq 0)$ the running maximum of $B$. To be more precise, this means $M_{t}=\sup_{s\leq t} B_{s}$ for all $t\geq 0$. In Problem 6.1 c) in ...
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1answer
28 views

Probability that a Brownian motion takes value 0 in an interval.

Given $W(t)$ a standard brownian motion. I.e. $W(0) = 0$. Find the probability that $W(t) = 0$ for $3 \le t \le 4$ The book I am using has an example where: $\displaystyle P(W(s) = 0, 1 \le s \le t) ...
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1answer
16 views

Variance of combination of Brownian Motions

Let $Z(t)=W(t)-\frac{t}{T}W(T-t)$ for any $0\leq t\leq T$ with $W(t)$ a Brownian motion, find the variance of $Z(t)$. My attempt: $Var(Z(t))=\mathbb{E}(Z(t)^{2})-\mathbb{E}(Z(t))^{2}$ $Z(t)=W(t)-\...
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0answers
18 views

Determine rating of change from floating data

For example in this set of data (in ascending order of time) [ 100, 98, 105, 91, 108, 106, 110, 109] It is clearly the trend is rising but i would like to know how to determine the rate of change, ...
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29 views

Brownian motion falling task.

Consider one dimensional brownian motion. Let's suppose it starts at point $a \in [0,L]$. If particles reach the bounds of segment, then they are falling down. Let's suppose that it goes a long time ...
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1answer
74 views

Probability that a Wiener process has no zeroes on $[a,b]$

Let $(W_t)_{t\ge0}$ a Wiener process. We want to find $p:=\mathbb{P}(W_{t} \text{ has no zeroes on $[a,b]$})$. I've considered $$p = \mathbb{P}(W_{t} > 0, t \in [a,b]) + \mathbb{P}(W_{t} < 0, ...
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1answer
347 views

Collection: Results on stopping times for Brownian motion (with drift)

The aim of this question is to collect results on stopping times of Brownian motion (possibly with drift), with a focus on distributional properties: distributions of stopping times (Laplace ...
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1answer
83 views

Why isn’t Brownian motion differentiable?

Intuitively, if increments become infinitesimally small, why doesn’t Brownian motion become a differentiable function?
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1answer
56 views

Ito's Lemma for a Brownian motion

I'm attempting to prove a lemma from a paper, in the context of optimal contracts. $r,\rho,\gamma,\alpha,\sigma$ are all known constants. $dR_t = (\alpha + r)dt + \sigma dZ_t$ where $Z_t$ is a ...
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1answer
111 views

Computing probabilities of standard Brownian motion

Let $W_t$ be a standard Brownian motion. a. Find $P(-3\leq2W_2-3W_5\leq5)$ b. Find the variance of $W_2-3W_3+2W_5$ c. Find $P(-2\leq W_2\leq 3\mid W_1=1)$ d. Find $P(-2 \leq W_3-...
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30 views

Marginal Distribution of Diffusion Process

Working on a problem that I'm having some trouble starting. I have $X_t = 2t + 3B_t$ for $t \ge 0$ where $B_t$ is a Brownian Motion. I want to find the marginal distribution of $X_t$, as well as $E(...
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0answers
51 views

Expected value of a Brownian motion before its first hitting time

Let $X_{t}$ be a Brownian motion with drift $\mu=0$ and variance $\sigma^{2}$. Also, let $X_{0} = a < b$. We know that the density of the first hitting time $H_{b} = inf \lbrace t: X_{t} = b \...
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30 views

Cumulative Distribution Function of Sequence Generated via Random Walk

Is it possible to generically describe the CDF of a finite length random sequence generated by storing the trajectory of a random walk? For example, assume $X$ is an iid random variable with the ...
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1answer
101 views

If $X(t)$ is a Brownian motion, show that $-X(t)$ is also a Brownian motion

Let $X(t), t \geq 0$ be a Brownian motion process with drift parameter $\mu$ and variance parameter $\sigma^{2}$ for which $X(0) = 0$. Show that $-X(t), t \geq 0$ is a Brownian motion process with ...
5
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1answer
1k views

Conditional expectation of Wiener process

I want to calculate $E(W_t | W_1)$, $E(W_t^2 | W_1)$ and $E(W_t^2 | W_1, W_2)$, where $(W_t)_{t\geq0}$ is a Wiener process. For the first one I used the conditional distribution formula for the ...
2
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1answer
100 views

Variance of a Brownian motion

Let $\{X(t), t \geq 0\}$ be a Brownian motion with drift parameter $\mu = 3$ and variance parameter $\sigma^2 = 9$. If $X(0) = 10$, find $P(X(0.5) > 10)$. First, I calculated the expectation and ...
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1answer
43 views

Introductory Brownian motion questions

Let $X(t), t \geq 0$ be a Brownian motion process with drift parameter $\mu = 2.5$ and variance $\sigma^{2} = 8$. If $X(0) = 20$, find (a) $E(X(3))$ (b) $\mathrm{Var}(X(3))$ (c) $P(X(...
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0answers
45 views

Conditional expectation of a function of brownian motion.

Assuming that $\{W(t) | t \geq 0\}$ is a Brownian motion, I am trying to find following conditional expectation $$\mathbb{E}\left[W^{2}(4) | W(1), W(2)\right]$$ My try: What I think is that I ...
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1answer
118 views

Including rotational motion into a reaction-diffusion model

The reference below describes a system of hypothetical sub-particle units or etherons, diffusing from a region of high to low concentration using Fick’s law of diffusion. How would one introduce ...
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2answers
123 views

Conditional expectation of geometric brownian motion

Given a geometric Brownian motion $S ( t ) = e ^ { \mu t + \sigma B ( t ) }$, I'm trying to calculate $E [ S ( t ) | \mathcal { F } ( s ) ]$ where $\mathcal { F } ( s )$ is the history of the process....
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99 views

Convergence in probability of solution of Geometric Brownian Motion

I am working on the following problem Given the solution to the Geometric Brownian Motion $$S_t=S(0)\exp\Big[(\mu-\frac{1}{2}\sigma^2)t+\sigma B_t\Big]$$ Where $\{B_t:t\geq 0\}$ is a Brownian ...
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0answers
40 views

derivative of integral of Brownian motion

\begin{eqnarray}\label{Bht} B^{H}_{t}=\int^{t}_{0}(t-s)^{H-1/2}dW_{s}\,, \end{eqnarray} where $W_{s}$ is a Brownian motion. Then, we can obtain \begin{eqnarray}\label{dBht} dB^{H}_{t}=(H-\frac{1}{2})\...
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0answers
18 views

exponential martingale, expectation of stopping time [duplicate]

Let $\{B(t): t \ge 0 \}$ be a linear Brownian motion. Show that, for $\sigma >0$, the process $\{exp(\sigma B(t)-\sigma^2 t/2): t \ge 0 \}$ is a martingale. 2.Show, by taking derivatives $$\...
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1answer
30 views

Brownian motion subtraction

Assuming that $\{ W ( t ) | t \geq 0 \}$ is a Brownian motion, I'm trying to determine the distribution of the random variable $W ( 1 / 2 ) - 3 W ( 4 )$. Here is my try: From properties of Wiener ...
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0answers
39 views

Generalizations of the Reflection principle for Brownian motion

The reflection principle for Brownian motion roughly states that a Brownian motion reflected a stopping time is also a Brownian motion. More precisely, if $W$ is a Brownian motion and $T$ a stopping ...
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58 views

Moments of a Wiener process evaluated at some random time.

Assume that we have a random variable $T$ that takes values in $[0, \infty)$, and we know that for any continuous integrable function we have that for all $k \in \mathbb{N}$ the following holds $$ \...
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0answers
110 views

Brownian motion probability

Let $(B_t)$ be a standard Brownian motion. How can I compute $P(B_2 > 0 | B_1 > 0)$? I know that $B_2-B_1$ follows a $\mathcal{N}(0,2–1)$, but I do not know how to compute $\int_0^{+\infty}P(B_2&...
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0answers
78 views

Density of hitting time of absolute value of a Brownian motion

I am interested in the probability density of $$ \tau =\inf\{t\geq 0: \vert W(t)\vert = 1\} $$ where $W(t)$ is a standard Wiener process. I have two approaches in mind: First approach: I could ...
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70 views

$B_t^3 - 3t B_t$ is a $L^2$ martingale ($B_t$ being a standard Brownian motion)

By Itô's formula I get that \begin{align} d(B_t^3 - 3t B_t) &= (3 B_t^2 dB_t + 3 B_t dt) - 3(B_t dt + 3 t d B_t) \\ &= (3 B_t^2 + 3t) d B_t \end{align} which seems related to martingale ...
2
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1answer
374 views

quadratic variation of Brownian motion $B(t)$.

Let $\{X_n\}_{n \geq 1}$ be a sequence of random variables with $\mathbb{E}[X_n] = u$. Suppose $\lim_{n \to \infty}\mathrm{Var}[X_n] = 0$. Do we have that $X_n$ converges to constant $u$ almost surely?...