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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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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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18 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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22 views

Freezing lemma and conditional expectation

Let $B$ be a brownian motion on a given filtration $(\mathcal{F}_t)_t$ and a given probability space $(\Omega, \mathcal{F}, P)$. let $\zeta$ be a positive r.v. independent from $B$. Let $X_t=B_{\zeta +...
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39 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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20 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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7 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
15 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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36 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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Find values of $a$ and $\lambda$ for which $Z_{0}e^{at+bW_{t}}-\lambda t$ is a martingale

Find values of $a$ and $\lambda$ for which $Z(t)=Z_{0}e^{at+bW_{t}}-\lambda t$ is a martingale. In here $W_{t}$ is a Brownian motion and $a,b\in\mathbb{R}$ can be positive as well as negative, since $...
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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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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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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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1answer
17 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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9 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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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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26 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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57 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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45 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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45 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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42 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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27 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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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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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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28 views

Correlated stock prices and geometric Brownian motion [migrated]

I have two uncorrelated stocks which follow geometric Brownian motion, as follows $$\begin{aligned} dS_a &= \mu_aS_adt + \sigma_aS_adW\\ dS_b &= \mu_bS_bdt + \sigma_bS_b dW \end{aligned}$$ ...
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1answer
42 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 ...
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1answer
29 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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27 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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2answers
40 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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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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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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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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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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1answer
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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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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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1answer
76 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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43 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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81 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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18 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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1answer
102 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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$B_t^3 - 3t B_t$ 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 ...
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Geometric Brownian Motion and Martingales?

Let Gt denote the canonical filtration of a Brownian motion Wt. Show that for any λ ∈ R, the S.P. Mt(λ) = exp(λWt − λ^{2}t/2), is a continuous time martingale with respect to Gt. Explain why its kth ...
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21 views

$\sum_{i=1}^{n} \Delta B_i^2\stackrel{ c.c. } \longrightarrow T $ for equally spaced interval?

{Q: [0, T] is spaced equally, as usual notation:$\Delta B_i=B_{t_i}-B_{t_{i-1}}$, $t_i-t_{i-1}=T/n$. Show $\sum_{i=1}^{n} \Delta B_i^2\stackrel{ c.c. } \longrightarrow T $} $Proof$: for all $\...
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22 views

Second order derivatives in Ito formula for Brownian motion and local martingale

Itô's formula for a $\mathcal{C}^2$ function of two variables F reads: \begin{align} F(X_t, Y_t) &= F(X_0, Y_0) + \int_0^t \frac{\partial F}{dx}(X_s, Y_s) \, dY_s + \int_0^t \frac{\partial F}{dy}...
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Interpretation of a Wiener process definition

people. So, I have studying an article about modeling a noise in my study field (Telecomunications) and I have found a mathematical notation that I never had found before. $\overrightarrow W(t) \...
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How to find the inverse or a tight bound on a series

If $ f(x)=1-\frac{4}{\pi}\sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1}e^{-\frac{\pi^2 (2k+1)^2}{8 x^2}}$, find $\min\{x:f(x)\geq 1-y \}$. The function $f(x)$ is increasing and its output falls in $[0,1]$...
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1answer
55 views

Brownian Motion problem 22

For a standard Brownian motion $\{W_t, t\geq 0\}$, find $\mathbb{P}(\max_{ t \in [0,1]}|W_t| <x)$. Page 79-80 of Billingsley, P., Convergence of probability measures, New York-London-Sydney-...
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1answer
23 views

Can we re-write the natural filtration of a Brownian Motion?

I'd be astounded if this isn't a duplicate, but I've yet to find anything equivalent. Whenever I've seen the natural filtration of a Brownian Motion stated explicitly, it's been in the form of $\...
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1answer
29 views

Probability of a Large Brownian Particle being at a Certain Position at a Certain Time

I am currently trying to follow along with my notes from a lecture and I am getting very lost in my professor's solution for determining the probability a Brownian particle sits at a specific position ...
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48 views

Expected Value, Brownian Motion, Ito Integral

We know $\mathcal{F}(t)=\sigma(\{W(s):0\leq s \leq t\})$ is the smallest $\sigma$ algebra for which the Brownian Motion $W(s)$ is measurable. We are given these definitions: $$ X(w) = \lim_{n\...
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1answer
43 views

martingale $X(t)=e^{-\lambda t}f(B(t))- \int_{0}^{t} e^{-\lambda s} (\tfrac{1}{2}\Delta f(B(s))- \lambda f(B(s)))ds$

Let $f: \mathbb{R}^d \longrightarrow \mathbb{R}^d$ be twice continuously differentiable and $\{B(t): t \ge 0 \}$ be a d-dimensional Brownian motion. Further suppose that, for all $t>0$ and $x \in \...