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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Covariance of stochastic integral

I have a big problem with such a task: Calculate $\text{Cov} \, (X_t,X_r)$ where $X_t=\int_0^ts^3W_s \, dW_s$, $t \ge 0$. I've tried to do this in this way: setting up $t \le r$ $$\text{Cov} \, (...
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0answers
41 views

Probability that Brownian motion falls between two piecewise constant functions

I'll first present the problem, and then describe my motivation: Suppose $a_j \in \mathbb{R}$, $b_j \ge 0$, and $0 = t_0 < t_1 < \cdots < t_J$ are time points. Let $W_t$ be a standard ...
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1answer
14 views

Show that $(B_t)$ and $(tB_{1_t})$ has the same distribution where ($B_t)_t$ is a brownian motion

Let $(B_t)_{t\geq 0}$ a brownian motion s.t. $B_0=0$. I want to show that $B_t$ and $tB_{1/t}$ has the same law. $$p\{tB_{1/t}\leq x\}\underset{u=1/t}{=}p\{\frac{1}{u}B_u\leq x\}=p\{B_u\leq ux\}$$ ...
3
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2answers
45 views

Mean and Variance Geometric Brownian Motion with not constant drift and volatility

I have to derive the Geometric Brownian motion (with not constant drift and volatility), and to find the mean and variance of the solution. $\quad \left\{\begin{aligned} & d X_t = \mu(t) X_t d t +...
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2answers
56 views

process with integral is martingale

How to show that the process $X_t=tW_t - \int_0^t W_s ds $ is a martingale? I guess I have to use the definition of martingale and properties of Wiener process, but I stack with this integral. Please,...
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1answer
36 views

Basic Question on Definition of Brownian Motion

I am quite new to discrete and continuous stochastic processes. It seems there is something I don`t understand about definition of Brownian motion. Let $\Omega, \mathcal{F}, \mathbb{P}$ be a ...
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0answers
21 views

Understanding of Second Arcsine law for Brownian motion

Ok I'm trying to understand the second arcsine law which states: Let $g_t:=\sup\{s\leq t:W_s=0\}$, then $$\mathbb{P}(g_t\leq s)=\frac{2}{\pi}\arcsin \left(\sqrt{\frac{s}{t}}\right )$$ This won't be ...
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1answer
17 views

Definition of Cylindrical Brownian Motion and Spatial Correlation

From Gawarecki and Mandrekar, Stochastic Differential Equations in Infinite Dimensions: We call a family $\{ W_t \}_{t\geq 0}$ defined on a filtered probability space $(\Omega, \mathcal{F}, \{\...
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1answer
20 views

Clarification regarding heat kernel for Brownian motion on a manifold

Let $X$ be Brownian motion on a Riemannian manifold $M$ starting at $x\in M$, D a domain and $f$ a bounded continuous function on $D$. Define $\tau_D$ to be the first exit time of $X$ from $D$. $u_f\...
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1answer
14 views

Concerning Covariance and Brownian Motion

Let $\{ X(t), t \ge 0\}$ be standard Brownian motion. How do I find Cov$[X(3) - 2X(2), X(4)]$? The answer is $-1$, but I can't seem to get there no matter what I hit it with. I know that $X(3) \sim ...
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1answer
41 views

Density of Running Maximum of Drifted Brownian Motion Computation

$\textbf{Proposition}$ The $pdf$ of the Maximum of a Brownian Motion with Drift is given by $$ f_{M_t}(m)={\sqrt{\frac{2}{\pi t}}} \mathrm{e}^\frac{(m-at)^2}{2t}-2a\mathrm{e}^{2am}\mathcal{N}\left (...
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0answers
36 views

Brownian motion on a manifold

If I have a manifold $M$ and a chart $\left(x,U\right)$, is it possible to simulate Brownian motion on that manifold by solving an SDE in the chart representation $x\left(U\right)$ and then use the ...
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1answer
29 views

SDE for Brownian motion on a circle [closed]

Brownian motion on a circle can be generated by $\left(\cos\left(B_t\right),\sin\left(B_t\right)\right)$ where $B$ is Brownian motion on the real line. My question is what SDE was solved to get this ...
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1answer
31 views

Maximum of standard brownian motion on an interval

I'm trying to find the probability that the maximum of standard Brownian motion on the interval $(t_1, t_2)$ exceeds a value $x$, i.e., $$P(max_{t_1 \le s \le t_2}B(s) \gt x)$$ I initially ...
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0answers
32 views

Stochastic process is brownian motion by Levy's characterization

I would like to know if $B_t=W_t-\int_0^t \frac{W_u}{u}du$ is a brownian motion. I know that $W_t$ is a brownian motion. For that i would like to use Levy's characterization, so I have to show that $[...
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1answer
58 views

Simple question on interpreting Geometric Brownian Motion SDE

I'm writing an overview that is more economic than mathematical and I want to explain shortly the stochastic differential equation of Geometric Brownian Motion as simple and clear as possible $$dS_t ...
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1answer
32 views

Probability using Brownian Motion

Assume that $B(t)$ is a Brownian motion and that $S(t)$ is defined as $S(t)=A\cdot e^{B(t)}$ for some positive constant $A$. Calculate the probability of the event ${S(3)>2S(1)}$. How could I go ...
3
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1answer
33 views

convergence in distribution of exponential of a brownian motion

If $(B_t)_{t≥0}$ is a standard Brownian motion, show that, as $t \to \infty$, $$ \left(\int_0^t e^{B_s} \, ds\right)^{1/\sqrt{t}} \text{ converges in distribution to} \ e^{M_1}, $$ where $M_1 = \sup_{...
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1answer
30 views

Geometric Brownian motion with exponential of sum of iid's

Glasserman's "Monte Carlo Methods in Financial Engineering" on p. 265 states that the geometric Brownian motion can be modelled with : $$S(t_n)=S(0) \exp(\sum_{i=1}^n X_i)$$ where $X_i$ are iid. I ...
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1answer
49 views

Show uncorrelated, with Brownian motions

I have $W_t$ is a Brownian Motion and $$B_t :=W_t-\int_0^t \frac{W_u}{u}du$$ is also a Brownian Motion. I have to show that these two are uncorrelated. I know for Brownian uncorrelated is ...
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1answer
36 views

How do you find the probability of a brownian motion?

If $B(t)$ is a brownian motion what do these two questions mean? 1. What is the probability of $B(2)$ 2. What is the probability of $B(2) \gt B(1)$ I know this is also called a Wiener Process and ...
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0answers
31 views

Brownian Motion with Levy's Characterization 2

Let W be a $\mathbb{R}$-valued Brownian motion. To prove that $(B_t)_{t\geq 0}$, where: $B_t:=W_t-\int_0^t\frac{W_u}{u}du$, is a Brownian Motion with respect to $\mathcal{F}^B$, I showed $[B]_t=t$ and ...
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1answer
15 views

Can Wiener process be axiomized without normal increments

A common characaterization of Wiener's process is the following which I took directly from Wikipedia: $W_0 = 0$ a.s. $W$ has independent increments: $W_{t+u} - W_t$ is independent of $σ(W_s : s ≤ t)...
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0answers
61 views

Brownian motion with Lévy’s Characterization

I want to show that: if for all $\lambda \in \mathbb{R}$ the process $\left(\exp\left(\lambda X_t-\frac{\lambda ^2}{2}t\right)\right)_{t\geq0}$ is a $\mathcal{F}^X$ local martingale, then the $\mathbb{...
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1answer
15 views

Strong Markov Property Brownian Motion for Non-Stopping Time

Let $B$ be a Brownian motion and let $\mathcal{F}^B$ be its natural filtration. Define the random variable $$ \tau = \inf\{ t \ge 0 \mid B_t = \sup_{0 \le s \le 1} B_s \}.$$ Now, $\tau$ is not an $\...
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0answers
16 views

Difference of Running Maximum of a Reflected Brownian Motion and the Reflected Brownian Motion

For a Brownian Motion $W_t$ and $M_t=\sup_{s<t} W_s$, we know $M_t-W_t$ is a reflected Brownian Motion. For a reflected Brownian motion $X_t=|W_t|$ and the running maximum $M'_t=\sup_{s<t} X_s$, ...
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0answers
17 views

Maximum of Brownian Motion and a constant

I am interested in the distribution of $Z(t) = \max\{B(t),m\}$ where $B(t)$ is a standard Brownian motion and $m$ is a constant. By distribution, I mean the distribution of $Z(t)$ for a given $t$. I ...
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1answer
19 views

Conditional expectation: when does $X_t=E[X_t\mid \mathcal{F}_s]$ for $s<t$

I came across a calculation (1$^\circ$ calculation, 2$^{nd}$ step) that stated, for $s<t$ $$E[B_s(B_t^2-t)]=E[B_sE[(B_t^2-t)\mid\mathcal{F}_s]]$$ I know the expectation here is zero, however, I ...
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0answers
17 views

Brownian motion conditional expectation: $E\left[(B_s-B_t)^3\mid\mathcal{F}_t\right]$

Compute: $E\left[(B_s-B_t)^3\mid\mathcal{F}_t\right]$, $s>t$. $B_t$ is standard 1D Brownian motion, and $\mathcal{F}_t=\sigma(B_t)$. Here is my attempt: $$ \begin{aligned} E\left[(B_s-B_t)^3\mid\...
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0answers
44 views

Hitting Times for Brownian Motion - Levy Process?

Let $X$ be a Brownian motion and let $$H_a = \inf\{ s \ge 0 \mid X_s = a \} \;\ \text{and} \;\ S_a = \inf\{ s \ge 0 \mid X_s > a \}.$$ Now, I've shown that $H_a$ and $S_a$ are equal almost surely (...
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1answer
44 views

How would you simulate Brownian motion with a die?

You can simulate Brownian motion on $[0, 1]$ for instance by splitting it into $K$ intervals and at each time $k \Delta t$ add $N(0, \Delta t)$ to your running total, where $\Delta t = 1/K$. If you ...
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46 views

Some Kind of Generalized Brownian Bridge

Let $\displaystyle X(t) = \int_0^t f(s)dB(s)$ where $B(t)$ is a Brownian motion and $f(t)\in L^2[0,1]$. What is a simple representation for $Y(t):=(X(t)|X(1))$ in terms of $B(t)$? Note, I am not ...
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1answer
51 views

Maximum Likelihood Estimation of Brownian Motion Drift

I'm looking at times series of stock movements over 10 minute windows, and am trying to measure the "trend" of these movements. Method A is to simply calculate $\Delta P$, the difference between the ...
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1answer
45 views

Martingale and local martingales

I have to show that $e^{B_t^1}\cos(B_t^2)$ is a martingale ($B=(B^1,B^2)$ is a two-dimensional Brownian Motion). I used Ito's formula and got $e^{B_t^1}\cos(B_t^2)=1+\int_0^t e^{B_s^1}\cos(B_s^2)dB_s^...
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0answers
20 views

Expectation related to Wiener process using strong Markov property

Can you help me to understand a result I found in Krylov's book "Introduction to stochastic calculus". First, I will introduce some notations: $w_t,t\ge 0$ denotes a Wiener process. $\mathcal{B}(...
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0answers
31 views

Expectation of an Exponentiated Integral of a Brownian Bridge

Given a Brownian bridge $X(t)$ where $X(0)=0$ and $X(1)$ equal to some given constant. What is $\displaystyle \mathbf E\Big[\exp\Big(\int_0^1X(t)dt\Big)\Big]$? I suppose I can always discretize the ...
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14 views

What is skew-product decomposition?

What is skew-product decomposition of Brownian motions referring to this paper Pauwels ,Rogers
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How can we proove that it's a Gaussian system?

$(W_1, W_2)$ are 2 independent Wiener processes and $$B_1= W_1, ~~~ B_2 = a W_1 + \sqrt{1-a^2} W_2,$$ where $a=(a(t, \omega))_t>0$ and is $(F_t=F_t^{(W_1,W_2)})$-measurable. $0<a<1$. It ...
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1answer
42 views

$tB_t$ Integral representation, Brownian Motion

I never learned stochastic differential equations, and so am trying to do some self study. I've arrive at this question: $tB_t\sim N(0,t^3)$? $B_t$ is standard brownian motion. $B_t\sim N(0,t)$, so ...
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0answers
15 views

Reflection principle for the modulus of the Brownian Motion

I have the following question. Suppose we define $M(t)=\sup_{0\le s\le t}|B(s)|$, where $B$ is an ordinary Brownian motion in $\mathbb{R}$. How can we compute $P(M(t)\ge a)$? Is it $2P(|B(t)|\ge a)$? ...
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18 views

Holder continuity of Brownian Motion

Can any one help me in this question please: By using the law of iterated logarithm, I have to show that $\forall t \in [0, T]$ ; where T>0, $\exists A_{t} \in \mathcal{A}$ with $P(A_{t})=1$, a random ...
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0answers
21 views

To show that $\dim Z=1/2$, why do I have to show that $p\{\dim Z=1/2\}=1$?

Let $(B_t)_{t\geq 0}$ a standard Brownien motion. I have to show that $\dim Z=\frac{1}{2}$ where $Z=\{t\in [0,1]\mid B_t=0\}$. Why to do this, I have to show that $$\mathbb P\{\dim Z=1/2\}=1\ \ ?$$ I ...
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0answers
56 views

Brownian Motion hitting time is finite yet has infinite expectation?

I've read that a hitting time of a Brownian motion (defined as $T_a = \inf\{t\ge0:W_t=a\}$ where $W_t$ is a standard Brownian Motion, i.e. a Wiener process), has the following two properties, which I ...
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1answer
31 views

Reference request for stochastic processes on manifolds

I'm looking for some references on stochastic processes on manifolds. The more introductory the better. Thanks.
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39 views

Ito's formula application

Let $ \alpha, \beta \in R$ and define $$ N(t)=e^{\beta t} \cos(\alpha W (t)) $$ I need to use Ito formula to compute $dN(t)$ Suppose $\alpha$ is fixed. What should $\beta$ be so that $N$ is a ...
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35 views

Yet another application of Ito's formula

Question : Let $dW^4(t) $ be the sum of an ordinary integral with respect to time and an Ito integral. Where $W^4(t)$ are standard Brownian motion. I am trying to apply Ito's formula to this, say ...
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0answers
18 views

bromnian motion and use of Lebesgue's differentiation theorem

Let $M$ be a Brownian motion with $M_0=0$ and $V\in L(M)$. Use Lebesgue's differentiation theorem to prove that there exists a predictable process $H\in L(M)$ such that $V\cdot M$ and $H\cdot M$ are ...
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0answers
35 views

show that a function with brownian motion is a martingale

Let $B=(B^1,B^2)$ be a two-dimensional Brownian motion w.r.t. the Filtration $\mathcal{F}^B$. Show that $(M_t^2)_{t\in \mathbb{R}_{+}}:=(e^{B_t^1} \cos(B_t^2))_{t\in\mathbb{R}_{+}}$ I've tried it ...
1
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1answer
37 views

Integral representation $B_T^3$

I have to find a $F_t$ such that $B_T^3=E[B_T^3]+\int_0^T F_t dB_t$. I have shown by ito formula that $B_T^3=\int_0^T 3 B_s^2 dB_s+\int_0^T 3 B_s ds$. Could you please help me?
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22 views

Brownian martingale as time-space changed brownian

Let $M$ be a true real martingale adapted to some brownian motion $B$. What are the most generic conditions on $M$ to find a deterministic map $\Phi:\mathbb{R}_+\times\mathbb{R} \to \mathbb{R}_+\...