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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34 views

Possible master thesis [on hold]

I am searching a topic for my master thesis. My interests are especially probability theory (something with brownian motion would be nice) and fourier analysis (also in an abstract Hilbert space ...
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1answer
39 views

Distributional equality

Let $(W_t)_{t\geq0}$ be a standard Brownian motion. I have to show that the following equality holds in distribution. Does someone has a good hint to show this? $\sup_{t \geq 0}( |W_t| -t) = \sup_{t \...
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1answer
26 views

a conundrum regarding integrated Brownian motion and fractals

Let $X(t)$ be a Brownian motion. I know that the integral \begin{equation} Y(t) = \int_0^t d\tau ~ X(\tau) \end{equation} is well-defined, since Brownian motion $X(\tau)$ is a.s. continuous. Thinking ...
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1answer
23 views

For a one-dimensional Brownian motion $B_t$ $Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}.$

A one-dimensional Brownian motion $B_t$ has exponential moments of all orders, i.e. $$Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}. (2.6)$$ This is given as a corollary to ...
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72 views

brownian bridge and supremum

I want to show that: $$ \sup_{u \geq 0} \frac{1}{u} \left( | B_u | - 1 \right) = \sup_{u \geq 0} \left( |B_u| - u \right) = \sup_{0 \leq u \leq 1 } b_u^2 $$ in distribution; with $ B_u = (1+u)b_{\frac{...
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1answer
39 views

brownian bridge definition [closed]

I am trying to solve an exercise and I have trouble with the definition of brownian bridge. "Let (bu , 0 ≤ u ≤ 1) be the Brownian bridge derived by conditioning a one-dimensional Brownian motion (Bu ,...
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26 views

Time-changed Brownian Motion

Let $B_t$ be a standard Brownian motion and let $\tau_{-1}= \inf \{ t \geq 0: B_t(\omega) = -1\}$. By the Continuous Time Stopping Theorem we know that \begin{align} Z_t = B_{t \wedge \tau_{-1}} \...
2
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1answer
26 views

Sharpen Doob's Maximal Inequality

Let $B_t$ be a Brownian motion, $B_T^* = \sup_{0\leq t \leq T} B_t$ and $\lambda > 0$. Applying Doob's maximal inequality gives: \begin{align} P(B_T^* \geq \lambda)\leq \frac{\mathbb{E}[B_T^p]}{\...
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0answers
26 views

Independence between the first exit time from an interval and the value of Brownian motion at this first exit time

Suppose you have an arithmetic Brownian motion (or Brownian motion with drift ) called X, started at a level x such that a < x < b, where a and b are two real points . Define tau as the first ...
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0answers
17 views

Given a set of 1-dimensional Brownian motions $\{a_i\}_{1\le i\le K}$, what is the average hitting time between $a_1$ and $\{a_i\}_{2\le i\le K}$?

Given $K$ Brownian motions $\{a_i(t), t\ge 0\}_{1\le i\le K}$ contained within interval $[0,s]$, the boundaries at 0 and s are reflected. Assume at time $t=0$, the initial locations are $a_1(0)\le a_2(...
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1answer
50 views

Correlation between stochastic processes

Question: if $W(t)$ is a standard Brownian motion with $W(0)=0$, what is the linear coefficient between the stochastic processes $W(t)$ and $I(t)=\int_0^t W(s)ds$? I argued as follows: what we want ...
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0answers
32 views

System of Stochastic Differential Equations (SDEs) from Diffusion on Manifold

I am looking at a system of SDEs due to Brownian motion on a 3d Riemannian manifold (see e.g. Ito, 1962, The Brownian Motion and Tensor Fields on Riemannian manifolds). I have reduced the associated ...
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1answer
69 views

Compute $\int_1^2 B_t \; dB_t$

I have to compute the following Ito integral: $$\int_1^2 B_t \; dB_t$$ where $(B_t)_{t \geq 0}$ is the 1-dimensional Brownian Motion. In the definition of Ito integral, the integral is taken from $0$ ...
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0answers
17 views

For showing measurability of Brownian motion, how does this set equality holds?

It is stated that the the following set equality easily comes from continuity of paths of Brownian motion $B_t$, but I can't seem to make sense of it - $$\{(\omega,t)\in \Omega\times (0,\infty) : ...
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0answers
11 views

On the conditional distribution of $B_{(s+t)/2}$ conditionally on $(B_t,B_s)$, for Brownian motion $B$

I've been reading stuff about Brownian motions and all that, and I came across the following statement: On proving that $B_{\frac{s+t}{2}}\sim N(\frac{x+y}{2},\frac{t-s}{4})$ conditionally on $B_s=x,...
2
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0answers
50 views

Sum of Wiener, limit in probablitity

Show that the sequence is convergence in probability and set the limit of it: $$\sum\limits_{k=n}^{2n-1}\left(W_{(k+1)/n}^2-W_{k/n}^2-\frac{1}{n}\right)\left(W_{(k+1)/n}-W_{k/n}\right).$$ If there ...
2
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0answers
40 views

martingale square integrable

Let $X_t=\int_0^te^{W_s}dW_s$ and $Y_t=\int_0^tW_sdX_s$. How to show that $X$ and $Y$ are martingale square integrable? ($W_t$ - Wiener) It it enough to show that $\mathbb{E}X_t^2<\infty$, $\...
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2answers
20 views

(locally) square integrable process

We are given a process $\left(X_t\right)_{t\geq 0} = \left(e^{aW_t^2}\right)_{t\ge0}$, where $W_t$ is Wiener process, $a > 0$. Check for which $a$: 1) $\mathbb{E}\int_{0}^{\infty} X_s^2 ds <\...
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0answers
31 views

Changing the order of integration for Brownian motion (outer integration over the range of inner integration)

$X_t$ is bounded Brownian motion and it can be even standard Brownian motion if you wish. I want to express $E[\int_{0}^{T}\int_{0}^{t}X^{n}dsdt]$ as a function of $E[\int_{0}^{T}X^{n}dt]$ For ...
3
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1answer
43 views

Stochastic differential equation substitution reasoning?

I need help to explain reasoning behind why we choose certain substitutions to solve SDE. After choosing the correct substitution the solution of the SDE are often quite easy. However I have trouble ...
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1answer
36 views

Compute expectation and covariance of Brownian bridge

Let $\{X(t), t \geqslant 0\}$ be a standard Brownian motion. That is, for every $t \gt 0$, $X(t)$ is normally distributed with mean $0$ and variance $t$. Then $\{X(t), 0 \leqslant t \leqslant 1 | X(1)...
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2answers
33 views

What is the distribution of the subtract of two random variables?

Definition) A stochastic process $\{X(t), t \geqslant 0\}$ is said to be Brownian motion process with drift coefficient $\mu$ and variance parameter $\sigma^2$, if it satisfies that $X(0)=0$. $\{X(t)...
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0answers
22 views

Why are Brownian Motion and Levy processes beginning “almost surely” at 0?

I am studying stochastic calculus, and I had a question about the definition of both Brownian motion, as well as Levy processes. So the formulation that I have seen both on Wikipedia and my textbook(...
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0answers
12 views

Please verify the solution about Brownian motion process.

Problem Let $Y(t)$ denote the amount of time by which the racer is ahead when $100t$ percent of the race has been completed. $\{Y(t), 0 \leqslant t \leqslant 1\}$ is modeled as a Brownian motion ...
2
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1answer
42 views

Solving Langevin equation

In a past exam paper that I am looking at, there is the following question: Given that the displacement, $\mathbf{x}$, of a particle in $3$-dimensional Brownian motion is given by: $$m\ddot{\...
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1answer
27 views

From brownian bridge to brownian motion proof

Let $B_t$ be a brownian motion. and let $\{W_t=B_t-tB_1:0\le t\le 1\}$ be a brownian bridge. Now let $Y_t=(1+t)W_{t\over 1+t}$. Proof that $Y_t$ is a brownian motion in $[0, \infty)$ My attempt: 1) $...
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1answer
37 views

Verifying a Brownian motion through the Laplace transform

Let $X(t)$ be a continuous stochastic process and $\mathcal G(t)$ be the $\sigma$-algebra generated by $\{X(\tau) : \tau\leq t \}$. Suppose that for any $0\leq s\leq t$ and $\lambda\in\mathbb C$ ...
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25 views

Is $B_{t\wedge H_a}$ bounded in $L^2$?

Let $a >0$, $(B_t)_{t\geq0}$ be a standard Brownian motion. Define the stopping time $$H_a := \inf\{t \geq 0 : B_t \geq a\}.$$ Then is the martingale $M_t$ where $M_t: = B_{t\wedge H_a}$ bounded ...
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0answers
22 views

Brownian Motion maximum process intuition

I am studying the maximum value of a Brownian Motion (BM) on an interval of time (as explained here between boxes 28 and 40) and I am having an issue aligning intuition with the mathematical result. ...
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0answers
29 views

Brownian motion independent RVs

Let $(W_t)_{t\in\lbrack 0,T\rbrack}$ be a standard Brownian motion. Does there hold that $W_s(W_t-W_s)$ and $W_k(W_l-W_k)$ for $0\leq s<t\leq k<l\leq T$ are independent RVs?
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1answer
30 views

What is the difference between these two formulas that price a stock? [closed]

What is the difference between these two formulas? They are both related to the price of a stock in the black-scholes model. The fact that the second one uses $t$ as a subscript which means it's not a ...
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0answers
23 views

Brownian motion - absolute value

I'm having some trouble integrating the equation in 8.2.5 (I'm trying to do 8.2.6). I need to do some form of u-substitution but I'm unsure of u=?. Also, once I've done the integration, to show that ...
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0answers
24 views

Reflected Brownian Motion probability

So I know that R(t) = |5 + B(t)| and that B(25) ~ N(0,25). I was told that P{R(t)>=10} = P{|5+B(25)|>=10} = P{B(25)>=5)+P{B(25)<=-15} but I'm not entirely sure how to get that. And I've been ...
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0answers
39 views

Brownian Moment Generating Function and Hitting Times

Here is my question. I've done the first part, but I'm stuck on the second. If I can work out (/be advised) how to do the second, then I hope to be able to do the third similarly. Please note: While ...
3
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2answers
84 views

Crossing of Brownian Motion Sample Paths

I would like to ask for a more rigorous statement and proof of Lemma on page 5 of this paper. In essence, it states that two distinct sample paths of a Brownian motion does not strictly cross (meaning ...
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0answers
24 views

What is the general Taylor Expansion for the following function of a function.

guys. I am stuck with a general form of Taylor Expansion of following function, which is defined as a function of a function: $$F(X(t+h))-F(X(t))=[X(t+h)-X(t)]\frac{dF}{dX}(X(t))+\\\frac{1}{2}[X(t+...
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0answers
22 views

System of SDEs and independence [on hold]

I am recently reading a paper that seems to claim the following fact without justification: $Y^1_t, \ldots, Y^n_t$ are stochastic processes defined on $\mathbb{R}$. Let $b: \mathbb{R}^2 \...
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0answers
39 views

Gaussian processes and bias

I would like to simulate two Gaussian processes along a time grid. Ideally, I would like to see the average of the samples, for each grid point, to be close to the mean. Using the antithetic method, I ...
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2answers
53 views

How to prove that the stochastic integral process is gaussian?

I would like to prove that for a $C^1$-function f and a Wiener process W, the integral process defined by $$ Y_t:= \int_0^t f (s)dW_s := f (t)W_t -\int_0^t W_s f'(s)ds $$ Is a centered gaussian ...
2
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0answers
37 views

Survival probability of a biased random walker

A random walker moves to $+1$ with probability $p$ and moves to $-1$ with probability $q=1-p$. If he starts at point $m$, what is the probability that he doesn't hit the point zero after $k$ steps, ...
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1answer
38 views

How to find the standard deviation from the given information and what is $B(0)$ equal to?

Assume that the risk free rate is $0$ and that the stock price is given by the equation $S(t)=6e^{2t+2B(t)}$ where $B(t)$ is the standard Brownian motion. Determine the price at time $0$ of the ...
1
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1answer
47 views

Maximum process of Brownian motion

Consider the linear standard Brownian motion $(B_t)_{t \ge 0}$. We define the maximum process $(M_t)_{t \ge 0}$ of $(B_t)_{t \ge 0}$ to be such that $M_t = \max_{0\le s \le t} B_s$. Prove that the ...
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0answers
38 views

How it is shown by the following integral?

Example: Ornstein-Uhlenbeck Process. Let $ dx=-\eta xdt+\sigma dz $ be an Ornstein-Uhlenbeck Process Write the moment-generating function for $x(t)$ as $$ M(θ,t)≡E(e^{-θx})=∫_\infty^∞ ϕ(x_0,t_0;x,...
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0answers
27 views

Verifying data came from a Wiener Process

From the Wiki article a Wiener Process has the properties that $$E[W_t] = 0$$ $$Var[W_t] = t$$ According to A Standard Wiener Process the Wiener Process is given by: $$W(t) - W(s) \tilde{} \sqrt{t-...
3
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1answer
38 views

Limit Brownian Bridge Integral

As a solution of the Brownian Bridge SDE, we arrive at the solution \begin{align} X_t = (1-t) \int_0^t \frac{1}{1-s}\ dB_S \end{align} defined for $0 \leq t <1$. In order to show that for any $g \...
3
votes
1answer
33 views

Wiener process and stochastic int

Let $h:[0,1] \rightarrow \left\{-1,1 \right\}$. How to show that $X_t=(\int_0^th(s)dW_s)_{t \in [0,1]}$ is a Wiener process? I know from the lecture that for every $h$ process $\int h \ dW_s$ is ...
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0answers
37 views

$n$ times integrated Brownian motion martingale process

According to this post, we found that a $n$ times integrated Brownian motion could be expressed as, \begin{align} V_n(t) = \int_0^t V_{n-1}(s)\ ds = \frac{1}{n!} \int_0^t (t-s)^n\ dB_s, \end{align} ...
3
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0answers
49 views

Find $P(B_3>0,B_6>0)$ where $(B_t)$ is a Brownian motion

Suppose that $B_{t}$ is a standard Brownian Motion. What is the probability that both $B_{3}$ and $B_{6}$ take positive values? This is what I've tried but then I get stuck and I'm not sure how to ...
0
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2answers
29 views

Integration by parts - Brownian motion and non-random function

Let $B$ be a standard one-dimensional Brownian motion. I want to show for a continuously differentiable non-random function $\phi$ that, \begin{align} \int_0^t \phi(s) dB_s = \phi(t) B_t - \int_0^t ...