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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Integrating deterministic function with respect to Brownian motion

I have looked everywhere for a satisfactory answer to this, including Shreve's textbooks, but I can't find one. If I want to integrate a some deterministic function f(t) with respect to brownian ...
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89 views

why the sigma algebra generated by null set and Brownian Motion is right continuous?

I mean why the generated one satisfies the definition of right continuous?
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63 views

first hitting time probability for a Brownian motion with variable diffusion

I am looking for the first hitting time probability of the following Brownian motion: $dX=\mu X dt+ \sigma (X) X dW$ assuming $X(0)=X_0$ and $\sigma(X)= \sigma_1$ if $X>X_1$ and ...
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143 views

infinitesimal generator of reflecting Brownian motion

Suppose $f\in C_0^{\infty}([0,\infty))$ and $f'(0)=0$. I'm having trouble proving that $$\frac{1}{t}E_x[f(|W_t|)-f(x)]\to\frac{1}{2}f''(x)$$ uniformly on $[0,\infty)$ as $t\downarrow0$. Showing the ...
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That Brownian Motion's increments are gaussian is “not surprising”?

In section 1 of chapter 1 of Continuous Martingales and Brownian Motion, the authors claim that the fact that the increments of of Brownian motion are gaussian random variables "is not ...
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186 views

Hitting probabilities for Brownian motion

Let $\mathbb D$ be the complex unit disk. Let $B$ be a standard complex Brownian motion started at $0\in \mathbb D$. Let $\tau = \inf\{ t : B_t \in \partial\mathbb D\}$. I am trying to show that if ...
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1answer
226 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
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1answer
123 views

maximize the expected value of the logarithm of the weighted average of random variables

I'm trying to do the following. $$\max_{m\in\mathbb{R}} \mathbb{E}\left[\log (wA + (1-w)B_m)\right],$$ where $0<w<1$ and $A, B_m > 0$ are correlated random variables. $A$ does not depend ...
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146 views

Intuition for the optimality of bold play

There is a standard result (I think originally by Dubins and Savage) that if one wants to maximise the probability of winning a certain amount in an unfair game of chance then an optimal strategy is ...
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1answer
504 views

Sum of 2 Brownian motions

Let's say, that $B_t$, $t\geq0$ is standard Brownian motion (Wiener process). Let's define process $$X_t=B_t+B_{t^2}\text{, }t\geq0$$ I need to find its variance, covariance, find out if it's ...
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2answers
98 views

Brownian motion or not?

Suppose that $(X_t , t\in [0;1])$ are independent normal r.v with mean 0 and variance $\sigma^2 _{t}$. Is this process brownian motion?
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128 views

why is the expected value of a Wiener Process = 0?

This section of wikipedia says that the expected value of a Wiener Process is equal to 0. Why is that?
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1answer
78 views

Measure of $\{t:B_t\in E\}$ for some null set $E$.

I am wondering if the following result can be found in any textbook or if you have a proof of it. When $E$ is a null set and $B_t$ is the Brownian motion, we have almost surely : ...
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1answer
113 views

Expectation of integral of involving geometric brownian motion

Compute $$\mathbb{E_P} \left( \exp{(\alpha W_t)} \int_0^t \exp{(\gamma W_u)} \,du \right)$$ where $\alpha$ and $\gamma$ are real numbers and $W_t$ is a Brownian Motion.
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100 views

Correlation function of Brownian motion. What am I doing wrong?

Can anyone tell me where I am going wrong here? (I am leaving out any random fluctuation forcings, because I don't think they are relevant to my problem.) 1: $\displaystyle \frac{dv(t)}{dt}=-\eta ...
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2answers
631 views

Conditional distribution in Brownian motion

I need to prove the following: Let $X$ be a Brownian motion with drift $\mu$ and volatility $\sigma$. Pick three time points $s < u < t$. Then, the conditional distribution of $X_u$ given ...
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1answer
467 views

Function of brownian motion is a martingale

Let $B_t,t\geq 0$ a brownian motion and $u(t,x)$ a function satisfying the following PDE $$\frac{\partial u}{\partial t}+\frac{1}{2}\frac{\partial^2 u}{\partial x^2}=0.$$ Then we prove that ...
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1answer
50 views

distribution of Brownian Motion involving integral

What is the distribution of $\int_{t}^{T} W(s)ds$? Given that W(t) is brownian motion. So far, I have the following, $\int_{t}^{T} W(s)ds$ = $(T-t)W(t) + \int_{t}^{T} (T-s)dW(s)$ Also, ...
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2answers
260 views

How to solve this stochastic integrals?

how can I solve these two stochastic integrals? $$\int_0^T B_t\,dB_t$$ $$\int_0^T f(B_t)\,dB_t$$ where B_t is the BM. Thank you very very much!
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1answer
499 views

Brownian bridge

Let $W = (W_t;F_t)$, $t \leq 0$ be a standard Wiener process, and let $(X_t)_{0 \leq t \leq 1}$ satisfy the stochastic differential equation $$ dX_t =- \frac{X_t}{1-t}dt+dW_t,\quad 0 \leq t \leq ...
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49 views

Exercise in brownian motion

Consider a system of n particles moving in three dimensional space under the action of an external force with $C^1$ potential V and coupled to a heat bath causing an external random effect. Then we ...
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1answer
1k views

Distribution of Sum of Two Brownian Motions

How do we find the distribution of the sum of two Brownian Motions? The questions was asked here: Distribution of Brownian motion, and was answered with We can write ...
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1answer
79 views

Stochastic differential equation problem and applying ito formula

I am given that for $b,a,\sigma >0$ and $x \in (-a,b)$ and $\nu \in \mathbb{R}$, I have the following stochastic differential equation: $$ dZ_t = \nu \,dt + \sigma\, dW_t$$ $$ Z(0) = x$$ and ...
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2answers
44 views

Independence of $T$ and $B_T$

Let $\{B_t:t\ge0\}$ be a real brownian motion such that $B_0=0$. Let $T=\inf \{t:B_t \notin (-a,a)\}$ with $a>0$. Are $T$ and $B_T$ independent? I tried the following and I would like your ...
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1answer
111 views

Brownian motion recurrence theorems and Hausdorff Dimension

I need help with proving: 1.If $d>1$ then d-dimensional Brownian motion starting at $x$ has 0 probability to actually hit $y$. Note that this is different from the usual notion of recurrence, ...
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1answer
670 views

Stochastic process, Gaussian, with zero mean is a Wiener process

Let $(\Omega, \mathcal F , \mathbb P)$ be a probability space and let $\mathcal F = \{\mathcal F_t\}_{t\ge} $ a filtration. Let $W=\{W_t;t ≥ 0\}$ be a stochastic process adapted to $\mathcal F$. ...
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1answer
761 views

How do you make dependent Brownian motions independent?

Can someone explain to me how to take 2 correlated Brownian motions and make them independent? I can't seem to grasp this process. Just assuming $dB_1(t)dB_2(t) = \rho dt $ From what was explained ...
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1answer
72 views

$P\{X_t=-X_t \}=1$

If we define that $X_t$ is Brownian motion over space $(\Omega,\mathcal F ,\mathcal F_t;P) $, then why is it true that the fact that $X_t$ is Brownian motion implies that $P\{X_t=-X_t \}=1$ is ...
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1answer
147 views

lower bound of expectation of stochastic differential equation

I'm looking for a lower bound on the expected value of a smooth, non-negative, increasing function $\mathbb{E}f(X_t)$, $f(0)=0$ of the solution to a stochastic differential equation $X_t = x + ...
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1answer
300 views

How is Brownian motion predictable?

Could someone please explain how Brownian motion is predictable? My understanding is that a predictable process is one that depends on information up to time t say but not t itself, therefore W_t has ...
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1answer
95 views

An application of Donsker's theorem.

Let ${X_i}$ be iid with $E[X]=0$ and $Var(X)=\sigma^2$ Let $S_0=0$ and $S_n=X_1+...+X_n$ for all $n \ge 1$. Show that $\lim_{n\rightarrow \infty}P(S_k>0 \space for \space k=n,n+1,\dots,2n)=1/4 $. ...
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282 views

Product of correlated brownian motions

Consider that the correlation between two standard brownian motions $dB_x$ and $dB_y$ be $\rho$. And we write $\mathtt{Cor} (dB_x,dB_y)$ = $\rho$. Show that $dB_xdB_y$ = $\rho dt$
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1answer
138 views

Let $W$ be a Wiener process and $X(t):=W^{2}(t)$ for $t\geq 0.$ Calculate $\operatorname{Cov}(X(s), X(t))$.

Let $W$ be a Wiener process. If $X(t):=W^2(t)$ for $t\geq 0$, calculate $\operatorname{Cov}(X(s),X(t))$
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1answer
78 views

Definition of regular point of a boundary with planar brownian motion

This is an exercise in G.Lawler's book Conformally invariant processes in the plane. First he defined regular point of a boundary using brownian motion: Suppose $D$ is a domain in $\mathbb{C}$ with ...
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1answer
99 views

The infinity version of Blumenthal's 0-1 law

Blumenthal's 0-1 law states that on the space of continuous maps with domain $[0, \infty)$ with the appropriate (Wiener) measure making the coordinate maps Brownian motions starting at $x$, any event ...
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2answers
120 views

Standard Brownian Motion

Let $\{X_t,t\ge 0\}$ be a standard Brownian motion. Compute the density of $X_t$ conditioned by $X_{t_1}$ and $X_{t_2}$ assuming that $t_1 <t<t_2$. Can anyone give me some hint to start the ...
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1answer
165 views

Show that $M$ is a martingale

Let $B$ be typical Brownian motion with $\mu >0$ and $x \in \mathbb{R}$. $X(t):=x+B(t)+\mu t$, for each $t\geqslant 0$, Brownian motion with velocity $\mu$ that starts at $x$. For $r \in ...
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1answer
228 views

A Boundary crossing result for discrete brownian bridge

Let $S_n$ be a random walk with gaussian increments with $S_0=0$, i.e. $S_n-S_{n-1}\sim N(0,1), n\geq 1$. Fix $a>0,b\in \mathbb{R}$ and $c<a+bn$. Define the new process $$ ...
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1answer
85 views

Harmonic Measure & Brownian Excursion

I have a disk of radius R removed from the plane. Brownian excursions are emanated from infinity. I am tasked to find a probability that the brownian motion impacts an arc of the disk boundary. I am ...
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1answer
183 views

Using a Brownian martingale to compute the second moment of a hitting time

Prove $ W_t=B_t^4 -6B_t^2t+3t^2$ is a martingale, and compute $E(T^2)$ where $T=\inf(t\ge0,B_t=-a, B_t=b)$ if $a=b$. Ok, if $0\lt t\lt s$, $W_t$ is a martingale if $E(W_s|[B_r]_{r\le t})=W_t$ So ...
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1answer
43 views

Basic brownian motion computation

Let $B_t$ denote a standard 1-d Brownian Motion. Find $P(B_2 \gt 2)$. My sol. $B_2 ~ N(0,2)$ so $P(B_2 \gt 2)=1-P(B_2\le 2)=1-\frac{\int_0^2e^{-\frac{x^2}{4}}}{\sqrt{4\pi}}$, but where do i go from ...
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1answer
85 views

Distribution of a Brownian motion with respect to $\mathbb{P}^x$

Let $(\Omega,\mathcal{A},\mathbb{P})$ a probability space and $(B_t)_{t \geq 0}$ a Brownian motion (started in $x=0$). Then one can define a probability measure $\mathbb{P}^x$, $x \in \mathbb{R}$, on ...
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1answer
76 views

Normal probability and Brownian motion

Let $X_t$ be a Brownian motion with parameter $\sigma$. Find the probability in terms of $$\Phi(x)= \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^x e^{- \frac{ \alpha ^2}{2}}d\alpha$$ How would I do this for ...
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1answer
196 views

Local Maximum of Brownian motion.

Given two positive rational number $a,b$. How to show that almost surely Brownian motion attains a local maximum at some time in $(a,b)$?
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3answers
98 views

Scaling and time inversion for Brownian motion basically the same?

Let $B(t)$ be a Brownian motion. For $a>0$, we have the scaling relation $$\hat{B}(t)=aB(t/a^2) \sim B(t)$$ and $\hat{B}(t)$ is also a Brownian motion. The time inversion formula states that ...
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199 views

Construction of Brownian Motion

In Wiener's construction of Brownian Motion, it is assumed that there exists a probability space $(\Omega,\mathcal F,\mathbb P)$ and random variables $X_n:\Omega\rightarrow\mathbb R$ for $n\in\mathbb ...
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1answer
173 views

Approximation of stochastic integral

Let $f \in C^2_C(\mathbb{R})$ and $$X_t = X_0 + \int_0^t \sigma(s) \, dB_s + \int_0^t b(s) \, ds$$ (1-dim.) Itô process where $\sigma,b: [0,\infty) \times \Omega \to \mathbb{R}$ progressively ...
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64 views

Rate of increase of maximum process of Brownian Motion

Suppose $M_t=\sup_{0\leq s\leq t}\{B_s\}$, where $\{B_t\}_0^{\infty}$ is a standard Brownian Motion. I would like to know if it is true that $M_t e^{-t}$ converges to 0 almost surely? Thanks!
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1answer
147 views

Laplace Transform of a Brownian motion

If $v(\omega,t) : \Omega \times [0,\infty) \to \mathbb{R}$ is a Standard Brownian motion, then for what values of $s,\omega$ does the Laplace transform $l(\omega,s) = \int_0^\infty e^{-st} v(\omega,t) ...
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78 views

Expected value of brownian motion for all positive paths

I've got this question but I can't figure it out. Derive the expected value of $B(t_1)$ of all paths that are positive $t_1$ and calculate the expectation for $t_1=1$ and variance$=1$? Thanks