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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117 views

On the quadratic variation

I understand that the Quadratic Variation of Brownian Motion $B_t$ is $[B_t,B_t]=t$ and I know that the equality is under the meaning of $\mathcal{L}^2$ convergence. Yet I saw in some book saying that ...
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210 views

Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
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0answers
183 views

Brownian motion with drift

I need help with the following problem: Let us denote the water level in a dam at time $t$ by $X(t)$, where $t$ is measured in months. We will assume that, at least until the first time that the dam ...
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2answers
425 views

First hitting time for a brownian motion with a exponential boundary

Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known? Trying to be more formal, if ...
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1answer
82 views

an issue with expectation

in book's Bernt.Øks SDE i read that book and i have some serious issues :( page 21 Example 7.4.2 ) Consider n-dimensional Brownian motion $W=(W_1, \ldots ,W_n)$ starting at $a=(a_1,\ldots,a_n) \in ...
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1answer
65 views

Strong markov property Brownian motion

I have a question regarding an argumentation, which is not clear to me: Let $B_t$ be the standard Brownian motion and $\tau$ a respective stopping time with finite mean. Fix $\epsilon>0.$ We can ...
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2answers
136 views

Progressive measurability of a specific set related to Brownian motion

Let $\{W_t: t \in R_+\} $ be a standard Brownian motion process on a given probability space. I am interested in assessing the progressive measurability of the following set: $Z(\omega) := \{t: ...
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0answers
122 views

Conditional expectation with three random variables

We have $N_1, N_2, N_3$ normally distributed random variables with $µ_i =E[N_i]$, $σ_{ij}=Cov(N_i,N_j)$. We also have $\tilde{µ}_i=E[N_i|N_2 = x] $, $\tilde{σ}_{ij}=Cov(N_i,N_j|N_2 = x)$ and $v^2 ...
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1answer
160 views

What is the expected value of the product of 3 Ito Integrals?

How can I calculate the expected value $\mathbb{E}(I_{110}^2 * I_{10})$, where $I_{110}$=$\int_{t_0}^T \int_{t_0}^{s_3} \int_{t_0}^{s_2} 1\, \, dW(s_1) dW(s_2) ds_3$ and $I_{10}=\int_{t_0}^T ...
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1answer
52 views

Inequality brownian motion

I have the probability $P(e^{σB_t+ αt}>Kf_t)$, where σ, α, K constants, f is a function of t and $B_t$ brownian motion. This probability must be independent of t. So why is $f_t$ chosen such as ...
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1answer
216 views

Conditional expectation with brownian motion

I want to find $E[e^{σB_t}|∫_0^1B_s ds]$. I make the notation $∫_0^1B_s ds = z$, and I know that: $E[B_t|Z]= 3t(1 - t/2)z $, $Var [B_t|Z] = t - 3t^2(1 - t/2)^2$ and $Var(∫_0^1B_s ds) = 1/3$. The ...
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0answers
48 views

Contradiction on equality with stochastic integrals

I want to compute $E[∫_0^tB_u \, du ∫_0^sB_u \, du]$ and I know from another source that should be equal to $ts^2/2$. But when I try to compute it like: $$\begin{align} & E\left[(tB_t- \int_0^tu ...
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1answer
93 views

Computing expectation of a stochastic integral

I need to compute the expectation $$E\left[\int_0^tu \, dB_u \int_0^s u \, dB_u \right].$$ Being that is my first question, how can I initialize MathJax if I have it on my hard drive.
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1answer
40 views

Probability of a Wiener process staying within a wedge

Given $m>0$, what is the probability that a one-dimensional Wiener process $W$ will satisfy $$\left|W_t\right|\leq mt$$ at all times during the period $t \in [0,1]$?
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1answer
116 views

Quadratic variation of $X(s)=W_{s+\epsilon}-W_{s}$

Let $W_s$ be a standard Wiener process. The quantity $W_{s+\epsilon}-W_{s}$ is another standard Wiener process when regarded as a function of $\epsilon$. Therefore, the quadratic variation of ...
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0answers
134 views

Brownian motion, rate of large events

Given the most simple brownian motion: $$ \dot x(t) = \sigma \eta(t)$$ where $\langle \eta(t)\eta(t')\rangle=\delta(t-t')$, I define as large event in a time-frame $\tau$ a portion of the trace ...
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1answer
122 views

Assistance with a BM exercise

A friend and I are attempting to answer part 3) of the exercise quoted below (from Continuous Martingales and Brownian Motion) regarding Brownian Motion (BM). We have some questions apropos thereof. ...
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0answers
99 views

Probability a geometric Brownian motion stays within an interval.

Let $X_s$ be a $(\mu,\sigma)$ geometric Brownian motion with $X_0 = x$. For some positive numbers $c < x < d$ and time $t$, what is the probability $X_s \in [c,d]$ for all $s \in [0,t]$? In ...
2
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1answer
266 views

Conditional hitting time distribution of a Brownian motion

This problem cropped up in some research I am doing. I imagine it is standard, but I cannot seem to find the answer. Let $W_t$ be a standard Brownian motion. Suppose there are four values $a < 0 ...
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1answer
900 views

How to integrate a Wiener process that freezes at a determined time?

I would like to calculate the expected variance of the average of a Wiener process from time $0$ to time $1$. The equation I believe I am trying to solve is: $$ \mathbb{E} \left[ \left( \int_0^1 W_t ...
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0answers
118 views

Leibniz Rule applied to Brownian integral

I am looking to take the partial derivative of an integral with respect to brownian motion. For Simplicity I will make it the same integral as in this post (don't have enough reputation to comment): ...
3
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1answer
68 views

The first two moments of $\int_0^1 B_s^2 \, ds$

I was trying to solve the following problem from Continuous Martingales and Brownian Motion by Daniel Revuz and Marc Yor, but got my solution back as the answer for variance was wrong. I have already ...
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1answer
68 views

Brownian Motion Question - Requires Verification

Suppose $Z(t)$ is a standard Brownian motion process with $Z(0)=0$, then calculate: $P(Z(3)>Z(2)>0)$ I have the following, but unsure if my rationale is correct: $Z(3)=X_1+X_2+X_3, ...
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1answer
101 views

Limit of a stochastic integral

Let $W_t$ be a one-dimensional Brownian motion and I would like to prove $$\lim_{\beta\rightarrow+\infty}\sup_{0\leq t\leq T}\left|e^{-\beta t} \int_0^te^{\beta s}\mathrm dW_s\right|=0$$ This is an ...
2
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0answers
757 views

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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1answer
91 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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0answers
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 ...
3
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0answers
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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2answers
140 views

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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0answers
189 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
231 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
125 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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0answers
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
523 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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2answers
129 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
79 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
115 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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0answers
101 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
644 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
491 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
51 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
267 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
513 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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0answers
51 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 ...
2
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1answer
81 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
115 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
694 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 $\mathcal F$. ...