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 $t-s$.

learn more… | top users | synonyms

1
vote
0answers
37 views

Conditional covariance

Just a simple question when we have $v_{st} = \operatorname{cov}(B_s, B_t\mid Z)$, where $B_t$ is a brownian motion. I know that the answer is $\min(s, t) - E[B_s Z]E[B_t Z]/E[Z^2]$ but i don't know ...
0
votes
1answer
59 views

Brownian Motion hitting random point

I got a problem that seems to be quite standard and easy, but I have lots of problems with it. I do already know that $T_a:=\inf\{t\geq 0: B_t=a\}$ is a stopping time for any $a\in\mathbb{R}$ fixed, ...
1
vote
1answer
145 views

hitting time of Brownian motion

I'm desperately trying to prove that for a standard BM $B_t$ the stopping time $T:=\inf\{t\geq0: B_t\geq\sqrt{1+t}\}$ is a.s. finite, i.e. $\mathbb{P}[T<\infty]=1$. I actually tried to play around ...
1
vote
0answers
27 views

Finding asymptotic behaviour

I have a problem in finding the asymptotic behavior of this sum: $$\sum_{i=0}^{n-1} \bigl|B^2 (t_{i+1})-B^2 (t_i)\bigr|$$ over $[0,T]$ when $h= t_{i+1}-t_i \to 0$ and $B$ is Brownian motion. The ...
4
votes
1answer
181 views

Exponentials of stochastic processes and Brownian motions

This is my first time looking at problems in stochastic calculus, so please bare with the simplicity of the question. As always, any help is greatly appreciated. 1) Given $X_t=\int_0^ur_sds$ for a ...
1
vote
2answers
79 views

can someone explain this notation to me?

$$ dz_t \sim O\left(\sqrt{dt}\,\right) $$ $z$ is a Brownian motion random variable, for reference. I just don't understand what the $\sim O$ part means. I've looked up the page for Big O notation ...
0
votes
1answer
74 views

Brownian motion at exponential time

I want to find the law of $B_T$, where $B_t$ is a brownian motion, $T$ is exp-distributed with parameter 1, with $B_t$ and $T$ being independent. My idea is to say that the density of $B_T$ is given ...
1
vote
2answers
146 views

Question on the proof of the simple Markov property of a Brownian motion

Today we proofed the (simple) Markov property for the Brownian motion. But I really don't get a crucial step in the proof. The theorem states in particular that for $s\geq0$ fixed, the process ...
1
vote
0answers
52 views

Bounding the expectation of monotone function of stopping times of Brownian motion

Let $X_t$ be a standard Brownian motion and let $Y_t:=X_t + \epsilon B_t$ where $B_t$ is an independent standard Brownian motion and $\epsilon>0$ is small. Let f be a monotone increasing function. ...
0
votes
1answer
234 views

Why is the Brownian motion a multivariate normal distribution?

I have seen in class that for some reasons I forgot, the Brownian Motion has a Multivariate normal distribution, but I am unable to prove it easily. Could someone tell me why it's true? From what I ...
1
vote
0answers
27 views

First hitting time for a brownian motion with two exponential boundaries

I asked a previous related question here: First hitting time for a brownian motion with a exponential boundary Now Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first ...
0
votes
2answers
348 views

Variance of stochastic integral of brownian motion

How do i compute this integral? $ Var [\int_0^T W(t)dW(t)] $ I know the following $E [\int_0^T W(t)dW(t)]$ is 0 but i'm not sure how to apporch the above
2
votes
1answer
84 views

Brownian bridge distribution: $\sup_{a \leq u \leq b} \frac{|W^0(u)|}{\sqrt{u(1-u)}}, 0 < a < b <1 $

If $W^0$ is a tied-down Wiener process (Brownian bridge) on the range $(0, 1)$, what is the distribution of \begin{equation} \sup_{a \leq u \leq b} \frac{|W^0(u)|}{\sqrt{u(1-u)}} \end{equation} ...
1
vote
1answer
87 views

Basic Martingale question

We know the following is a martingale, and is commonly used to represent Ito's Integral. If $W$ is a brownian motion. $ \int_0^T W(u) dW(u) = \frac{1}{2} W^2 (t) - \frac{1}{2}t, \ for \ \ T \ge t \ge ...
0
votes
1answer
106 views

Geometric Brownian motion problem

Here's the question: Let $S(t)$, $t \geq 0$ be a Geometric Brownian motion process with drift parameter $\mu = 0.1$ and volatility parameter $\sigma = 0.2$. Find $P(S(3) < S(1) > S(0)).$ Is ...
0
votes
1answer
102 views

Question regarding Ito integral

I have a question regarding Ito integral, in particular, when I am trying to prove the normality of Ito integral, I encountered the following differential equation I need to solve: $$dX_{t} = ...
0
votes
1answer
224 views

Joint Distribution of two correlated ito integral

I have a question regarding finding the joint distribution of two process$$dX_{t}=a_{t}dB_{t}$$$$dY_{t}=b_{t}dW_{t}$$where $B_{t}$ and $W_{t}$ are two Brownian motions with correlated increments, in ...
1
vote
0answers
111 views

Brownian motion hitting probability

Let $B_t$ be a brownian motion and $g(t)$ a function of the time $t$. $B_0=0$. Let $\Phi$ be the c.d.f. of a normal distribution. At time $t$, the probability that $B_t > g(t)$ equals ...
2
votes
1answer
120 views

Question regarding Ito Process

I am new to Ito Process, so I have a following question. Consider a standard Ito Process, $$X_t=X_0+\int_0^t\mu_sds+\int_0^t\sigma_sdW_s$$ where W is the m-dimentional Brownian motion and X is a ...
0
votes
1answer
49 views

Is filtration necessary for continuous random variables?

Define $(\mathscr{F}_t)_{t\geq 0}$ being the natural filtration induced by the Brownian motion $(B_t)_{t\geq 0}$. That is $$\mathscr{F}_t=\sigma(B_s\mid 0\leq s\leq t), \forall t\geq 0,$$ i.e. ...
0
votes
3answers
108 views

Brownian motion and adapted

I'm reading Bernt Oksendal's "Stochastic Differential Equations" and "adapted" is one of the concept that I could not understand. First, "adapted" is defines at Ch3.1, page 25 (sixth edition): ...
3
votes
0answers
193 views

How to prove Brownian motion is Gaussian Process?

I'm reading Bernt Oksendal's "Stochastic Differential Equations" and this is one of the proof that I'm totally lost. This is from Ch2.2, page 12-13 (sixth edition). First, Brownian motion is ...
2
votes
1answer
106 views

Calculating the probability of following event involving Brownian motion

I have a big time trouble in evaluating the following probability. It is related to brownian motion and measure, so I am asking experts from both fields for help! Denote $B_t$, $t\in [0, T]$ be ...
1
vote
0answers
47 views

Bounded Brownian Bridge

I am trying to calculate the following expectation value of $$ E[exp(-\int_{t_0}^{t_1} X_s ds)] $$ in which Xs is a bounded Brownian bridge, which means $X(t_0)=a$, $X(t_1)=b$ and $A<X(t)<B$. ...
0
votes
1answer
125 views

Local maxima and minima of Brownian motion

I have trouble understanding why the Brownian motion is nowhere differentiable, and I found somewhere that after showing the total variation of the Brownian motion is $+\infty$, the author claims that ...
0
votes
1answer
201 views

Mean and variance of a brownian bridge

I am trying to compute mean and variance of the stochastic process $X_t$, which is a Brownian bridge from x to y, in the time-interval $[t,T]$. $$X_t = y + ...
0
votes
1answer
97 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 ...
7
votes
0answers
145 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 ...
1
vote
0answers
126 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 ...
2
votes
2answers
238 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 ...
1
vote
1answer
81 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 ...
1
vote
1answer
60 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 ...
2
votes
2answers
102 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: ...
1
vote
0answers
114 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 ...
0
votes
1answer
129 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 ...
0
votes
0answers
61 views

Expected value of the product of multiple Stratonovich Integrals

I want to calculate: $\mathbb{E}(J_1* J_{10}* J_{10} *J_{110})$ where $J_1=\int 1 dW$, $J_{10}=\int\int 1 dW dt$, $J_{110}= \int \int \int 1 dW dW dt$ are multiple Stratonovich Integrals over the ...
0
votes
1answer
40 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 ...
0
votes
1answer
154 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 ...
1
vote
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 ...
1
vote
1answer
85 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.
0
votes
1answer
37 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 $t \in [0,1]$?
2
votes
1answer
107 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 ...
1
vote
0answers
132 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 ...
1
vote
1answer
121 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. ...
1
vote
0answers
92 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
votes
1answer
187 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 ...
1
vote
1answer
509 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 ...
1
vote
0answers
85 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): ...
0
votes
0answers
50 views

Arbitrage with fractional Brownian motion

I need some help to understand L.C.G. Rogers' paper "Arbitrage with fractional Brownian motion" (1997). The fractional Brownian motion $(X_t)_{t\in \mathbb{R}}$ with self-similarity parameter ...
3
votes
1answer
65 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 ...