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

0
votes
2answers
33 views

Brownian and Brackets

A continuous martingale with deterministic bracket must be a Brownian motion. Is this statement ture or not, please? If true, how to show it? If not, what is a counter example?
0
votes
1answer
17 views

Martingale property of negative Brownian motion

Let $B_t$ be Brownian motion, with $B_0=0$. Next define $M_t=-B_t$. Have I understood it correctly that $M_t$ is not a Martingale? $E[M_t]=0$ $E[M_{t+1}|M_t]=-M_t$ and therefore not a Martingale? ...
4
votes
1answer
47 views

Why is the canonical filtration of a Brownian motion left-continuous?

Let $\{W_t, t\geq 0\}$ be a Brownian motion, and has a.s. continuous sample paths. Let $\{\mathcal{F}^W_t, t\geq 0\}$ be the canonical filtration, i.e. $\mathcal{F}^W_t=\sigma(W_s, 0\leq s\leq t)$. ...
0
votes
2answers
38 views

simple stochastic differentiate

someone can help me to differentiate $$a(t-1)+bt+(1-t)\int_{0}^{t}\dfrac{dB_s}{1-s}?$$ I've tried but I really don't know how to do with the last part.. Thank you somuch for your help
1
vote
2answers
61 views

Stratonovich integral

I'm having some troubles to calculate the Stratonovich integral $I(sin)(t)=\int_{0}^{t}\sin{B_{s}}dB_{s}$. I've tried with the limit of ...
2
votes
1answer
46 views

Conditional Integral of Square of Brownian Motion?

I am struggling to compute the expectation and variance of the following, where $W(s)$ is a standard Brownian motion: $$ X := \int_{0}^{A}W(s)^2ds$$ $$ Y:= \int_0^AW(s)ds $$ $$E[X\mid Y] = \space ?$$ ...
2
votes
1answer
36 views

Proving Brownian Motion has Stationary Increments

In Oksendal's 'Stochastic Differential Equations', we define Brownian Motion as follows: Fix $x\in\mathbb{R}^n$ and define for $y\in\mathbb{R}^n$: $$p(t,x,y)=(2\pi ...
1
vote
1answer
132 views

The Brownian motion process in Sheldon M. Ross

Today I study Brownian Motion and Geometric Brownian Motion using textbook: An Elementary Introduction to Mathematical Finance, Third Edition by Sheldon M. Ross but I missed the class because I was ...
0
votes
1answer
90 views

How to solve a linear stochastic differential equation?

I don't know how to find a solution of this stochastic differential equation: $dX_{t}=(1+\delta \mu X_{t})dt+\delta X_{t}dB_{t}$ Where $B_{t}$ is a standard Brownian motion and $\mu$ and $\delta$ ...
0
votes
1answer
54 views

Prove $A_t := W_t^3-3t W_t$ a martingale

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let $(A_t)_{t \geq 0}$ where $A_t = W_t^3 - 3tW_t$. ...
0
votes
0answers
18 views

How can a Jump Diffusion Process be defined on White Noise rather than Brownian Motion

Jump diffusion processes are a Brownian process with randomly occurring jumps: http://en.wikipedia.org/wiki/Basic_affine_jump_diffusion Can such a process be simulated using white noise rather than ...
0
votes
1answer
24 views

Cauchy distribution for Brownian motion

This is homework so no answers please Problem: Find distribution of $(B_{1}(T_{a}),B_{2}(T_{a}))$, where $T_{a}=inf_{t\geq 0}\{B_{2}(t)=a\}$ Any mistakes: $T_{a}=inf_{t\geq 0}\{B_{2}(t)=a\}$ has ...
1
vote
1answer
53 views

Random walk and Occupation measure

This is homework so no answers please I want to find for some $A\subset \mathbb{R}$ the limit $$\lim_{n\to \infty}\mu_{n}(A)=\lim_{n\to ...
1
vote
1answer
27 views

Help understanding a proof of non-differentiablity of Brownian motion

The following statement and proof are taken from the book Brownian Motion by Peter Morters and Yuval Peres. Since I initially didn't fully understand the proof I added some clarifications and I was ...
0
votes
0answers
47 views

Showing $E[e^{-\lambda \tau_{a}\wedge\tau_{-a}}]=sech(a\sqrt{2\lambda})$

This is homework so no answers please. For $\tau_{a}=inf_{t}(B_{t}=a)$ , we already know $E[e^{-\lambda \tau_{a}}]=e^{-\sqrt{2\lambda}a}$. By $B_{t}$ I mean Brownian motion. The question is to show: ...
1
vote
1answer
46 views

Covariance of m-fold integrated Wiener process

The problem I'm trying to perform a Bayesian approach to the Maximum Likelihood Estimation procedure of Wecker and Ansley (1983). To this end, I need to compute the full likelihood of the data given ...
2
votes
0answers
90 views

Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
0
votes
2answers
53 views

Conditional expectation of Brownian motion given its absolute value.

Assume that $W_t$ is Brownian motion (1-D) and that $t<T$. How can I compute $$E(W_t||W_T|),$$ the conditional expectation of $W_t$ given $|W_T|$, i.e. with respect to the $\sigma$-algebra $F$ ...
0
votes
2answers
44 views

Itô process and covariance of two Brownian motion

I'm a novice in studying the stochastic different equation, and didn't know whether I have describe the question correctly. Here is the question: Suppose $$\begin{array}{rcl} ...
1
vote
1answer
27 views

mean hitting time of a level and growth rate of maximum process

Let $X_t$ be the absolute value of Brownian motion starting at $0$, let $\tau_x$ be it's first hitting time of the level $x>0$, and let $M_t$ be it's running maximum up to time $t$. Suppose we knew ...
1
vote
0answers
56 views

Expectation of absolute value of Brownian motion

I'm working on this problem that I can't seem to figure out. The problem involves a 1-dimensional Brownian motion, $B_t$, where the subscript denotes the time, and it asks me to show that the ...
1
vote
0answers
37 views

Ito's integral from the definition

I am doing Oksendal's book exercises one by one. I got stuck in 3.2. I need to prove, from the definition that $$\int_{0}^{t}B_s^2\text{d}B_s=\frac{B_s^3}{3}-\int_{0}^{t}B_s\text{d}s,$$ where ...
1
vote
1answer
23 views

Probability Brownian Motion doesn't hit a point in the limit.

This is a question from Revuz and Yor (exercise 3.18) for which I seem to get a different answer. Show that $\lim_{t \to \infty}\,t^{1/2}\,\mathbb{P}\{B_s\leq1\,\forall\, s\in[0,t]\}=\sqrt{2/\pi}$. ...
1
vote
1answer
24 views

Reference for the Construction of Brownian Motion

A common method for constructing Brownian motion is referred to as the Levy construction, the Levy-Ciesielski construction, the Ciesielski construction and sometimes seems to be attributed to Wiener ...
1
vote
0answers
10 views

Holder continuity, brwonian motion [duplicate]

Let $B$ stand for a brownian motion on a finite interval $[0,1]$. If i am not wrong, i think that there exists a positive constant $c$, such that almost surely, for h small enough , for all $0< t ...
0
votes
0answers
39 views

Deriving Spectral density of White noise from Brownian motion

This is homework so no answers please Here is the problem and my answers (so please tell me if I made any mistakes): I am not asking you to compute the sum at the end, but to tell me if I made any ...
0
votes
1answer
29 views

Distribution of hitting position of line by brownian motion.

What is known about the distribution of the hitting position of a line by a 2d brownian motion? I've tried to make some simulations of a 2d brownian motion where every computational step has a ...
0
votes
1answer
59 views

Brownian motion - Hölder continuity

Let $B$ stand for a Brownian motion on a finite interval $[0,1]$. If I am not wrong, I think that there exists a positive constant $c$, such that almost surely, for $h$ small enough , for all $0< t ...
1
vote
1answer
29 views

$\forall p>0$ we get $\displaystyle \lim_{n\to \infty} n^{a_{p}}\sum _{i=0}^{n}|B_{\frac{i+1}{n}}-B_{\frac{i}{n}}|^{p}=c_{p}$

This is a Homework question, so please do not answer it. Find real constants $a_{p},c_{p}$ s.t. $\forall p>0$ we get $\displaystyle \lim_{n\to \infty} n^{a_{p}}\sum ...
2
votes
0answers
21 views

Separation of variables and Fourier transformation

I know there's another question very similar to this argument. In the book "Probabilità e modelli aleatori" of Enzo Orsingher, at pag 134, it shows that the transiction function of an absorbing ...
0
votes
0answers
25 views

Brownian motion and proximity to a set

Here is the problem: Given two square planes $S_{1},S_{2}\subset \mathbb{R}^{3}$ oppositely positioned to the origin along the x-axis and $S_{1},S_{2}\perp x$-axis. Also, let ...
0
votes
0answers
20 views

Hitting time and its distribution

÷I'm reading an italian book about casual process (Probabilità e modelli aleatori of Enzo Orsingher). At pag 105 there's the probability of the stopping time $T_\beta$. $$P\{T_\beta \leq ...
0
votes
1answer
13 views

Exclusion-Inclusion principle for Hitting times of two disjoint sets

Consider disjoint sets A, B and Brownian motion $B_{t}$ with $B_{0}\notin A\cup B$. Let $T_{A}:=inf_{t>0}\{B_{t}\in A\}$. Then, do we get $P(T_{A\cup B}<\infty)=P(T_{A}<\infty)+P(T_{ ...
1
vote
1answer
37 views

Simple question about the definition of Brownian motion

I have a question concerning the definiton of Brownian motion. Usually (e.g. on Wikipdia) one demands a brownian motion $\lbrace B_t\rbrace_{t\in[0,\infty)}$ to satisfy the following condition: ...
1
vote
1answer
32 views

Intuition underlying stopped martingales

Let $X$ be a martingale and $T$ a stopping time. Define the stopped martingale $X_{\min\{T,n\}}$. What is the intuition underlying this process? It is quite confusing here. $X$ is random and $T$ is ...
3
votes
1answer
91 views

Expectation of Square of Stopping Time

Let $B_t$ be standard Brownian motion and $a < 0 < b$. Define stopping time $T$ as follows. $$T = \min \{t \geq 0: B_t \in \{a, b\} \}.$$ The expectation of $T$ is $\mathbb ET = |a|b$ and can be ...
1
vote
1answer
24 views

Solve parameter from stochastic integral

how can I solve $\rho$ from the following: $\int_0^T dV_t = \int_0^T \kappa (\theta - V_t) dt + \int_0^T \sigma \rho \sqrt{V_t} dW_t + \int_0^T \sigma \sqrt{1-\rho^2} \sqrt{V_t} dZ_t$, where $W_t$ ...
1
vote
0answers
44 views

$E[e^{\lambda X_t}|\mathcal{F_s}]$, where $X_t=\int_0^t(W_s-\frac{s}{t}W_t )ds$

I was trying to compute $E[e^{\lambda X_t}|\mathcal{F_s}]$, where $X_t=\int_0^t(W_s-\frac{s}{t}W_t) ds$, $\mathcal{F}$ is associated to $W$. I tried the following. 1) Splitting the integral ...
-3
votes
1answer
52 views

Are $W_s$ and $\int_s^tW_s \, ds$ independent? [closed]

Let $W_s$ be a Brownian motion, I have to compute $E[W_s\int_s^tW_sds]$. Are they independent?
0
votes
1answer
46 views

$E[W_s\int_s^t W_sds]$, $W_s$ is a brownian motion

Let $W_s$ be a brownian motion, I found $E[W_s\int_s^tW_sds]$ in a much longer exercise but I don't know how to compute it. Any suggestion?
0
votes
1answer
45 views

For a Poisson process prove that (a) $N (t) -λt $ and (b) $e^{(\log(1-u) N (t) + uλt)}$, are martingales

For a Brownian motion ${z (t)}$ and for any $β ∈ R$, be $V (t) = \exp\{ βz (t) - (t β ^ 2) / 2\}, t≥0 $ Show that ${V (t)}$ is a martingale with respect to a Brownian filtration. Also ${N (t)}$ be a ...
2
votes
3answers
42 views

Is $W_{2t}-W_t$ a brownian motion?

Is $W_{2t}-W_t$ a brownian motion? $(W_t)_{t\geq 0}$ is a brownian motion, I have to show that $X_t:=W_{2t}-W_t$ is a brownian motion as well. $$W_{2t}= 1/\sqrt{2} W_t$$ (by scaling property) then ...
1
vote
1answer
24 views

Distribution of two-sided boundary stopping time of Brownian motion.

If $B_t$ is a Brownian motion, and a one-sided boundary stopping time is given by: $\tau_a=\inf\{t:B_t=a\}$ the distribution of $\tau_a$ is given by: $f_{\tau_a}(t)=\frac{|a|}{\sqrt{2\pi ...
3
votes
1answer
76 views

Determine the distribution of $\int_0^t (W_s-\frac{s}{t}W_t) ds$, where $(W_s)_{s\geq 0}$ is a brownian motion

I have to find the distribution of $X_t:=\int_0^t (W_s-\frac{s}{t}W_t) ds$ where $(W_s)_{s\geq 0}$ is a brownian motion. I already showed the first integral $\int_0^t W_s ds$ is ...
1
vote
0answers
54 views

Hyperbolic vs Euclidean Brownian Motion

In this article, page 4 of the linked pdf file, Lalley and Sellke claim that a hyperbolic Brownian motion can be obtained by time-changing a 2-dimensional Euclidean Brownian motion, conditioned to ...
2
votes
1answer
43 views

Invariance Properties of Brownian Motion

I am trying to make sense of the Scaling-Invariance and Time-Inversion properties of Brownian motion by producing a sample path. For the record, I am using the following definitions. Let $B(t)$ be the ...
4
votes
2answers
167 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
2
votes
1answer
31 views

Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
0
votes
1answer
40 views

Stopping times problem: $ \tau_+ = \inf \{t \ge 0 \mid W_t>0\}$

Stopping times problem, $\tau_+ = \inf \{t \ge 0 \mid W_t>0\}$ I can not prove the following : P/S: When I look at the stopping time, I feel that $\{W_0 > 0\} = \{\tau_+ = 0\}$ , is that ...
1
vote
0answers
18 views

Stopped supremum of the Brownian local time still $L^p$ bounded in space?

Let $B_t$ be a standard Brownian motion and $L_t^x$ its local time in $x$ at time $t$. For fixed $t$ and $p>1$, it holds that $$ \sup_{x \in \mathbb{R}} \operatorname{E} [ (L_t^x)^p ] < ...