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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2answers
94 views

Find a parameter for which a process is martingale

Find $\beta \in \mathbb{R}$ for which $$2W_t^3+\beta tW_t$$ is a martingale, where $W_t$ is standard Wiener process. My attempt: $$E(2W_t^3+\beta tW_t|F_s)=2E(W_t^3|F_s)+\beta ...
2
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2answers
53 views

Show that process satisfy given equation

I have to show that process (1) $$X_t=e^{-bt}X_0+\int_0^te^{-b(t-s)}\sigma dW_s$$ satisfies the following equation (2) $$dX_t=-bX_tdt+\sigma dW_t$$ My attempt: Multiply both sides of (1) by $e^{bt}$ ...
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0answers
65 views

brownian motion and stopping time

I have an exercise about Brownian motion which I don't understand completely. Let $(B_s)_{s\geq0}$ be a standard real Brownian motion. For $t > 0$, we define the random times $g_t ...
-1
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1answer
117 views

Brownian Motion Probability calculation P[Z(s)<a, Z(t)<b]

Is there a closed form solution for $P[Z(s)<-a, Z(t)<-b]$ where $Z$ is a Brownian motion and $0<a<b$ are constants? In the post Probability Brownian Motion - dependence the user Did gave ...
2
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0answers
99 views

Almost sure non differentiability of Brownian Motion

Problem: Let $t>0$, show that the standard Brownian motion is almost surely not differentiable a $t$ Now, through a Borel Cantelli argument I proved that, almost surely $$\limsup_{\epsilon ...
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0answers
90 views

An application of the Dambis-Dubins-Schwarz theorem. Is my argument correct?

I attended a lecture today, in which the professor went through an example with a lot of tedious calculations to show something which I'd think would follow directly from the Dambis-Dubins-Schwarz ...
4
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1answer
71 views

Laws and Moments of two dimensional brownian motions

I am a bit rusty on this. So let us consider the following two dimensional standard Brownian motion issued from zero defined on the probability space $(\Omega, \mathcal{F},\mathbb{P})$ (note that, in ...
4
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0answers
47 views

2-D exponential functional brownian motion

I'm looking for the distribution of $X = \int_0^T e^{-W_t} dt \int_0^T e^{W_t}dt$ and $Y = \frac{\int_0^T e^{-W_t} dt}{ \int_0^T e^{W_t}dt}$ (where $W_t$ is a standard brownian motion) On most ...
0
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1answer
59 views

Geometric Brownian motion - Share Prices

The current share price quoted to 30 €. The volatility is 25% per annum. The drift of 5% per annum 1) How is the share price in 6 months probabilistic distributed? 2) The expected value ...
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0answers
28 views

Convolution of Brownian function with characteristic function

Given Brownian function defined on the interval $[0,1]$. Our aim is to filter this function, one may use the low band pass filter. The idea is to cut the high frequency of its Fourier transform by ...
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1answer
43 views

$E \left [B^2_s \left( \int^t_s B_u dB_u \right)^2 \right]$

I am trying to solve following expectation $E \left [B^2_s \left( \int^t_s B_u dB_u \right)^2 \right]$ with $0 \leq s \leq t \leq T$ and $B_t$ a 1-dim. Brownian motion. Further using $E \left[ . ...
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0answers
48 views

given SDE how to find martingale measure

I've been stuck with the question how to find a measure to make a discounted price a martingale. I cannot use Girsanov because I am only given the SDE for which an unique strong solution exists but ...
1
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0answers
48 views

Probability that the value at time T from one geometric Brownian motion process is greater than the value from another GBM

I am having a competition between $n$ people (starts at time $t$=0), each who accumulates points on a daily basis, which I assume is a geometric Brownian motion process with parameters $\mu_i$, ...
3
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1answer
90 views

Expectation of brownian motion at hitting time

Am i correct in my derivation? I want to calculate $\mathbb{E}B_{\tau_a}$. From the definition of the hitting time i get $B_{\tau_a}=a$, so $$\mathbb{E}B_{\tau_a}=\mathbb{E}a=a$$ I am new to the ...
1
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0answers
239 views

Running maximum for Geometric Brownian Motion

Can anyone provide the expression and source for the running maximum $M_t$ for geometric Brownian motion $X_t$ as a function of the initial value $X_0$, drift $\mu$ and diffusion $\sigma$? $X_t$ ...
1
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1answer
94 views

Quadratic covariation of Itô processes

I haven't found any similar question in the forum, so I trust some of you will find this thought-provoking (at the very least). Perhaps you can help me. Let's consider first the two following ...
2
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1answer
173 views

Brownian Bridge as a Gaussian Process

Let $B=\{B_t:t\geq 0\}$ be a standard Brownian motion. Define the Brownian brige $X=\{X_t:t\geq0\}$ as $$ X_t=B_t-tB_1\quad t\in[0,1] $$ Show that $X$ is (i) Gaussian and find its (ii) mean and (iii) ...
2
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1answer
81 views

Show that this is a stopping time

Show that $\sigma=\inf \{ t\ge 0 : |B_t|= \log t \}$ is a stopping time with respect to $(\mathcal F_t^B)_{t\ge0}$. I've been trying to put the set $\{\sigma\le t\}$ equal to a countable union and ...
0
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1answer
79 views

expectations of Brownian motions

Let $B_t$ be a standard Brownian motion started at zero, and let $M_t$ be a stochastic process defined by $M_t=3\int_0^{t^{1/9}} s^4dB_s$ Compute $E\left[1+\int_0^t(1+M_s)^4 dM_s\right]$. Compute ...
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1answer
56 views

Show that a process is no semimartingale

_Hello everyone! I got a little question about how to show that the process $X_t:=|B_t|^{\frac{1}{3}}$ is NOT a semimartingale. So far I tried to apply Ito. Since if $X_t$ was a semimartingale so is ...
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2answers
79 views

Solution of $\frac{\partial f}{\partial t}(t,x) = 2 \frac{\partial^2f}{\partial x^2}(t,x)$

Consider the PDE $$\frac{\partial f}{\partial t}(t,x) = 2 \frac{\partial^2f}{\partial x^2}(t,x)\tag{1} $$ with $t\ge0,\ x\in\mathbb R,\ f(0,x)=e^x$. I want to find $f(t,x)$. I know that the heat ...
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0answers
54 views

Trying to understand geometric brownian motion with example

I've been spending the past week trying to understand geometric brownian motion/log-normal distribution/geometric processes/ without too much success. The problem is all the summaries online begin by ...
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0answers
62 views

Indicator function of an infinitesimal set

While reading a paper related to functional of brownian motion I came across the following notation $1(B_t \in dx)$, where $1(A)$ is the indicator function of the set A, and $B_t$ is a standard ...
3
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1answer
263 views

Expectation of stochastic integrals related to Brownian Motion

I'm trying to solve a problem that's now doing my head in a bit. I'll share with you the question and let's see if somebody can shed some light into the matter: Let B be a standard Brownian Motion ...
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0answers
36 views

Are my estimates of parameters of geometric brownian motion correct?

I wrote a simulation of a geometric Brownian motion which works like this: $t_i - t_{ i-1 } \sim Exp(\lambda )$ $Z_i \sim N(0,1)$ $Y_i \sim e^{ \sigma \sqrt { t_i - t_{ i-1 } } Z_i +\left( \mu ...
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1answer
76 views

Expected Value of SDE

how do I compute the expected value and variance of a stochastic process where I only know the SDE? In particular for the process $dB_t = ( 1/B_t - B_t/(T-t)) dt + dW_t$ where $W$ is a standard ...
1
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1answer
122 views

General question about expected value of brownian motion

I was wondering if anyone could tell me a little about expectation of brownian motions and how it is connected with normal distribution. I know that B(t) is N(mean*t,variance*t) and that the expected ...
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1answer
47 views

A Special case of geometric Brownian motion

I'm sitting with a special case of GBM where $B(t)$ is Brownian motion with drift $\mu$ and variance $\sigma^2$. I want to find the expected value of $A(t)$, where $A(t)$ is given by ...
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0answers
63 views

Independence in Brownian Motion

I've read two times in different lecture notes that for a Brownian Motion $(B_s)_{s\ge 0}$ and $t<u$ the random variable $B^t_{u}:=B_{t+u}-B_t$ is independent from ${\cal F}_t:=\sigma\{B_s:s\le ...
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1answer
47 views

$B(t)$ is brownian motion. I want Find $d(M(t))^2$,where $M(t)=e^{B(t)-\frac{t}{2}}$,

let $B(t)$ is brownian motion. Find $d(M(t))^2$,where $M(t)=e^{B(t)-\frac{t}{2}}$,
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2answers
43 views

I want Find $d(\frac{X(t)}{Y(t)})$ where $B(t)$ is a brownian motion and $X(t)=tB (t)$ and $Y(t)=e^{B(t)}$.

Let $B(t)$ is a brownian motion and $X(t)=tB (t)$ and $Y(t)=e^{B(t)}$. Find $d(\frac{X(t)}{Y(t)})$ Thanks for help.
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1answer
56 views

Fake Brownian Motion

Does there exist a martingale which has Marginal distributions same as Brownian Motion marginals but the process itself not being Brownian motion? Any references are highly appreciated. Thanks.
4
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0answers
105 views

Time scaling of Brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion and $A_t$ be an increasing continuous process adapted to the filtration generated by the Brownian Motion and $A_0 = 0$. I am trying to prove ...
1
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1answer
45 views

Wiener process analytic expression from geometric brownian motion

The solution to the SDE $dx= -kx\ dt + cx \ dW$ is $x(t) = x_0 e^{(c - k^2/2)t}e^{-k W}$ with mean $\langle x(t) \rangle = x_0 e^{(c - k^2/2)t}$ where $W(t)$ is the Wiener process. Im ...
3
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1answer
57 views

The limit supremum of a function involving Brownian motion

I would like, for some $\delta>0$ and a Brownian motion $B$, to calculate $\displaystyle\limsup_{t\to\infty}\left(\exp\left( (1+\delta)t\right)\cdot\exp\left(-B_t-\frac{t}{2}\right)\right)$ ...
7
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1answer
141 views

Must $n$ independent Wiener processes be simultaneously positive at some time?

Consider $n$ independent one-dimensional Wiener processes $(W_i)_{1\leqslant i\leqslant n}$. Is there with probability $1$ some time $t\in[0,1]$ such that $W_i(t)>0$ for every $1\leqslant ...
1
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1answer
116 views

Exercise 3.3.25 of Karatzas and Shreve

This is the Exercise 3.25 of Karatzas and Shreve on page 163 Whith $W=\{W_t, \mathcal F_t; 0\leq t<\infty\}$ a standard, one-dimensional Brownian motion and $X$ a measurable, adapted process ...
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0answers
41 views

When is a continuous path stochastic process be representable as diffusion or Ito process?

When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
2
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1answer
62 views

I want to show $E(B(t)-B(s))^4=3(t-s)^2$

Let $B(t)$ and $B(s)$ are brownian-motion I want to show $$E(B(t)-B(s))^4=3(t-s)^2$$ thanks for help.
1
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1answer
59 views

Problem 3.2.28 of Karatzas and Shreve

It's the Problem 2.28 of Karatzas and Shreve on Page 147: Let $M=W$ be standard Brownian motion and $X\in\mathcal{p}$. We define for $0\leq s<t<\infty$ $$\zeta_t^s(X)\triangleq\int_s^t X_u ...
2
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1answer
39 views

Distribution of Difference of Independent Random Variables

Usually in the development of the theory of Brownian motion, one makes the assumption that $X_t$ (the coordinate functions on $(\mathbb{R}^*)^{[0,\infty)}$). have normal distributions with mean $0$ ...
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1answer
84 views

Let B(·) and W (·) are two independent Brownian motions. Show two integrals have the same distributions.

Let B(·) and W (·) are two independent Brownian motions. How to show that the distributions of $\int_{0}^{1}(B(t)+W(1-t))^2dt$ and $\int_{0}^{1}((B(t))^2+(B(1)-B(t))^2)dt$ are the same? I think that ...
2
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1answer
53 views

Distribuation Max - Min of Brownian motion

I'm looking for the distribuation of $M_X(t) - m_X(t)$ of the brownian motion and not the joint distribuation. where $m_X(t) = \min\limits_{0\leq s\leq t}X(s)$ and $M_X(t) = \max\limits_{0\leq s\leq ...
2
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2answers
64 views

Ito's Isometry for three factors

Ito's Isometry states the following: If $\{W_t\}_{t\ge0}$ is a Brownian motion and $\{\phi_t\}_{t\ge0},\{\psi_t\}_{t\ge0}$ are two non-anticipative piecewise-continous processes with $\mathbb ...
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0answers
13 views

Distributions related to 1-dimensional Brownian Motion

Let $W_t$ be a one-dimensional Brownian Motion. Then how do we go about finding the distribution of $ \int_0^{1} W_t dt$ ? Moreover, assume $W_t = (W^1_t,W^2_t)$ is a 2-dimensional Brownian motion. ...
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0answers
85 views

Verifying a standard Brownian Motion?

Let $\{X_t, t\ge 0\}$ be a standard Brownian motion process. For a fixed positive number s and all $t\ge 0$, we define $Y_t = X_{t+s} - X_s$. Is $\{Y_t, t\ge0\}$ a standard Brownian motion? Attempt: ...
5
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1answer
287 views

$\int_0^tB_s^2\ dB_s$ - Gaussian Process and independent increments?

For $(B_t)_{t\ge0}$ a standard Brownian motion (Wiener process) define the stochastic process $X_t:=\int_0^tB_s^2\ dB_s$. I am currently trying to assess if $(X_t)_{t\ge0}$ is a Gaussian process and ...
2
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1answer
78 views

Expectation of product of correlated Brownian motions at different time points

Given the information about the correlation of two Brownian motions as $E[dW_1 dW_2] = \rho dt$ and knowing that $E[W_1(t)W_1(t')] = \min(t,t')$, I want to compute $E[W_1(t)W_2(t')]$ I interpret ...
0
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1answer
144 views

Blumenthal 0-1 law

Let $(B_t)$ be a Brownian motion. Consider the event : $B(n)>a \sqrt n $occuring infinitely often. I want to prove that this event has probability 1. we can see that, by rescaling property, ...
0
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2answers
123 views

Blumenthal zero-one law

How to prove $\limsup\limits_{n \to \infty} \frac{1}{\sqrt n}B(n) = +\infty$ using Blumenthal zero-one law, where B is a Brownian motion?