Questions tagged [brownian-motion]

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 $t-s$.

Filter by
Sorted by
Tagged with
2
votes
0answers
21 views

Does the Cramer-Wold theorem hold for random elements?

Recall the Cramer-Wold theorem; If $(X_n)$ and $X$ are $d$ dimensional random variables, then $X_n\to_d X$ iff $c\cdot X_n\to_d c\cdot X$ for all $c\in R^d$. Does there exist a version of this for ...
0
votes
1answer
28 views

Independent increment of Brownian motion (mistakes wikipedia definition ?)

I know that Brownian motion has the property that if $0\leq t_1\leq t_2\leq ...\leq t_n$ then $$B_{t_1}, (B_{t_2}-B_{t_1}),...,(B_{t_n}-B_{t_{n-1}})\tag{*}$$ are independents. In wikipedia they say ...
0
votes
3answers
35 views

If $E[B_{t}]=0$ then why is $E[B_{t}^{2}]=t$

Let B be a brownian motion. I know that a brownian motion includes the fact that for a family $(B_{t})_{ t \in [0,\infty[}$, the increments have the normal distribution: $B_{t}-B_{s}$ ~ $\mathcal{N}(0,...
0
votes
0answers
18 views

Using Brownian motion to obtain pairwise distances

I am a biologist seeking a math solution to my question. My data consists of a data matrix on m species, for each of which I have a measured value X on each of k traits (that have been normalized so ...
0
votes
1answer
41 views

Question about bounded convergence theorem and martingale

Q) Let $B_t$ be a brownian motion and let $\tau = \text{inf}\{t:B_t=a+bt\}, a>0$. Use the martingale $\text{exp} (\theta B_t-\theta^2t/2)$ with $\theta = b+(b^2+2\lambda)^{1/2}$ to show $$E_0\text{...
1
vote
1answer
23 views

Expectation of a local martingale

How do I prove that $\mathbb{E}[\int_o^t B^4_s(\omega) ds ]$ is finite for any $ t \geq 0$? Here $B_t$ is the standard Brownian Motion. If I can change the expectation with the integral the result ...
1
vote
0answers
26 views

Determine $P(B_{s} > 0, B_{t} > 0)$ where $s < t$

Let B be a brownian motion, and set $t < s$. Determine $P(B_{s} > 0, B_{t} > 0)$. Seeing as though this is a brownian motion I assume we have to use the increments, so: $P(B_{s} > 0, B_{...
0
votes
2answers
45 views

Expectation of an integral of a function of a Brownian motion

$B_t$ is a Brownian motion and $Y_t:=e^{aB_t+bt}$. For which $a$ and $b$ is $Y_t\in M^2$? I found a theorem that says that sufficient for $Y_t\in M^2$ would be $E[\int_0^\infty Y_t^2 dt]<\infty $ ...
0
votes
0answers
28 views

Asymptotic Behavior of Stochastic Differential equations

I have system of three SDE's. They have the following form: $$dx=x(zF_1(x,z)-d)dt+\sigma_1 dW_1(t)$$ $$dy=y(zF_2(x,z)-d)dt+\sigma_2 dW(t)_2$$ $$dz=((x+y)d-z(xF_1(x,z)+yF_2(x,z)))dt+\sigma_3 dW_3(t)$$...
2
votes
1answer
75 views

Conditional expectation of a function of brownian motion.

Assuming that $\{W(t) | t \geq 0\}$ is a Brownian motion, I am trying to find following conditional expectation $$\mathbb{E}\left[W^{2}(4) | W(1), W(2)\right]$$ My try: What I think is that I ...
1
vote
1answer
20 views

3 Brownian motion and conditional expectation

I'm supposed to compute $E\left(W_6 | W_2 , W_4\right)$ knowing that $W$ is a standard Brownian motion. I found $W_2$ but it's weird... Any help ?
1
vote
0answers
64 views

Is this function of a Brownian motion a martingale?

$B_t$ is a Brownian motion and $Y_t:=e^{aB_t+bt}$ for $t\ge0$. For which $a, b\in\Bbb R$ is $Y_t$ a martingale? My calculations with the martingale property lead me to the presumption that it is not ...
1
vote
1answer
51 views

General stochastic integration by parts.

For $W_t$ the Brownian motion, Let $X_t$, $Y_t$ be two diffusions, then since $$d(X_tY_t) = X_tdY_t+Y_tdX_t+dX_tdY_t$$ the generic stochastic integration by parts formula is given by $$\int_a^bX_t\,...
3
votes
0answers
59 views

What is the meaning of a space-time white

For reference this is from page 27 here Consider the stochastic heat equation $$ \begin{cases} \dfrac{\partial u}{\partial t}(x,t)=\dfrac{\partial ^2 u}{\partial x^2}(x,t) +f(u(x,t))\dot{W}(x,t) &...
4
votes
0answers
106 views

Explanation of White Noise

I have been trying to understand what a White Noise is and also a White noise Process and I have been trying to piece together different definitions that I have found online. I was wondering if my ...
2
votes
1answer
28 views

Law of Logarithm Type Result for Standard Brownian Motion

Let $W_t$ be a standard Brownian motion. Show that almost surely we have: $$ \limsup_{t \to \infty} \frac{W_t}{\sqrt{t \log t}}<\infty $$ The approach suggested by the writer of the problem is to ...
0
votes
3answers
37 views

Covariance of squares of Brownian motion

$B(t)$ is a Brownian motion. Calculate $Cov[B(s)^2,B(t)^2]$ for $s,t\ge0$. For this I would need $E[B(s)^2B(t)^2]$. Without the squares that wouldn't be a problem but I have no idea how to calculate ...
1
vote
0answers
40 views

How to show that $E[e^{W_s + W_t}] = e^{\frac{t+s}{2}}e^{\min(s,t)}$?

So I am trying to show that $E[e^{W_s + W_t}] = e^{\frac{t+s}{2}}e^{\min(s,t)}$ Where $W_s, W_t$ are Brownian motions. I had an idea that I could express each Brownian motion as a function of $Z \...
0
votes
0answers
117 views

Martingale related to exponential martingale

Hello everyone if have the following task: Prove that the process $Y_t=(B_t+t)\exp(-B_t-t/2)$ is a martingale. I know that the stochastic exponential $X_t=\exp(B_t-t/2)$ is a martingale, where $B_t$ ...
0
votes
1answer
46 views

Covariance and Brownian motion

B(t) is a Brownian motion and $0<s<t<u<v$. Calculate $Cov[(B(t)-B(s))^2,(B(v)-B(u))^2]$ I know the formula for the expected value of a product of 2 B-terms. But here I need the expected ...
1
vote
0answers
27 views

Prove that $(B_t)$ is a brownian motion $\iff$ $(B_t)$ and $(B_t^2-t)$ are continuous martingale.

I would like to prove that $(B_t)$ is a brownian motion $\iff$ $(B_t)$ and $(B_t^2-t)$ are continuous martingale. I did the implication, but I have difficulties for the converse. Continuity is fine. ...
1
vote
2answers
33 views

If $(B_t)$ is a Brownian motion and adapted to the filtration $(\mathcal F_t)$ does $B_t-B_s$ is independent of $\mathcal F_s$?

Let $(B_t)$ a Brownian motion adapted to the filtration $(\mathcal F_t)$. Q1) If $t> s$, does $B_t-B_s$ is necessarily independent of $\mathcal F_s$ or $(\mathcal F_t)$ must be the natural ...
4
votes
1answer
92 views

Consistency of a family of probabilities in the construction of a pre-brownian motion

I'm trying to understand how to construct the pre-brownian motion and I'm stuck at trying to prove the consistency of a family to be able to apply the Kolmogorov extension theorem. Definition. Let $...
-2
votes
1answer
63 views

A student takes a test with biased assesment and answers randomly. What is the most probable score?

A student takes a test consisting of 100 questions in which the following mark pattern has been set: +4 for a correct answer -1 for an incorrect answer 0 for an unattended question His marks` range ...
2
votes
1answer
45 views

Probability that the product of Brownian motion at two dates is positive

Let $(B_t)_{t \ge 0}$ be a standard Brownian motion, $B_0 = 0$. $0 \le t_1 \le t_2$ As the title says, I am trying to calculate the following probability : $\text{Pr}(B_{t_1}B_{t_2} \ge 0)$ Any ...
4
votes
1answer
72 views

Sharpness of Kolmogorov-Chentsov

The Kolmogorov-Chentsov continuity theorem is a general way to estimate the Hölder continuity of a process $X$ (up to taking a different version $\tilde{X}$ of $X$ ). However, I would like to know if ...
0
votes
0answers
42 views

Can Brownian Motion be considered a Markov chain?

Let's say we have a $1$ dim. Brownian motion, $(B_t)_{t \in [0,\infty)}$. Please note the continuous time. How could one prove that $B_t$ is a Markov chain ? Do we define a stochastic matrix using ...
2
votes
1answer
32 views

Wiener process identities

Let $W(t)$ be the Wiener process. Prove the following identities by taking limits of the Forward Euler discretization\begin{gather*} \int_0^Tt\,dW(t)=TW(T)-\int_0^T W(t)\,dt\\ \int_0^T W(t)\,dW(t)=\...
2
votes
1answer
82 views

Continuous-time Martingale and Brownian Motion Supremums

I am reading through Le Gall's book on Brownian Motion, Martingales, and Stochastic Calculus. I just read through the chapter on optional stopping of martingales, but I cannot solve the first exercise:...
1
vote
0answers
20 views

Using the Strong Markov property, show $Z_{t} := B_{t+\tau}-B_{\tau}$ is a Brownian motion and independent with $\mathcal F_{\tau}$

Strong Markov property states that for bounded jointly measurable process $Y_s(\omega)$ on $ [0,\infty) \times \Omega$ and a stopping time $\tau$, $$E^x(Y_{\tau} \circ \theta_{\tau})= \phi (B_{\tau},...
1
vote
2answers
2k views

Compute expectation and covariance of Brownian bridge

Let $\{X(t), t \geqslant 0\}$ be a standard Brownian motion. That is, for every $t \gt 0$, $X(t)$ is normally distributed with mean $0$ and variance $t$. Then $\{X(t), 0 \leqslant t \leqslant 1 | X(1)...
0
votes
1answer
83 views

Autocorrelation of exponential to the power of Wiener-noise

I want to find the autocorrelation $\langle x(t)x(t+\tau)\rangle$ of the following stochastic function $$ x(t) = e^{-\beta W(t)}W(t) $$ I have so far $$ \begin{align} \langle x(t)x(t+\tau) \rangle &...
6
votes
0answers
86 views

Confusion over L2 Spaces

I'm new to stochastic calculus, so hopefully this isn't too silly of a question. Setup: Let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a filtered probability space on which a Brownian ...
0
votes
0answers
24 views

Comparing geometric brownian motion at stopping time

Suppose we have 2 geometric brownian motions, $(X^{(1)}_t)_{t \geq 0}$ and $(X^{(2)}_t)_{t \geq 0}$ driven by the same Brownian motion and started at the same position. Suppose they have the same ...
5
votes
1answer
79 views

Issue concerning the uniqueness of the SDE of Bessel process

I'm reading the book Bessel Processes, Schramm-Loewner Evolution, and the Dyson model of Makoto Katori. In chapter 1, the $D$-dimensional Bessel process $R_t^x$ is introduced and is defined as follows ...
3
votes
1answer
2k views

Partial Derivative of an Integral

If $f(t)$ is a deterministic function of $t$ and $B_{n}$ is a brownian motion and: $Z =\displaystyle\int^t_0 f(s)d\left(B(s)\right)$ How does one take the partial derivatives wrt to $t$ and $B_n$ on ...
0
votes
1answer
40 views

Solving for two conditions on a stochastic process.

This is a stochastic calculus question being asked by a physicist who has not previously delved deeply into stochastic calculus. I apologise therefore for (presumably) mangling the terminology. I am ...
1
vote
1answer
62 views

Solving nonlinear stochastic differential equation

I'm trying to solve the following nonlinear stochastic differential equation: $$ dx = 3a(x^{1/3}-x)dt + 3\sqrt{a}x^{2/3}dW $$ According to my TA I'm supposed to transform variables to $y=x^{1/3}$, ...
3
votes
1answer
42 views

Prove that $(\tilde W_t)$ is a Brownian motion where $\tilde W_t=2\alpha -W_t$ if $t>\tau_\alpha $

Let $(W_t)$ a Brownian motion and $\tau_\alpha =\inf\{t\geq 0\mid W_t=\alpha \}$. I would like te prove that $$\tilde W_t= W_t\boldsymbol 1_{t\leq \tau_\alpha }+(2\alpha -W_t)\boldsymbol 1_{\tau_\...
1
vote
1answer
27 views

Problem to understand the proof of the reflexion principle of Brownian motion in wikipedia

We want to prove that $$\mathbb P(\sup_{0\leq s\leq t}W_s\geq a)=2\mathbb P(W_t\geq a).$$ Here is the proof. My mistakes are in the proof of $$\mathbb P(\sup_{0\leq s\leq t}W_s\geq a, W_t<a)=\...
1
vote
1answer
71 views

Why is a fractional Brownian motion not a semi-martingale?

I am wondering what the simplest explanation is of why a fractional Brownian motion is not a semi-martingale. Also, it would be great if the answer could explain which of the assumptions going into ...
2
votes
1answer
37 views

Why $\mathbb P(B_{T_a+(t-T_a)}-B_{T_a}<0\mid T_a<t)=\frac{1}{2}$?

Let $(B_t)$ a Brownian motion starting at $0$, and $T_a=\inf\{t\geq 0\mid B_t=a\}$. So, I know that $(B_{T_a+t}-B_{T_a})_t$ is a Brownian motion independent of $\mathcal F_{T_a}$ (where $\mathcal F_t$ ...
1
vote
0answers
20 views

Joint Density function of Brownian motion with drift and it's running maximum

pretty new to this kind of maths, it's a little outside my usual area so would appreciate sense checking something. I have read that the joint density function for Brownian motion and it's running ...
1
vote
1answer
47 views

Contradiction Optional stopping theorem : If $\tau=\inf\{t\geq 0\mid B_t=1\}$ why $\mathbb E[B_\tau]=0$?

Let $(B_t)$ a standard Brownian motion. So, it's a Martingale. Let $$\tau=\inf\{t\geq 0\mid B_t=1\}.$$ In particular, $$B_{\tau}=1\ \ a.s.,$$ therefore, we could expect that $\mathbb E[B_\tau]=1.$ But ...
1
vote
1answer
26 views

Is there a concrete meaning of Brownian motion $W_t$ has variance $\sigma ^2t$?

Let $(W_t)$ a Brownian motion. In particular, $W_t=\mathcal N(0,\sigma ^2t)$. So the fact that $\mathbb E[W_t]=0$ mean that the process $(W_t)_t$ gravitate around $0$. But how could w interpret $Var(...
0
votes
1answer
57 views

Ito formula $f(t,x)=e^{ax+bt}, t\in\mathbb R_+, x\in \mathbb R$

Let $$f(t,x)=e^{ax+bt}, t\in\mathbb R_+, x\in \mathbb R$$ and $Y(t)=f(t,W_t), t \ge 0$. Then $(Y(t),t\ge0)$ is an Ito-process. I shall use Ito's formula and write $Y$ as a sum of initial value, ...
1
vote
1answer
713 views

Variance of integral

I am trying to understand stochastic calculus and got stuck calculating the following. I need the distribution of a zero bond under the black model, so I am deriving the variance using the second ...
1
vote
2answers
2k views

Brownian Motion is Martingale

I'm reading a book, and they say Brownian Motion is martingale then show it with the following calculation: Suppose $(B_t)$ is brownian motion which generates the filtration $\mathcal F_t$ (for all $...
1
vote
1answer
20 views

two sided brownian motion hit time +1 -1

Given standard brownian motion a common way to calculate expectation of hitting time $t$ of either +1 or -1 is considering martingale $W_t^2-t$. Thus we get to $\frac{1}{2}((+1)^2-t) +\frac{1}{2}((-1)^...
1
vote
1answer
37 views

Let $W(t)$ be a standard Brownian motion. Define $X(t)=e^{W(t)}$. Let $0<s<t$. Give the expression for $Cov(X(s),X(t))$

Let $W(t)$ be a standard Brownian motion. Define $X(t)=e^{W(t)}$. Let $0<s<t$. Give the expression for $Cov(X(s),X(t))$. Here is my work: $$Cov(X(s),X(t))=E[X(s)X(t)]-E[X(s)]E[X(t)]$$ Then ...