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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1answer
97 views

Variance for the distance between two Brownian particles vs. a Brownian particle and a stationary particle

I have two Brownian particles, $B_1$ and $B_2$ (with diffusion coefficients $D_1$ and $D_2$), at coordinates $P_1$ and $P_2$ in a three-dimensional fluid. I let the system evolve for $t$ seconds. ...
2
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1answer
40 views

Motion of the centroid of $k$ Brownian particles?

Imagine we have $k$ Brownian particles diffusing in a three-dimensional solution, where each particle has the same diffusion coefficient $D$ (measured in $\mu^2/sec$). Now imagine that we have a ...
1
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1answer
51 views

Convergence in $L^{2}(\Omega)$

Let $T>0$ and $P^{n}:=\lbrace0=t_{0}^{n}<t_{1}^{n}<...<t_{m_{n}}^{n}=T\rbrace$ be the $n$- th division of the interval $[0,T]$ such that $\delta(P^{n})\to0$, as $n\to\infty$, where ...
1
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1answer
114 views

Convergence to Brownian motion integral

Let $X_i$ be i.i.d with $\mathbb{E}(X_i) = 0$ and $Var(X_i) =1, \, S_n = \sum_{i=1}^n X_i$. I would like to show that $\sum_{i=1}^n \frac{f(S_i/\sqrt{n})}{n}$ converges to $\int_0^1 f(B_t)dt$ in ...
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2answers
124 views

The probability of a Brownian particle traveling a distance $L$ before returning to its point-of-origin

What's the probability that a Brownian particle diffusing along a one-dimensional interval returns to its point of origin before traveling a distance $L$? We know that in the limit of a random walk ...
3
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1answer
392 views

Brownian Bridge Representation

Let $B_t$ be a Wiener Process, then $U_t=B_t-tB_1,~0\le t \le 1$ is a Brownian bridge. Show that $X_t=(1+t)U_{{t}/({1+t})}$ is a Wiener Process. I'm not quite sure how to start this off. Any help ...
3
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1answer
358 views

linear combination of two Wiener processes

I have a question concerning the linear combination of two Wiener processes (please see http://en.wikipedia.org/wiki/Wiener_process for a definition). Let $W$ and $\tilde{W}$ be two Wiener processes ...
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0answers
46 views

Simulating of GBM

I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem: Given a GBM of the form $dS(t) = \mu S(t) dt + ...
3
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1answer
56 views

two r.v sharing the same law

I have a question: Let $X=B^{+}$ or $X=|B|$ where $B$ is the standard Brownian motion. Set $$J_p=\sup_{t\geq 0}(X_t-t^{\frac{p}{2}})$$ where $p>1$ and $q$ its conjugate ...
0
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1answer
294 views

Is the following a Wiener process?

This is a worked example on Wiener processes. Question: Pick a normally distributed random variable $Z \sim N(0,1)$, then define $W(t) = Z\sqrt{t}$. Is $W(t)$ a Wiener process? Answer: ...
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2answers
161 views

Mean and variance of this random variable

How can we compute the mean and variance of $e^{W_tW_s} $ where $(W_t)_{t \geq 0} $ is a Brownian motion? If we want to compute $ \mathbb{E}(W_tW_s) $, the usual thing to do is to assume that $ s ...
1
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1answer
122 views

Absolute continuity of the distribution of $X_t=aB_t+bt$, $Y_t=a(t)B_t$ with respect to the Wiener measure

Let $B_t:$ 1-dimensional Brownian motion, $P:$ its distribution on the Wiener space $C([0,1],\mathbb{R})$ $X_t=aB_t+bt\text{; }t \in [0,1]$, $P_{a,b}$ its distribution $Y_t=a(t)B_t\text{; }a:[0,1] ...
6
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1answer
358 views

Computing the limit of the expectation of a function of a stochastic process (phew!)

I state my problem in a few lines then describe what I have already done. I have a quite simple stochastic differential equation (SDE): $dx=-2x \, dt+\sqrt{1-x^2} \, dW$ with $W$ a brownian. I ...
3
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1answer
83 views

Given the SDE: $dX_t=dB_t+b(X_t) dt$ with $(x,b(x)) \leq 0, \forall x \in \mathbb{R}^n$, prove that $E[|X_t|^2] \leq nt+E[|X_0|^2]$

I'm working on this problem: Given a solution $X_t$ to the SDE $$dX_t=dB_t+b(X_t) dt$$ where $B_t$ is an $n$-dimensional Brownian motion, and $b:\mathbb{R}^n \to \mathbb{R}^n$ a Lipschitz ...
1
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1answer
409 views

d-dimensional Brownian motion and martingales

I was solving questions from the Martingales chapter in "Stochastic Processes" by Richard Bass. There was a question regarding d- dimensional Brownian motions(BM): Let $(W_t^1,...,W_t^d)$ be a d ...
2
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2answers
71 views

Cost for hedges under a Wiener process

I'm trying to estimate the hedging costs relating to a financial derivative which moves like a Wiener process, and I'm struggling to find the correct setup to solve the problem. Suppose I have a ...
4
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1answer
106 views

How to show that the following process is a submartingale

Suppose we have a filtration $(\mathcal{F}_t)$ satisfying the usual conditions. Let $W$ be a Brownian Motion with respect to that filtration. We define the two processes $X_t:=W^2_t$ and ...
3
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1answer
242 views

Fixed-Time Brownian Motion Exit Probabilities

A standard computation using martingale techniques allows us to compute probability that a Brownian motion started at zero exits the interval $[-a,b]$ ($a, b > 0$) at $-a$ or $b$. It appears to me ...
3
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1answer
180 views

convergence ito integral

It is easy to calculate the integral $\int_0^T B_t \, dB_t=\frac{1}{2}B_T^2-\frac{1}{2}T$ That means I showed that $\int_0^T S_n \, ...
0
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1answer
84 views

How can we easily compute $\mathbb{E} [ \left|W_t\right| ^\alpha]$?

How can we easily compute $\mathbb{E} [ \left|W_t\right| ^\alpha]$, where $\alpha \in \mathbb R^*_+ $ and $W = (W_t)_{t \geq 0}$ is the one dimensional standard Brownian motion (or wiener process)?
6
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1answer
936 views

Expectation of an integral w.r.t. Brownian Motion

I know the following statement: if $f$ is a deterministic function and continuous, i.e. $f\in C^0([0,T],\mathbb{R})$, then $\int f(s)dW_s$ is normally distributed with mean zero and variance $\int ...
2
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0answers
140 views

Independence of Brownian Motion with respect to a stopping time

Let $B_t$ be a brownian motion, $B_0=0$, and $\gamma \in \mathbb{R}$. Now, let's build the following stopping time: \begin{equation} T = \inf \{ t \geq 0 : |B_t + \gamma t| = 1 \}. \end{equation} If ...
2
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1answer
100 views

Fractional Brownian motion as integral, mean zero

Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $$X(t)={1\over ...
2
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1answer
178 views

Translational invariance of Brownian motion

Let $(\Omega,\mathcal{A},\mathbb{P})$ a probability space, $(X_t,\mathcal{F}_t)_{t \geq 0}$ a time-homogeneous Markov process. A paper I read defines a probability measure $\mathbb{P}^x$ by ...
3
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1answer
136 views

Brownian motion interesting question

I found this interesting question on the internet, but unfortunately I could not solve it. What is probability that Brownian motion (starting at origin) has value 1 before having value -2?
3
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1answer
175 views

Brownian motion and hitting frequency

Suppose we have a Brownian motion $B_t$ with $B_0 = 0$ and $B_t - B_s \sim N(0,t-s)$. Every time $B_t$ hits $\pm h$, where $h$ is some "barrier" $>0$, I pay someone £1 and the brownian motion ...
2
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1answer
206 views

How to show that $X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$ (“inverse brownian”) is a martingale?

Consider $$X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$$ where $ \left(B_{t }\right)_{t \geq 0}$ is a $ \mathcal F_t$- brownian motion in $\mathbb R ^3$, null at ...
2
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1answer
103 views

Show that $M_t = \int_0 ^t \exp{((B_2(s)^2)} dB_1(s)$ is not a continuous square integrable martingale

Consider the following $\mathcal F_t$- (continouous) local martingale $$M_t = \int_0 ^t \exp{((B_2(s)^2)} dB_1(s)$$ where $\left(B_t\right)_{t\geq0} =\left(B_1(t),B_2(t)\right)_{t\geq0}$ is ...
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1answer
998 views

How to prove the martingale?

How to prove that the integral $\int_{0}^{+\infty}\upsilon e^{-ru}S_{u}dW_{u}^{Q}$ is a martingale under Q where $S_{t}$ is a martingale under Q and $\mathbb{E}^{Q}[\int_{0}^{+\infty}|\upsilon ...
3
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1answer
150 views

Autocorrelation of wrapped Wiener process

Let $\phi(t)$ be a Brownian Walk (Wiener Process), where $\phi\in[0,2\pi)$. As such we work with the variable $z(t)=e^{i\phi(t)}$. I would like to calculate $$E(z(t)z(t+\tau)).$$ This is equal to ...
7
votes
1answer
252 views

Integral of the positive part of a Brownian motion

Let $X(t)$ be the standard Brownian motion, I need to find the distribution of $S=\int_{0}^T(X(t))^+dt$, where $(x)^+=\max\{0,x\}$. I want to use the distribution to get a concentration bound for ...
5
votes
2answers
278 views

Brownian Motion Covariance: max instead of min

It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion. Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that ...
4
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1answer
453 views

Covariance of Brownian-motion-like processes

We know that $\operatorname{Cov}(B_s,B_t)=\min(s,t)$ if $B_t$ is Brownian motion. What is $\operatorname{Cov}(B_{f(s)},B_{f(t)})$ for some injective $f$? How can I write $B_{f(t)}$ in an Ito ...
3
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1answer
2k views

Covariance of Brownian Bridge?

I am confused by this question. We all know that Brownian Bridge can also be expressed as: $$Y_t=bt+(1−t)\int_a^b \! \frac{1}{1-s} \, \mathrm{d} B_s $$ Where the Brownian motion will end at b at $t ...
2
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2answers
138 views

Relation between $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$

Let $B_t$ be a standard Wiener motion. What can we say about $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$? Is there a relation?
2
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1answer
172 views

What is the conditional distribution of $B(s)\mid B(t_1)=x_1,B(t_2)=x_2$ for $0<t_1<s<t_2$?

Given that $\{B_t,t\ge0\}$ is a standard Brownian process. What is the conditional distribution of $B(s)$ given $B(t_1)=x_1$ and $B(t_2)=x_2$, for $0<t_1<s<t_2$? My try: First i tried to ...
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1answer
858 views

How do you show this is a martingale?

How do you show the following process is a martingale? My notes say it is a martingale by I can't work it out. $$ E[e^{\sigma B(t) - \frac{\sigma ^2 t}{2}} | \mathscr{F}(s)] $$ I tried to multiply ...
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2answers
579 views

Conditional Expectation of integral of Wiener process

Let $W_t$ be a standard Wiener process. How can we calculate: $$\mathbb{E}\left[\int_0^t|W_r|^2\text{d}r \ |\ \mathcal{F}_s\right]$$ where $(\mathcal{F}_s)_{s\geq0}$ is the natural filtration?
4
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1answer
2k views

Expectation of Stopping Time w.r.t a Brownian Motion

How do you take the expectation of a stopping time with respect to a Brownian motion? The specific question is: $$ \tau = \inf\{ t \ge 0: B(t) \in \{-a, b\}\} $$ I understand the optional stopping ...
4
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1answer
210 views

Applying Ergodic Theorem on fractional Brownian motion

For a fractional Brownian motion $B_H$ consider the sequence for $p>0$ $$Y_{n,p}={1\over n}\sum\limits_{i=1}^n \left|B_H(i)-B_H(i-1)\right|^p.$$ By the Ergodic Theorem it is ...
3
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2answers
622 views

Show that this process is a martingale

Let $B_t$ be a Brownian motion and $M_t=\max_{0\leq s\leq t}B_s$. Show that: $$(M_t-B_t)^4-6t(M_t-B_t)^2+3t^2$$ is a martingale for $t\geq0$.
4
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1answer
201 views

Show that $M_t$ is a Standard Brownian Motion

Let $M=(M_t)_{t\geq0}$ with $$M_t=\int_0^{\log\sqrt{1+2t}}e^s\text{d}B_s$$ where $(B_t)_{t\geq0}$ is a Standard Brownian Motion. Show that $M$ is also a Standard Brownian Motion and compute ...
6
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1answer
520 views

Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
2
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0answers
99 views

A problem with regard to Wiener process

Let $W$ be a Wiener process and $U_x$ is the amount of time spent below $x$ during time interval $(0,1)$. Hence $U_x=\int\limits_0^1I_{\{W(t)<x\}}dt$. My question is: what is the probability ...
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1answer
692 views

Expectations of multiplied Wiener processes.

I wish to evaluate the following: $E[W(t-1)W(t)^2]$ $E[W(t)^3]$ where $t > 1$, $W$ is a standard Brownian motion and we are at $\mathscr{F}_0$ now. I know that $E[W(t-1)W(t)] = \min{(t-1,t)} ...
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2answers
438 views

expectation of brownian motion squared with regard to stopping time

let $T_1$ be the first occurrence of a Poisson process at rate $\lambda$, and $X(t) = \sigma B(t) + \mu t$ be another independent Brownian motion with drift, calculate $E(X(T_1))$ and ...
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0answers
456 views

Show that this semimartingale is a local martingale

Let $B_t$ be a standard Wiener motion, $I_t=\int_0^t|B_s|^2\!\text{ds}\ $and $S_t=\max_{0\leq s\leq t}B_s$. Let also $F:\mathbb{R}^2_+\times\mathbb{R}\times\mathbb{R}_+\rightarrow\mathbb{R}$ a ...
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0answers
112 views

Is this geometric Brownian Motion?

The SDE for GBM is usually specified as: $$dX(t) = X(t)[\mu dt + \sigma dW(t)]$$ If we model diffusion as stochastic, is the following still GBM? $$dX(t) = X(t)[\mu dt + \sigma_t dW(t)]$$ ...
2
votes
1answer
86 views

Clarke Ocone representation formula

Let $(B_t)_{t}$ a Brownian motion and $F \in L^2(\Omega,\mathcal{F}_T,\mathbb{P})$. Then we know by Itô's representation theorem that there exist a process $X$ such that $$F=\mathbb{E}F+\int_0^T X_s ...
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0answers
101 views

Fractional Brownian motion, selfsimilar

Let $0<H<1$. A real-valued Gaussian process $\left(B_H(t)\right)_{t\geq 0}$ is called fractional Brownian motion (fBm) if $\ \mathbb{E}[B_H(t)]=0$ and ...