Tagged Questions

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
144 views

Specific use of reflection principle for Brownian motion

Let $B$ be a Brownian motion with $B(0)=0$, $x,y>0$ and $B^*$ the Brownian motion reflected at $-x$. I came across the following: $$ \mathbb P_0(\inf_{s\in [0,t]}B(s)<-x, B(t)\geq y-x) = ...
2
votes
1answer
208 views

Doob's stopping time theorem with unbounded stopping time

Let $(X_t)_{t\geq0}$ be Brownan motion on $\mathbb R$, and $\tau$ is a stopping time adapted with the natural filtration generated by the Brownian motion. If $X_0=0$, $E(e^{\tau/2})<+\infty$. ...
0
votes
2answers
75 views

Identity for exponential of Brownian motion using scaling relation

Let $B$ be a Brownian motion and $s\wedge t := \min\{s,t\}$, $s\vee t := \max\{s,t\}$. I stumbled over the following identity: $$ \mathbb E[\exp(B(s\wedge t) + B(s\vee t))] \\=\mathbb ...
4
votes
1answer
280 views

Stopping time and Brownian motion (specific example)

Let $B$ be a Brownian motion. I want to show that $$ \inf\{t\geq0 \mid B(t)=\max_{x\in [0,1]}B(s)\} $$ is not a stopping time w.r.t. the standard filtration. How can one intuitively see that this ...
1
vote
1answer
536 views

Checking for Martingales on Stochastic processes

I am confused about how to check whether a process is Martingale. I know, I have to check for clear drift but a bit confused about to approach this problems. I need to apply Ito's first i think. For ...
0
votes
1answer
267 views

Expectation of a stochastic exponential

In class a while ago we used the following simplification: $$ \mathbb E \left[ \exp\left(\langle \boldsymbol a,\mathbf W_t\rangle \right) \right] \quad =\quad \exp\left(\frac12 |\boldsymbol a|^2 ...
0
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0answers
229 views

Geometric Brownian Motion

Consider asset price $S$ that evolves according to Geomtric Brownian Motion with constant $\mu$ and $\sigma$ $$dS = \mu Sdt + \sigma SdX$$ Show by the application of Itô's Lemma to function $\log S$ ...
0
votes
1answer
84 views

Squared increments of Brownian motion in $L^2$

I have encountered an $L^2$ limit, which I am not sure of: Let $B_t$ be a Brownian motion and $a=t_0<t_1<\dots <t_n=b$. Show that $$\lim_{max(t_{i+1}-t_i)\mapsto 0} \sum_{0\leq i \leq n-1} ...
2
votes
1answer
86 views

Two Questions about Brownian Motion

How do you show $B_T\in\mathcal{F}_T$ for T is a stopping time? Note the filtration is generated by the Brownian motion (and not necessarily completed, in particular, ...
2
votes
1answer
51 views

Existence Brownian Motion

I'm reading through a proof of the existence of a Brownian motion and at some point they state that for $0\leq t_{0}<t_{1}...<t_{n}$ there exist multivariate normal distributions with covariance ...
2
votes
0answers
114 views

Independence of Brownian motion-related stopping times

Let $(B_t,\mathcal{F}_t)_{t \geq 0}$ a Brownian motion on a probability space $(\Omega,\mathcal{A},\mathbb{P})$. For $a \in \mathbb{R}$ define a stopping time $\tau_a$ by $$\tau_a := \tau(a) := ...
0
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0answers
83 views

How to prove this inequality involving integration with respect to Brownian motion?

If $B_t$ is the Brownian Motion, I have to verify that $$E\left\lvert\int_s^t G(t,w)\,dB_t\right\rvert^6\leq 15^2\cdot (t-s)^2\cdot\int_s^t E\lvert G(t,w)\rvert^6\,dt$$
1
vote
2answers
671 views

variance of square of brownian motion increment

In other words, $$\text{Var}\left\{ [W(t) - W(s)]^2 \right\} = \mathbb E \left\{ (W(t) - W(s))^4 \right\} - \left[ E\left\{ (W(t) - W(s))^2 \right\} \right]^2 $$ How is this equal to $(t-s)^2$ ...
2
votes
1answer
175 views

Distribution of the integral of a diffusion process

Suppose $X(t)$ is a diffusion process with $E[X(t)]=0$ and variances $\sigma^2_t$ concave in time. If $X$ is also a Brownian motion, then the distribution of $\int_0^T X(t) dt$ is known to be ...
0
votes
1answer
70 views

How to prove that for Brownian motion in $(a, b)$ $\mathbb{E}^x[\min(H_a, H_b)] = (x-a)(b-x)$?

i'm wondering if anyone can help me with proving the fact that for BM in the interval $(a,b)$ and with $$H_y = \inf\{t>0: X_t = y\},$$ the following is true: $$\mathbb{E}^x[\min(H_a, H_b)] = ...
3
votes
1answer
93 views

Limit of occupation times for Brownian motion

Let $B_t$ be a standard Brownian motion on $\mathbb R$ started at $0$. For $A\subset\mathbb R$ Lebesgue measurable, let $\mu_T(A) = \frac{1}{T} m(t \leq T: B_t \in A)$, where $m$ is Lebesgue measure. ...
1
vote
1answer
388 views

Running maximum of Wiener process

The joint distribution of the running maximum $ M_t = \max_{0 \leq s \leq t} W_s $ and $W_t$ is $f_{M_t,W_t}(m,w) = \frac{2 ( 2 m - w)}{t\sqrt{2 \pi t}}e^{-\frac{(2m-w)^2}{2t}}, m ...
0
votes
1answer
112 views

Independent increments of $X_t:=\int_0^t\phi(s) dW_s$

Motivated through the following question Can we prove directly that $M_t$ is a martingale, I want to ask this in a separate question. Suppose we have a deterministic function $\phi$ which belongs to ...
0
votes
1answer
90 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
votes
1answer
37 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
vote
1answer
50 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
vote
1answer
101 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 ...
1
vote
2answers
104 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
votes
1answer
345 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
votes
1answer
272 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 ...
1
vote
0answers
43 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
votes
1answer
54 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
votes
1answer
232 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: ...
3
votes
2answers
157 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
vote
1answer
114 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] ...
5
votes
1answer
316 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
votes
1answer
76 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
vote
1answer
334 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
votes
2answers
66 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
votes
1answer
94 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
votes
1answer
208 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
votes
1answer
164 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
votes
1answer
68 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
votes
1answer
743 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
votes
0answers
121 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
votes
1answer
92 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
votes
1answer
153 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
votes
1answer
129 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
votes
1answer
160 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
votes
1answer
170 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
votes
1answer
98 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 ...
-3
votes
1answer
801 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
votes
1answer
135 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
227 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
261 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 ...