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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93 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 ...
4
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1answer
350 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 ...
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3answers
535 views

Wiener Process $dB^2=dt$

Why is $dB^2=dt$? Every online source I've come across lists this as an exercise or just states it, but why isn't this ever explicitly proved? I know that $dB=\sqrt{dt}Z$, but I don't know what ...
4
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1answer
108 views

Ornstein-Uhlenbeck process: increments

I'm new to the forum so I hope this first question goes well. Let the Ornstein-Uhlenbeck process be defined as: $$ dV_t = - \beta V_t dt + \sigma dW_t $$ with $V_0 = v$, where $W_t$ is a Wiener ...
4
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1answer
775 views

Brownian motion - characteristic function

Let me remind first the construction of Brownian motion. Fix a vector $x \in \mathbb{R}^n$ and define $p(t,x,y) := (2\pi t )^{-\frac{n}{2}} \cdot \exp{\left( - \frac{|x-y|^2}{2t} \right)},$ for $y ...
4
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1answer
322 views

Hausdorff dimension of graphs of one-dimensional Brownian motion

First question here, my apologies if it is a duplicate or inappropriate. There is a page on Wikipedia listing fractals by Hausdorff dimension and it includes the graph of a "regular Brownian ...
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1answer
50 views

Understanding of Brownian Motion

My background is functional analysis rather than probability, but I would like to understand what is a Brownian motion. Below I'm giving my current understanding, can anyone verify whether I'm ...
4
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1answer
164 views

Lookback option with floating strike: boundary condition

I am trying to make sense of one of the boundary conditions of a look-back option with floating strike. Some notation first: let $v(t,x,y)$ denote the price at time $t$ of the option under the ...
4
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1answer
46 views

Expectation of $e^{-4B_\tau}$, where $\tau$ is an extended stopping time

This is an specific example so with a bit of luck I can get some general methodology from your answers. I have this stopping time: $$ \tau = \inf\{t \geq 0; B_t < t-2 \} $$ This is a clear ...
4
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1answer
434 views

Brownian bridge

Let $W = (W_t;F_t)$, $t \leq 0$ be a standard Wiener process, and let $(X_t)_{0 \leq t \leq 1}$ satisfy the stochastic differential equation $$ dX_t =- \frac{X_t}{1-t}dt+dW_t,\quad 0 \leq t \leq ...
4
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1answer
120 views

maximum of a brownian motion and its integral

Let $W_{t}$ be a brownian motion and $$ W^{*}_{t} = \max_{s<t} W_{s} $$ Then can you please explain why we have this: $$ (W^{*}_{t} - W_{t})dW^{*}_{t} = 0 $$
4
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1answer
360 views

Solutions to stochastic differential equations

I'm in a bit of trouble with my homework and was wondering if anyone could help me find the solutions to these two stochastic differential equations. Would really appreciate it! Thanks in advance! :) ...
4
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2answers
284 views

Brownian motion introduction

I didn't get any answers to my previous question; so I am trying a different tack. I am familiar with a first course in probability theory using measure theory, to the extent of proving the Central ...
4
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1answer
73 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 ...
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1answer
163 views

Application of central limit theorem for triangular arrays

A (1-dim) Brownian motion $(B_t)_{t \geq 0}$ satisfies the following properties: (B0): $B_0=0$ a.s. (B1): $(B_t)_t$ has independent increments (B2): $(B_t)_t$ has stationary increments, ...
4
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1answer
182 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 ...
4
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1answer
568 views

Brownian hitting time of a _very_ simple linear boundary

I realize that general results on the hitting times of a curve are practically nonexistant, but I am hoping that someone can string together a sequence of tricks to tell me what $$ \Pr\left( ...
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0answers
134 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
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0answers
48 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 ...
4
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0answers
111 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 ...
4
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1answer
258 views

Stochastic integrals and new probability measures

Let $B$ be a standard Brownian motion on $(\Omega, \mathcal{F}, P, ({\mathcal{F}_t})_{t\ge0})$, where the filtration is the one generated by $B$. Fix a time interval $[0,T]$. Define the process $X$ as ...
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2answers
242 views

Question about an exercise in Revuz/Yor

I'm solving exercise 2.28 in Revuz/Yor. I was able to prove 1). Unfortunately at 2) I got stuck. I have to show: Let $B$ be a d-dimensional Brownian motion and $A\in \mathcal{A}:=\cap_t ...
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0answers
158 views

Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion

Let $ 0 < r < 1$, fix $x > 1$ and consider the integral $$ I_{r}(x) = \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}.$$ In the investigation of ...
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1answer
315 views

Stochastic integral inequality

Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that ...
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2answers
120 views

Brownian motion: Show $\lim \sum W_{i} (W_{i+1}-W_{i})=\frac12 W^2_t-\frac12 t$ in probability.

Let $\{t_i\}_{i=1}^n$ be a partition of $[0,t]$ and $W$ a standard Brownian motion. Write $W_i$ for $W_{t_i}$. Show $$ \lim \sum W_{i} (W_{i+1}-W_i)=\frac12 W^2_t-\frac12 t $$ where the limit is in ...
3
votes
1answer
97 views

$\varlimsup_{t\rightarrow\infty} \frac{B_t}{\sqrt{t}}>0$

I am trying to prove the following statement about the standard Brownian Motion: $\varlimsup_{t\rightarrow\infty} \frac{B_t}{\sqrt{t}}>0$. I know that it is trivial to prove the above statement by ...
3
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2answers
126 views

That Brownian Motion's increments are gaussian is “not surprising”?

In section 1 of chapter 1 of Continuous Martingales and Brownian Motion, the authors claim that the fact that the increments of of Brownian motion are gaussian random variables "is not ...
3
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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 ...
3
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1answer
466 views

Solving SDE: $dX(t) = udt + \sigma X(t)dB(t)$

Solve the SDE: $dX(t) = udt + \sigma X(t)dB(t)$ Provided Question The SDE is $dX(t) = udt + \sigma X(t)dB(t)$. Find $X(t)$, where $X(t)$ is some stochastic process and $B(t)$ is a Wiener process. ...
3
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1answer
283 views

Proving the reflection principle of Brownian motion

The reflection principle of Brownian motion states that Brownian motion reflected at some stopping time $\tau$ is still a Brownian motion. The proof found in Mörters & Peres (as well as in ...
3
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1answer
53 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 ...
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1answer
1k 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 ...
3
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1answer
253 views

beginner's question about Brownian motion

I have just started learning about stochastic processes and I am confused with the notion of Brownian motion. The text defines (linear) Brownian motion under measure $\mathbb{P}$ as $B=(B_t; t\geq 0)$ ...
3
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1answer
149 views

Probability for brownian motion

How can I prove it? For $b>a>0$, show that $$ \operatorname{Pr}\left({\sup_{t\geqslant 0}\left(\frac{b+X(t)}{1+t}\right)\geqslant a}\right)=e^{-2a(a-b)} $$ where $X(t)$ is a Brownian ...
3
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1answer
395 views

$\mathcal{F_t}$-martingales with Itô's formula?

I need a little help with a problem. I am given some stochastic processes and supposed to show that they are $\mathcal{F_t}-$martingales. The first one is this, and they all look similar: ...
3
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1answer
221 views

a question about Wiener process

I don't quite understand a property of the Wiener process. Such process has the property that $W(t) - W(s) \sim \mathcal{N}(0, t-s)$ where $t > s > 0$. What I don't understand is this. As the ...
3
votes
1answer
57 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 ...
3
votes
1answer
81 views

Strong Markov property of Brownian motion

I was able to understand Brownian Motion $\{B(t):t\geq0\}$ has Strong Markov Property i.e. For any stopping time $\tau$, $P(B(t+\tau)\leq y | \mathcal{F}_{\tau})=P(B(t+\tau)\leq y|B(\tau))$ a.s. , $y ...
3
votes
1answer
210 views

Proof on Brownian Bridge

PROBLEM Let $U_t$ be a Brownian bridge on $[0,1]$ and let $Z$ be a standard normal random variable independent of $U_t$. $(a)$ Prove that the process $W_t = U_t + tZ$ is a brownian motion. $(b)$ ...
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2answers
259 views

Is the condition “sample paths are continuous” an appropriate part of the “characterization” of the Wiener process?

Wikipedia has separate articles on "Brownian motion" and "Wiener process" (http://en.wikipedia.org/wiki/Brownian_motion and http://en.wikipedia.org/wiki/Wiener_process ). I am not an expert, but that ...
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1answer
74 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 ...
3
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1answer
124 views

Are these 2 random variable independent???

Assume $\{B_t:t\ge0\}$ be a brownian motion process. Is $B_s-\frac{s}{t}B_t$ and $B_t$ independent given that ($s\le t$)
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1answer
55 views

$E \left\{ \left( \sum_{i=0}^{n-1} \left[ B_{c_i} \left( B_{t_{i+1}} - B_{t_i}\right)\right] \right)^2 \right\}$, where $c_i \in [t_i, t_{i+1}]$

Let $B$ be a standard Brownian motion and $\{t_i\}_{i=0}^n$ a partition of $[0,t]$. Define $c_i= (1-c)t_{i+1}+ct_i$, for some $c \in [0,1]$. Write $B_i$ for $B_{t_i}$ and $$ S_n=\sum_{i=0}^{n-1} ...
3
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1answer
249 views

Are hitting times of Brownian motion independent?

Suppose that $B_t$ is a standard Brownian motion. And $T_a$, $T_b$ are the hitting time whereas $a<0$, $b>0$. Then are these two random variables independent?
3
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1answer
97 views

Integrating brownian motion times exponential function

I am trying to calculate $$\int_0^tB_se^{\lambda s}ds$$ but I am unsure of how to start the computation. The motivation behind this is that I read (and am now trying to prove) that ...
3
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1answer
101 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 ...
3
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1answer
63 views

The limit supremum of a function involving Brownian motion

I would like, for some $\delta>0$ and a Brownian motion $B$, to calculate $\displaystyle\limsup_{t\to\infty}\left(\exp\left( (1+\delta)t\right)\cdot\exp\left(-B_t-\frac{t}{2}\right)\right)$ ...
3
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2answers
444 views

Conditional distribution in Brownian motion

I need to prove the following: Let $X$ be a Brownian motion with drift $\mu$ and volatility $\sigma$. Pick three time points $s < u < t$. Then, the conditional distribution of $X_u$ given ...
3
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1answer
79 views

Distribution of a Brownian motion with respect to $\mathbb{P}^x$

Let $(\Omega,\mathcal{A},\mathbb{P})$ a probability space and $(B_t)_{t \geq 0}$ a Brownian motion (started in $x=0$). Then one can define a probability measure $\mathbb{P}^x$, $x \in \mathbb{R}$, on ...
3
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1answer
126 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?