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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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 ...
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435 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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772 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 ...
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1answer
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Compute the distribution of $\int_0^1 B_t dt$

I need an help with the following: let $(B_t)_t$ a Brownian motion. Compute the distribution of $X:=\int_0^1 B_t dt$. Integrating by parts we have that: $$\int_0^1 B_t dt=B_1-\int_0^1 t dB_t.$$ Now, ...
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Dominated convergence problems with Wald's identity for the Brownian Motion

In the course of proving Wald's second identity $E(B^2_T)=E(T)$, where $(B_t)_{t\geq0}$ is the Brownian motion and $T$ is a stopping time with $E(T)<\infty$, I got stuck with the following problem. ...
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180 views

Brownian motion, reproducing kernel Hilbert space, and the Laplace operator

Consider the standard Brownian motion on $[0,1]$: $$ dB_t, \; B_0 = 0, $$ defined on the probability space $(\Omega, P)$. It covariance function is $K(s,t) = \min \{s , t\}$ on $[0,1] \times ...
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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 ...
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Quadratic Variation of Brownian Motion

Quadratic Variation of a Brownian motion $B$ over the interval $[0,t]$ is defined as the limit in probability of any sequence of partitions $\Pi_n([0,t])=\{0=t^n_0<\cdots<t^n_{k(n)}=t\}$ of the ...
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218 views

Convergence of Ornstein-Uhlenbeck process as a scaled Brownian Motion

Let $W$ be a standard Brownian motion. Let $\alpha,\sigma^2 >0$, and let $X_0$ be a $\mathbb{R}$-valued random variable with distibution $\nu$ that is independent of $\sigma(W_t,t\geq 0)$. Now ...
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proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
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Accelerated Eigenfunction Expansions of Random Functions

I am interested in eigenfunction expansions of random functions. We know that the autocorrelation of brownian motion, $\{ B_t \}_{t \geq 0}$, is given by $$ E[B_t B_s] = \min\{s,t \}, $$ which can ...
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The probability that a linear Brownian motion will hit a curve

Summary I am trying to estimate the probability that a standard linear Brownian motion will hit some curve. To make things a bit simple, I can assume that the curve is a graph of a function, that is ...
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0answers
166 views

Intuition for the optimality of bold play

There is a standard result (I think originally by Dubins and Savage) that if one wants to maximise the probability of winning a certain amount in an unfair game of chance then an optimal strategy is ...
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expected value of brownian motion

How can you find this expected value? $$ \mathbb{E}[|W_{t}^2 - t|] $$ where $W_{t}$ is a brownian motion.
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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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371 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 ...
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Show that $X(t)=t W(1/t)$ is a Brownian motion if $W(t)$ is a Brownian motion.

I am trying to solve a past exam question for which I have its answers. I've got to the end, but the very last and simplest line has confused me. I've spotted some errors and corrected them, but I ...
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1answer
96 views

Probability distribution of $\int_0^t \frac{W_s}{s} \,ds$

I am currently working on an exercise that requires the knowledge of the distribution of $\int_0^t \frac{W_s}{s} \,ds$, where $W$ is a Brownian motion. I can compute the distribution of $\int_{0}^T ...
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1answer
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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 ...
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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 ...
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Variance of product of Brownian motions

Let $\{B_{t}\}_{t\geq0}$ be Brownian motion. What is the variance of $B_{t}B_{s}$?
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How to derive the Ornstein-Uhlenbeck Stochastic Integral Equation?

I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-a\int^t_0 X_s\,ds +B_t$. B is the Brownian motion. And its analytic ...
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A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
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Distribution related to brownian bridge

Let $B(t)$ be a Brownian Bridge and $U$ is uniformly distributed on $(0,1)$. I wish to know the distribution function $B(U)$. Is it possible? As we know, $B(t)\sim N(0,t(1-t))$. But, I haven't a clue ...
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231 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 ...
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1answer
342 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 $B=(B_t; t\geq 0)$ ...
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Implementing Ornstein–Uhlenbeck in Matlab

I am reading this article on Wikipedia, where three sample paths of different OU-processes are plotted. I would like to do the same to learn how this works, but I face troubles implementing it in ...
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154 views

Is $\mathbb{E}\exp \left( k \int_0^T B_t^2 \, dt \right)<\infty$ for small $k>0$?

Suppose that $B$ is a Brownian motion. Does it hold that \begin{equation} \mathbb{E}\left[\exp\left(k\int_0^T[B(t)]^{2}\,dt\right)\right] <\infty\text{ ?} \end{equation} for some positive constant ...
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Density of first hitting time of Brownian motion with drift

I just started learning about Brownian motion and I am struggling with this question: Suppose that $X_t = B_t + ct$, where $B$ is a Brownian motion, $c$ is a constant. Set $H_a = \inf \{ t: X_t =a ...
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Sum of Brownian Motions

I've got a little problem: if $X_{t}$ and $Y_{t}$ are two indipendent Brownian motions, is then $$Z_{t}:=X_{t}+Y_{t}$$ a Brownian motion too? I've got some troubles only with showing that $Z_t$ is ...
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1answer
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Is the graph of a Brownian motion over an interval measurable?

Let $n \in \mathbb{N}_1 := \{1, 2, \dots\}$ and let $B:\Omega \times [0, \infty) \rightarrow \mathbb{R}^n$ be a standard, $n$-dimensional Brownian motion over the probability space $(\Omega, ...
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1answer
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Exponentials of stochastic processes and Brownian motions

This is my first time looking at problems in stochastic calculus, so please bare with the simplicity of the question. As always, any help is greatly appreciated. 1) Given $X_t=\int_0^ur_sds$ for a ...
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1answer
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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 ...
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1answer
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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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1answer
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Itô symmetry for elementary predictable stochastic processes

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
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1answer
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Joint distribution of $(W(1),W(3),W(3)-W(2))$ for a Brownian motion $(W(t))_{t \geq 0}$

Let $(\Omega,\mathcal{F},P)$ be a probability space, $(W(t),t \ge 0)$ a Brownian motion and $(\mathcal{F}_t,t \ge 0)$ its natural filtration. What is the joint probability distribution of ...
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886 views

Distribution of stochastic integral

Assume that $\mathrm{d}S = \sigma \, \mathrm{d}W$ with initial level $S(0)$ and where $\mathrm{d}W$ is usual Brownian motion. Now $$A(T) = \frac{1}{T} \int_0^T S(t) \, \mathrm{d}t.$$ ...
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1answer
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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 ...
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1answer
755 views

Absolute value of Brownian motion

I need to show that $$R_t=\frac{1}{|B_t|}$$ is bounded in $\mathcal{L^2}$ for $(t \ge 1)$, where $B_t$ is a 3-dimensional standard Brownian motion. I am trying to find a bound for ...
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1answer
32 views

Brownian motion: Strong Markov versus translation invariance

In the proof of the reflection principle in Durrett's textbook (Probability: Theory and Examples (4e), Theorem 8.4.1, page 317), there's a step which I'm a little shaky on. Basically, this proof ...
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Prove identity in law for stochastic process driven by Brownian Motion

Let $B = (B_t)_{t\geq 0}$ be a standard brownian motion started at $0$. Consider the two following stochastic equations: \begin{equation} \begin{split} dX_t &=& (13 + 2X_t)\,dt + (6 + ...
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Stopped process of Brownian motion

I am baffled about the following problem: Let $(B_t)$ be a standard Brownian motion. Let $$ \tau:= \inf\{ t \geq 0 :B_t = x \} \wedge \inf\{ t \geq 0 :B_t = -y \}$$ be a stopping time, where $x,y ...
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1answer
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Why is the canonical filtration of a Brownian motion left-continuous?

Let $\{W_t, t\geq 0\}$ be a Brownian motion, and has a.s. continuous sample paths. Let $\{\mathcal{F}^W_t, t\geq 0\}$ be the canonical filtration, i.e. $\mathcal{F}^W_t=\sigma(W_s, 0\leq s\leq t)$. ...
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1answer
130 views

Expectation of Square of Stopping Time

Let $B_t$ be standard Brownian motion and $a < 0 < b$. Define stopping time $T$ as follows. $$T = \min \{t \geq 0: B_t \in \{a, b\} \}.$$ The expectation of $T$ is $\mathbb ET = |a|b$ and can be ...
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1answer
204 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 ...
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1answer
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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 ...
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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 ...
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1answer
419 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 ...
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1answer
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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: $$ (W^{*}_{t} - W_{t})dW^{*}_{t} = 0 $$
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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! :) ...