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 $t-s$.

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14 views

Evaluate $\int_0^r e^{ku}\; dW_u$.

I want to know how to find $$\int_0^r e^{ku}\; dW_u$$ where $W_u$ is a Wiener process with mean 0 and variance $u$.
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1answer
29 views

Simplifying a Stochastic Equation

Let $$dG_t=\mu(t)dt+\sigma(t)dW_t$$ where $W_t$ is standard Brownian motion and define $$f(t,u)=E[e^{-r(u-t)}(\mu(u)-rG_u)|F_t]$$ where $(F_t)_{t\in [0,T]}$ is a filtration, $T\ge u\ge t$ and $r$ is a ...
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1answer
813 views

Skorokhod Embedding theorem proof

It occurs in Durrett's proof of Skorokhod embedding that he needs the following. Suppose that we have a Brownian motion $B_t$ in $1$-d that starts at $0$. (To be clear, it is not necessarily ...
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1answer
16 views

Simple Exploitation of Markov Property for Brownian Motion

The elementary Markov Property as I learned it reads: For a positive real number $a$ the stochastic processes $(B_t)_{0 \leq t \leq a}$ and $(W_t)_{t \geq 0}:=(B_{t+a}-B_a)_{t \geq 0}$ are ...
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1answer
96 views

Revuz and Yor's Book “Continuous Martingales and Brownian Motion” - Chapter 1 - Exercise 1.19

Context : This post is the second of a series of posts taking their origins from the exercises in the Revuz and Yor's Book "Continuous Martingales ans Brownian Motion". The reason for doing so is ...
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41 views

What is $\mathbb E\left[\exp\left(\int_0^t B_s^2ds\right)\right]$?

Let $(B_t)$ a Brownian motion. I know that $$\mathbb E[f(X)]=\int_{\mathbb R}f(x)\mu_X(dx),$$ So, logically if $$f(B_s)=\exp\left(\int_0^t B_s^2ds\right)$$ then $$\mathbb E[f(B_s)]=\int_{\mathbb R}e^{...
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1answer
50 views

Calculate expected value of $H_t=exp(W_t/(1+t))$

If $W_t$ is a Wiener Process/Brownian motion standard, what is the expectation of: $H_t=exp(W_t/(1+t))$ I know the answer is: $E(H_t)=exp(t/(2(1+t)^2))$ But the solution I find is $1/2*exp(t/(2(1+...
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1answer
46 views

Running Maximum of Brownian motion is singularly continuous?

Let $W_t$ be the standard Brownian motion, and define the running maximum of Brownian motion as $$M_t\doteq \max_{0\leq s\leq t} W_s.$$ Then $M_t$ is non-decreasing function. We can regard this non-...
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29 views

Brownian motion and Beta distribution

I am interested in the distribution of the time that the standard Brownian $W_t$ motion on $[0,1]$ satisfies the following inequality: $$W_t \ge stW(1)$$ For different values of $s$. I conjecture that ...
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29 views

Probability of Brownian Motion

I've studied a proof from the book An Introduction to Stochastic Differential Equations concerning the nowhere differentiability of the Brownian motion and I'm stuck at the following proof: In the ...
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1answer
34 views

Relation between $\sup_{0\leq s\leq t}|W_s|$ and $\sup_{0\leq s\leq t}W_s$ for Brownian motion $W$

Let $W$ be a Brownian motion. In a calculation, I have to compute $$\mathbb P \left(\sup_{0\leq s\leq t}|W_s|>a\right).$$ My idea would be to use the reflexion principle that says that $$\mathbb P\...
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22 views

How to show that $(e^{-rt}S_t)_{t\in [0, T]}$ is a $Q$ martingale?

We say $(W_t)_{t\in[0,T] }$ is a Brownian Motion on $(\Omega, F, F_W, P)$. For $\lambda\in \mathbb{R}$, we define $$Q(A):=E[1_AM_T], \quad A\in F,$$ where $$M_t:=exp(-\lambda W_t - \frac{1}{2}\lambda^...
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16 views

Is running maximum of Brownian motion is absolutely continuous?

Given the standard Brownian motion $W_t$, define the running maximum $M_t$ as $$M_t \doteq \max_{0 \leq s \leq t} W_s.$$ Is this function absolutely continuous?$~~$ That is, could we represent it as $...
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1answer
12 views

Quotient of two assets following a GBM

I have X and Y following the dynamics $dX(t)=X(t)(rdt+\sigma_{X}dW(t)$ $dY(t)=Y(t)(rdt+\sigma_{Y}dW(t)$ The components of the BM are independent. What is the distribution of $ln(\frac{X(t)}{Y(t)})...
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1answer
36 views

If $T_{U,V}=\inf\{t:B_t \notin (U,V)\}$ does $\int E[B_{T_{U,V}}|U,V]dP=\int E[B_{T_{u,v}}]dP_{U,V}$

I have the following question: if $T_{U,V}=\inf\{t:B_t \notin (U,V)\}$ does $$\int E[B_{T_{U,V}}|U,V]dP=\int E[B_{T_{u,v}}]dP_{U,V}$$ - or if it makes the difference does $\int E[1_{\{-\infty ,x]} ...
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1answer
52 views

Conditional expectation of exponential Brownian motion

Let $f\in L^2[0,T]$. Show that conditional expectation $$ \mathbb{E}\bigg[ e^{\int_{0}^T f(s)dB_s}\bigg| \mathscr{F}_t\bigg] =\exp \left(\int_0^t f(s)\ dB_s + \frac{1}{2} \int_0^T |f(s)|...
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30 views

Expectation of stochastic process continuous in starting point.

Let $(B_t)$ be a Brownian motion started in $x$ and $f_1,\dots,f_n$ be $\mathbb{R}$-valued continuous bounded functions. Consider the map $x \mapsto \mathbb{E}^x \big( f_1(B_{t_1}) \cdots f_n(B_{t_n}) ...
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44 views

Expectation of the exponent of a constant times exiting time of a Brownian motion (i.e. $\mathbb{E}_x[e^{n\sigma}]$)

Suppose $a, n>0$ and $B_t$ is a Brownian Motion, define $$\sigma=\inf\{t:B_t\in\{-a,a\}\}.$$ I want to find $\mathbb{E}_x[e^{n\sigma}]$. (Notice since $n>0$ it is $\textbf{not}$ Laplace ...
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1answer
39 views

Brownian motion and Running Maximum

Take B$_t$ as a standard Brownian motion such that B$_0$ = 0. And M$_t$ is the corresponding running maximum. i.e. M$_t$ = max$_{0\leq s \leq t}$ B$_s$. My goal is to compute: (i) Quadratic ...
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11 views

Deriving Ito process with a drift from geometric Brownian motion.

Please help me solve this question. Thank you. Let the Geometric Brownian motion be: $$ \frac{\Delta S}{S} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t} $$ $\Delta S$ = change in stock price (s) ...
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Is there anything known about $\sup_{s\leq t} \vert B_s \vert - f(s)$, where $B$ is a Brownian motion and $f$ measurable.

If we consider the process $$ Y^f_t := \sup_{s\leq t} \vert B_s \vert - f(s)$$ is there anything known about the distribution or at least the probability $\Bbb P (Y^f_t \leq 0 )$ ? Of particular ...
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18 views

Book-recommendation: Numerical method for stochastic differential equations

Speaking of numerical stochastic differential equations, the book of Peter Kloeden 1992 Numerical Solution of Stochastic Differential Equations is a quite famous and standard reference. But when I ...
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1answer
31 views

What is the significance of $[t/ \Delta t]$ in Ross' definition of Brownian motion?

Picture is from Ross' Introduction to Probability Models, 11th ed. I understand the definition of $[t/\Delta t]$, I just don't see how it connects to the position at time $t$ (eq. 10.1). For $\Delta ...
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1answer
16 views

reflection principle brownian motion

In this proof they use the strong markov property but i don't understand why we need it. Could anyone explain it to me? Thank you
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1answer
15 views

Characterization of Brownian Motion (Problem Karatzas/Shreve)

In the book "Brownian Motion and Stochastic Calculus" by Karatzas/Shreve, they state the following problem (chapter 5, problem 4.4): A continuous, adapted process $W= \{W_t,\mathcal{F}_t;0\leq t < ...
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1answer
23 views

Brownian motion expected value of $W(1)$.

I know that if $W(t)$ is a Brownian motion on the interval $[0,t]$, then $$\mathbb{E}[W(t)]=0$$ but why $$\mathbb{E}[W(1)]=0?$$ Lets say that $W(1)=a$, then why we do not have $$\mathbb{E}[W(1)]=\...
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7k views

Prove the time inversion formula is brownian motion

Let $B=(B_t)_{t\geq 0}$ be a brownian motion. Show the time inversion formula $\hat{B}=(B_t)_t\geq0$ is a brownian motion, where for $t \geq 0$ we set $\hat{B}=0$ for $t=0$ and $\hat{B}=tB_{1/t}$ for $...
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1answer
858 views

Density of the Absorbed Process

The curiosity arose while reading the Ch.18 of Arbitrage Theory in Continuous Time 3/ed, dedicated to pricing Barrier Options. Definition 18.1 For any $y\in R$, the hitting time of y, $\tau(X,y)$, ...
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2answers
46 views

Probability of $B_t < 0$ if $B$ is Brownian Motion

Let's consider Brownian motion $B_t$. We assume that $B_0 = 0$. I am to show that $$P(\inf{ \{t>0: B_t <0 \}} = 0) = 1.$$ It seems pretty obvious. I don't know how to start the proof however. ...
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1answer
41 views

$E[(\int_{0}^{\infty}f(t)dW_t)^2]$ for $f(t)=(W_2-W_1)1_{[2,3)}(t)+(W_3-W_1)1_{[3,5)}(t)$

Let $f(t)=(W_2-W_1)1_{[2,3)}(t)+(W_3-W_1)1_{[3,5)}(t), t \ge 0$. $(W_t)_{t\ge0}$ is Brownian motion. What's the 'best' method to calculate $\mathbb E[(\int_{0}^{\infty}f(t)dW_t)^2]$? I would ...
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1answer
20 views

Product of two independent brownian motion is a martingale

Given $X, Y$ independent Brownian motions, I'd like to show that $XY$ is a martingale. This seems to be a fairly easy result, but I can't work it out, nor find anywhere that gives a proper proof. I ...
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1answer
956 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
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1answer
27 views

First variation on Brownian motion

The first variation of function $f(t)$ on interval $[0,T]$ is defined as $$ FV(f) = \lim_{\|\pi\| \rightarrow 0} \sum_{i=0}^{n-1} |f( t_{i+1} ) - f( t_i)|.$$ How can we estimate the first variation ...
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1answer
40 views

Markov property for 2 dimensions and example

As I try to study Markov processes I struggle in understanding how to extend the Markov property for one dimension \begin{align*} \mathbb{P}\left[X_{t} \in A | \mathcal{F}_{s}\right]=\mathbb{P}\left[...
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10 views

Understanding where the Mandelbrot Van Ness representation of fractional brownian motion comes from

I would like to know more about this way of representing the fBm process. Define: $$K_H(t,u) = (t-u)^\kappa_+ - (-u)^\kappa_+,$$ where $\kappa = H - 1/2$. The Mandelbrot Van Ness representation of ...
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14 views

Showing Ito's formula gives a semimartingale decomposition for brownian motion

I'm trying to show that for a standard Brownian motion and some twice continuously differentiable function $f$ that $f(B_t)$ is a local martingale iff $f'' = 0$. Applying Ito's formula gives, and ...
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2answers
172 views

Distribution of $\max_{t \in [0,1]} |W_t|$ for Brownian motion

For a standard Brownian motion $\{W_t, t\geq 0\}$, find $\mathbb{P}(\max_{ t \in [0,1]}|W_t| <x)$. Page 79-80 of Billingsley, P., Convergence of probability measures, New York-London-Sydney-...
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1answer
33 views

W_t^3 martingale or not? two arguments puzzle me.

I want to study whether $W_t^3$ is a martingale or not? where $W_t$ is the standard Brownian motion. I have method 1 argument, but I also got second argument which implies different conclusion. ...
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1answer
14 views

Question on Construction of Brownian motion with Kolmogorov-Centsov Theorem

I am studying the construction of Brownian motion with Kolmogorovs Extension Theorem. In the attached section from Karatzas and Shrive (Brownian Motion and Stochastic Calculus) it says that $C([0,\...
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1answer
21 views

$\mathbb E[W_s^2W_t^2]$ for Brownian motion

Consider a Brownian motion $(W_t)_{t\ge0}$ on $(\Omega ,\mathcal F, \mathbb P)$. How can I calculate $\mathbb E[W_s^2W_t^2]$? I know that $\mathbb E[W_sW_t^2]=0$ but I don't know if that helps here.
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22 views

Distribution of $(X,Y,Z)$, $X=W(3)-W(1), Y=W(5)-W(2), Z=W(7)-W(4)$. (Brownian motion)

Let $W(t),t \ge 0$ be a Brownian motion on $(\Omega,\mathcal F,P)$ and let $(\mathcal F(t),t\ge0)$ be the natural filtration of $W$. Let $X=W(3)-W(1), Y=W(5)-W(2), Z=W(7)-W(4)$. I should ...
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1answer
25 views

Show a stochastic process is a Martingale

B$_t$ is a Brownian motion starting from 0. For any fixed constant $\sigma$ $>$ 0, X$_t$ = e $^{\sigma B_t - \sigma^2t/2}$, t$>$0 is a martingale w.r.t. the filtration generated by Brownian ...
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1answer
400 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
103 views

Hitting times by Brownian motion

[Edited] Suppose that $A$ is a (Borel) measurable set and $X$ is an Ito diffusion, i.e., $dX_{t}=\mu(X_t)dt+\sigma(X_t)dB_t$. Consider a hitting time $\tau_A$ of the given set $A$ by the process $X$:...
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1answer
55 views

Conditional expectation of Brownian motion's first hitting time

Let $T_x$ be the first hitting time of $x$. Let $B_t$ be a Brownian motion started at $x\in [0,R]$. Show that $$E[T_R \mid T_R < T_0]=\frac{R^2-x^2}{3}.$$ By using the fact that $B_t^2 - t$ is a ...
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1answer
59 views

Solution to Dirichlet is necessarily given by Brownian Motion (problem with stopping times)

Let $D \subset \mathbb{R}^n$ be a bounded open set and $f: \partial D \rightarrow \mathbb{R}$ a continuous function. The Dirichlet problem consists of finding a continuous function $U: \overline{D} \...
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1answer
58 views

Calculating the expecation of the supremum of absolute value of a Brownian motion

I got a Brownian motion $B(t)$ that starts in $0$ and want to calculate the expectated value of the supremum on the interval $[0,1]$ of the absolute value of it, i.e. $E \left (\sup \limits_{t \in [0,...
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1answer
17 views

brownian translation

I have a bit of struggle with understanding what it means to have the same law as a brownian. For instance, how can i prove that $$\sup_{t\in[k-1,k]} |W_t - W_{k-1}| \stackrel{\mathcal{L}}{=} \sup_{t\...
3
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1answer
80 views

Revuz and Yor's Book “Continuous Martingales and Brownian Motion” - Chapter 1 - Exercise 1.11 (again)

Context : This post is the first of a series of posts taking their origins from the exercises in the Revuz and Yor's Book "Continuous Martingales ans Brownian Motion". The reason for doing so is ...
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0answers
22 views

Quadratic variation of $w^2(t)$

What is the quadratic variation of ($w(t))^2$, where $w(t)$ denotes a Wiener- process and what is the expected value of it? What is $E([(w(t))^2])$? My result is: $2(t^2)$, but knowing myself, I am ...